SYSTEMATICS OF THE COULOMB BARRIER CHARACTERISTICS RESULTING FROM M3Y NUCLEON-NUCLEON FORCES FOR REACTIONS WITH HEAVY IONS Текст научной статьи по специальности «Физика»

ФИЗИКА АТОМНОГО ЯДРА И ЭЛЕМЕНТАРНЫХ ЧАСТИЦ

Известия Саратовского университета. Новая серия. Серия: Физика. 2023. Т. 23, вып. 2. С. 157-166

Izvestiya of Saratov University. Physics, 2023, vol. 23, iss. 2, pp. 157-166

https://fizika.sgu.ru https://doi.org/10.18500/1817-3020-2023-23-2-157-166, EDN: DNUYIV

Article

Systematics of the Coulomb barrier characteristics resulting from M3Y nucleon-nucleon forces for reactions with heavy ions

1.1. Gontchar1, M. V. Chushnyakova2H, N. A. Khmyrova1

1 Omsk State Transport University, 35 Marxa prospect, Omsk 644046, Russia 2Omsk State Technical University, 11 Mira prospect, Omsk 644050, Russia

Igor I. Gontchar, vigichar@hotmail.com, https://orcid.org/0000-0002-9306-6441

Maria V. Chushnyakova, maria.chushnyakova@gmail.com, https://orcid.org/0000-0003-0891-3149

Natalya A. Khmyrova, nata_ruban@mail.ru, https://orcid.org/0000-0003-0690-9138

Abstract. In the literature, often the capture cross sections for spherical heavy-ions are calculated by virtue of the characteristics of the s-wave barrier: its energy, radius, and stiffness. We evaluate these quantities systematically within the framework of the double-folding model. For the effective nucleon-nucleon forces, the M3Y Paris forces with zero-range exchange part are used. The strength of this part is modified to fit the barrier energy obtained with the density-dependent finite-range exchange part. For the nucleon density, two options are employed. The first one (V-option) is based on the experimental charge densities. The second one, C-option, comes from the IAEA data base; these densities are calculated within the Hartree-Fock-Bogolubov approach. For both options, the analytical approximations are developed for the barrier energy, radius, and stiffness. The accuracy of these approximations is about 3% for the barrier energy and radius and about 10% for the stiffness. The proposed approximations can be easily used by everyone to estimate the capture cross sections within the parabolic barrier approximation.

Keywords: double-folding model, nucleon densities, characteristics of Coulomb barrier Acknowledgements: This work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".

For citation: Gontchar 1.1., Chushnyakova M. V., Khmyrova N. A. Systematics of the Coulomb barrier characteristics resulting from M3Y nucleon-nucleon forces for reactions with heavy ions. Izvestiya of Saratov University. Physics, 2023, vol. 23, iss. 2, pp. 157-166. https://doi.org/10.18500/1817-3020-2023-23-2-157-166, EDN: DNUYIV

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC0-BY 4.0)

Научная статья УДК 539.17.01

Систематика параметров кулоновских барьеров, вычисленных с использованием M3Y нуклон-нуклонных сил, в реакциях с тяжёлыми ионами

И. И. Гончар1, М. В. Чушнякова2н, Н. А. Хмырова1

1 Омский государственный университет путей сообщения, Россия, 644046, г. Омск, пр. Маркса, Д. 35

2Омский государственный технический университет, Россия, 644050, г. Омск, пр. Мира, д. 11

Гончар Игорь Иванович, доктор физико-математических наук, профессор кафедры «Физика и химия», vigichar@hotmail.com, https://orcid.org/0000-0002-9306-6441

НАУЧНЫЙ ОТДЕЛ

Чушнякова Мария Владимировна, кандидат физико-математических наук, доцент кафедры «Физика», mvchushnyakova@omgtu.tech, https://orcid.org/0000-0003-0891-3149

Хмырова Наталья Анатольевна, кандидат физико-математических наук, доцент кафедры «Физика и химия», nata_ruban@mail.ru, https://orcid.org/0000-0003-0690-9138

Аннотация. В литературе сечения слияния (захвата в орбитальное движение) для сложных сферических ядер часто вычисляются с помощью характеристик барьера, соответствующего лобовому столкновению: высоты барьера, его радиуса и жёсткости. В настоящей работе мы рассчитываем эти величины систематически в рамках модели двойной свёртки. В качестве эффективного нуклон-нуклонного (NN) взаимодействия используется парижское M3Y взаимодействие с нулевым радиусом обменной части. Её амплитуда варьируется так, чтобы воспроизводить высоту барьера, полученного при использовании обменной части с конечным радиусом взаимодействия и плотностной зависимостью NN-взаимодействия. Для нуклонных плотностей использовано два варианта. Первый (V-опция) основан на экспериментальных зарядовых плотностях. Второй (C-опция) - это протонные и нейтронные плотности, вычисленные с помощью подхода Хартри-Фока-Боголюбова и опубликованные МАГАТЭ. Для обеих опций нами построены аналитические аппроксимации высоты барьера, его радиуса и жёсткости. Точность этой аппроксимации составляет около 3% для высоты барьера и около 10% для жёсткости. Предложенные аппроксимации могут быть полезны всем для быстрой оценки сечений захвата с помощью модели параболического барьера.

Ключевые слова: модель двойной свёртки, нуклонные плотности, параметры кулоновского барьера Благодарности: Настоящая работа была поддержана грантом Фонда развития теоретической физики и математики «БАЗИС». Для цитирования: Гончар И. И., Чушнякова М. В., Хмырова Н. А. Систематика параметров кулоновских барьеров, вычисленных с использованием M3Y нуклон-нуклонных сил, в реакциях с тяжёлыми ионами // Известия Саратовского университета. Новая серия. Серия: Физика. 2023. Т. 23, вып. 2. С. 157-166. https://doi.org/10.18500/1817-3020-2023-23-2-157-166, EDN: DNUYIV Статья опубликована на условиях лицензии Creative Commons Attribution 4.0 International (CC-BY4.0)

Thus, for the fast evaluation of the capture cross section by means of Eq. (1), it is sufficient to know B0, RB0, CB0• For finding these quantities, one needs the s-wave nucleus-nucleus potential which consists of the Coulomb UC(R) and nuclear Un(R) terms.

The nuclear term (Strong nucleus-nucleus Potential, SnnP) is a crucial ingredient for any theoretical description of the capture process. Often for SnnP they use the Woods-Saxon profile [15-19]. The parameters of this profile (depth, diffuseness, and radius) are varied more or less arbitrary to fit Gth to the above barrier experimental capture cross sections. Obviously, the Woods-Saxon profile represents the SnnP only qualitatively.

The much better founded proximity potential [20, 21] is employed every now and again [2226]. This potential includes a universal dimension-less function independent of the colliding nuclei. Yet some parameters of this potential can be varied individually for a given reaction.

The single-folding potential [27, 28] is more rigorous. For evaluating this potential one needs to know: (i) the distributions of the nucleon centers of mass (the nucleon densities) for both colliding nuclei; and (ii) the interaction energy between the whole target (projectile) nucleus and a nucleon of the projectile (target) nucleus. For the nucleon densities, the two-parameter Fermi profile was applied in [27-29]. The parameters of the profile might be obtained from the electron scattering data [30]. However, the electron scattering is only sensitive to the Coulomb interaction. Therefore, in such experi-

1. Introduction

Collision of two complex nuclei (heavy ions) resulting in the capture of them into orbital motion is the first step for formation of new superheavy chemical elements and/or isotopes [1-6]. The theoretical cross sections of the capture process are subject of significant uncertainties [6-8]. Often the capture cross sections for spherical colliding nuclei are evaluated as follows [9-12]: 2

Oth = -^h— £ (2J +1) Tj, (1)

2MrEc .m. j

Tj =\ 1 + exp

2П (Bj - Ec.m. )

(2)

h ®bj

Here, Ec.m. stands for the collision energy; J is the angular momentum in units of h; mR is the reduced mass of colliding ions; BJ and K>BJ are the Coulomb barrier energy and "frequency", respectively. For the J-dependence of the barrier energy, the following approximation is often applied [10, 13, 14]:

h2 J2

BJ = B0 +-^, (3)

2mRRBo

where B0 and RB0 are the s-wave barrier energy and radius, respectively. The J-dependence of K>BJ is usually neglected, so

®bj =J-Cb0 (4)

mR

where CB0 is the second derivative of the nucleus-nucleus potential with respect to the center-to-center distance.

1

ments the charge density distribution is measured, not the nucleon density. The direct experimental information of the nucleon density is scares [31-33]. For the second ingredient of the single-folding potential being the nucleon-nucleus potential, usually the Woods-Saxon profiles are used. Parameters of these profiles are extracted from the fit of the elastic scattering data [28]. Thus, the single-folding approach still has six individual fit parameters for a given reaction.

Employing the effective nucleon-nucleon forces (NN forces) seems to be the next step towards a more realistic description of the nucleus-nucleus potential. This step is realized in the double-folding (DF) model [34, 35]. The nucleon densities of the colliding nuclei are one more ingredient of this model. Three such DF potentials using different effective NN forces are known in the literature: i) with the M3Y ones [36,37]; ii) with the relativistic mean-field ones [8, 38, 39]; iii) with the Migdal forces [40, 41].

The aim of the present work is to calculate systematically the characteristics of the heavy-ion s-wave Coulomb barriers B0, RB0, CB0 obtained within the framework of the DF approach with the M3Y NN forces and to explore whether there are any regularities in their behavior versus the approximate barrier energy

BZ

ZpZt

Al/3 + Al/3

/ip + /1 t

MeV.

(5)

The present paper is organized as follows. Section 2 is devoted to the DF model applied for the calculation of the nucleus-nucleus potential. The nucleon densities are discussed in Sec. 3. Sections 4 and 5 represent the results obtained for two sorts of densities. Conclusions are collected in Sec. 6.

2. The double-folding model

The nucleus-nucleus s-wave potential U0 versus the distance R between the centers of mass of spherical projectile (P) and target (T) nuclei reads

Uo (R) = Uc (R) + UnD (R) + UnE (R)• (6)

Here UC is the Coulomb term, UnD and UnE stand for the direct and exchange parts of the SnnP, respectively. These three terms read

UnE = JdrP j drTPap (rP) Ve (s) Pat (tt) • (9)

Here pAP and pAT (pqP and pqT) denote the nucleon (charge) densities, rP and rT are the absolute values of the radius-vectors of the interacting points of the projectile and target nuclei. Vector s connects two interacting points and is determined by vectors R, rP, and rT (see Fig. 2 in [42, 43] or Fig. 1 in [44, 45]). The point-point Coulomb potential is denoted as vC (s).

In Eqs. (7), (8), (9), we neglect the possible time-dependence of the densities. This so-called frozen density approximation (FDA) seems to work reasonably well unless the density overlap of the colliding nuclei is about 1/3 of the saturation density 0.16 fm-3. Recently, the FDA was inspected carefully and compared with the adiabatic density approximation (ADA) in Ref. [46].

The direct part of the NN-interaction vD (s) consists of two Yukawa terms [47, 48]:

Vd (s) = £ Got

t=i

—s exp( —

' vt

(10)

For the exchange part vE (s), one finds in the literature two options: an advanced and complicated one with a finite range and a simpler one with zero range [42, 48]. Equation (9) is valid for the latter version for which

Ve (s) = Ge 8 (s).

(11)

Uc = j drp J drT pqP (rp) VC (s) PqT (rT) , (7) Pf (r) = p0F

UnD = J drp j drT Pap (rp) Vd (s) Pat (гт ), (8)

It has been demonstrated recently [44] that varying the value of GE with respect to its standard value -592 MeV fm3 from Ref. [37] down to -1040 MeV fm3 allows to reproduce the Coulomb barrier energies resulting from the option with the finite range exchange force. Computer calculations with zero-range option are significantly faster than those ones with the finite range option. In the present paper, we apply Eqs. (9), (11) with GE =

= -1040 MeV fm3. +

3. Nucleon densities

In the present study, we employ two prescriptions for the nucleon densities coming from Refs. [30] and [49]. We denote them as V-densities and C-densities, respectively. In both sources [30] and [49], the density is approximated by the three-parameter Fermi formula (3pF-formula)

1 - wFr2/R2F 1 + exp [(r - Rf) /aF] • (

Here RF corresponds approximately to the half central density radius, aF is the diffuseness, p0F is

defined by a normalization condition. In Ref. [30], the 3pF-formula (or its version with wF = 0 called 2pF-formula) is applied to approximate the experimental charge density (in this case the subscript F takes the value Vq). We use the same 3pF-formulas for proton (F = Vp) and neutron (F = Vn) densities for a given nucleus. The parameters RVq, aVq, and wVq of the charge density are taken from Ref. [30]. The half-density radii for protons RVp and neutrons RVn as well as wVp and wVn are taken to be equal to RVq and wVq, respectively, whereas the proton and neutron diffusenesses, aVp and aVn, are calculated via the charge diffuseness aVq [42, 43]:

avp — avn

5

N

V-^l0-76-0Л1 z )• (13)

We use all spherical nuclei for which Eq. (12) is available in Ref. [30]. The values of the parameters are presented in Table 1.

In Ref. [49], theoretical proton and neutron densities calculated within the Hartree-Fock-Bo-golubov approach are approximated by Eq. (12) with wF = 0. In this case, the subscript F takes the

values Cp, Cn, and Cq. We take RCq

Rep and

aCq

- . „2

aCp + 7П2

N

0.76 - 0.11 —

z

(14)

The values of the parameters are again presented in Table 1.

4. Results: V-densities

In Fig. 1, a, we present the calculated s-wave Coulomb barrier energies B0V. These calculations are performed for four groups of the reactions induced by: 16O, 40Ca, 58Ni, and 88Sr (symbols). This allows to cover a wide range of BZ = 12 ^ 300 MeV. The line in the figure corresponds to B0 = BZ. One sees that BZ is indeed a good approximation for the DF M3Y barrier energies.

The fractional difference between these two quantities

^ — ^ - 1 SB Bz

(15)

is displayed in Fig. 1, b. The symbols correspond to DF M3Y-calculations where the curve represents

Table 1. Parameters of Eq. (12) for the V- and C-densities for the spherical nuclei involved in the considered reactions. RVq, wvq, aVq of the charge V-densities are taken from Ref. [30] and avp for the proton and neutron V-densities calculated according to Eq. (13). Rep, acp, Rcn, acn for the proton and neutron C-densities are taken from Ref. [49] and acq for the

charge C-densities are calculated using Eq. (14)

5

Nuc. Rvq, fm WVq avq, fm avp, fm RCp, fm aCp, fm RCn, fm aCn, fm aCq, fm

16 O 2.608 —0.051 0.513 0.465 2.699 0.447 2.652 0.460 0.497

40 Ar 3.730 —0.190 0.620 0.582 3.657 0.480 3.564 0.532 0.525

40 Ca 3.766 -0.161 0.585 0.543 3.564 0.532 3.685 0.481 0.575

48 Ca 3.737 -0.030 0.524 0.481 3.887 0.467 3.989 0.493 0.512

48Ti 3.843 0.000 0.588 0.548 3.942 0.477 3.979 0.478 0.523

52 Cr 4.010 0.000 0.497 0.449 4.064 0.467 4.085 0.470 0.514

54Fe 4.075 0.000 0.506 0.458 4.145 0.463 4.127 0.463 0.511

58Ni 4.309 -0.131 0.517 0.470 4.241 0.467 4.156 0.512 0.515

60Ni 4.489 -0.267 0.537 0.492 4.274 0.471 4.128 0.532 0.518

62 Ni 4.443 -0.209 0.537 0.495 4.318 0.468 4.177 0.532 0.514

64Ni 4.212 0.000 0.578 0.538 4.362 0.465 4.298 0.567 0.511

88Sr 4.830 0.000 0.449 0.496 4.911 0.480 4.971 0.488 0.525

112 Sn 5.375 0.000 0.560 0.518 5.404 0.463 5.331 0.555 0.509

116 Sn 5.358 0.000 0.550 0.508 5.458 0.459 5.396 0.568 0.505

118 Sn 5.412 0.000 0.560 0.519 5.484 0.457 5.428 0.574 0.503

120 Sn 5.315 0.000 0.576 0.537 5.508 0.455 5.458 0.546 0.501

124Sn 5.490 0.000 0.534 0.492 5.556 0.452 5.570 0.552 0.497

142 Nd 5.774 0.000 0.513 0.468 5.872 0.466 5.865 0.534 0.511

148 Sm 5.771 0.000 0.596 0.558 5.9548 0.4721 5.9360 0.5575 0.517

206Pb 6.610 0.000 0.545 0.504 6.6800 0.4666 6.6999 0.5542 0.511

the following approximation

^BV = 0.106 - 0.219 • exp

-Bz

-0.230 • exp

16.1 MeV -Bz \ 206 MeW .

(16)

Fig. 1. (a) Calculated s-wave Coulomb barrier energies B0V, (b) fractional barrier differences <z)B (see Eq. (15)), and (c) errors eBV (see Eq. (17)) are shown as functions of the approximate barrier energy BZ for four groups of the reactions induced by: 16O, 40Ca, 58Ni, and 88Sr (symbols). The line in panel (a) corresponds to B0 = BZ, the line in panel (b) is the approximation (see Eq. (16)) (color online)

Thus, employing Eqs. (5), (15), (16) one obtains the value of B0 with the typical accuracy of

1-2% (see Fig. 1, c). We define the error of approximation for quantity x as

=

xfit xcalc

1.

(17)

Let us go over to the stiffness of the barrier, CB0. The calculated values are shown by symbols in Fig. 2, a, their linear fit reads

CB0V = -0.755 MeV - 0.0494BZ

(18)

(line in Fig. 2, a). Accuracy of this fit is typically within 10% (see Fig. 2, b) although for lighter reactions it reaches —20% due to smaller values of the stiffness.

Fig. 2. (a) Barrier curvatures CB0V and (b) errors 8CV (see Eq. (17)) are shown as functions of BZ for four groups of the reactions (symbols). The line in panel (a) is the approximation (see Eq. (18)) (color online)

The calculated barrier radii versus BZ are shown by symbols in Fig. 3, a. However, their dependence upon Ap/3 + AT/3 is simpler and more regular (see Fig. 3, b). The linear fit of this dependence reads

RB0V/fm = 3.89 + 0.918 ( AP + A

,1/3 , Л/3

(19)

(line in Fig. 3, b). Typical error of this fit is within 2% (see Fig. 3, c).

T

Fig. 3. (a) Barrier radii RB0v as functions of BZ, (b) the same

RB0V and (c) errors£RV (see Eq. (17)) versusAp/3 + Aj3 are shown for four groups of the reactions (symbols). The line in panel (b) is the approximation (see Eq. (19)) (color online)

5. Results: C-densities

The same procedure, as in Sec. 4, was performed for C-densities. Results are shown in Figs. 4-6. The approximate formulas read

^bc — 0-0591 - 0.212 • exp -0.197 • exp

-B

16.1 MeV -Bz \ 155 MeV / '

CB0C — -0.752 MeV - 0.04648BZ

(20)

(21)

Rb0c/fm = 3.80 + 0.95 (Ap/3 + A1/^ (22)

The quality of the fits is approximately as for the case of V-densities (see Figs. 4, c, 5, c, 6, c).

These three approximations for the characteristics of the s-wave Coulomb barrier obtained for V-and C-densities are compared in Fig. 7. Obviously, the trends for two versions of densities coincide with each other in all three panels. As reactions become heavier, the difference appears to be more significant. Although the pairs of curves are close to each other in Fig. 7, one should remember that several percent difference in the barrier energy might influence the fusion cross section substantially, especially for the near- and sub-barrier energies.

Fig. 4. Same as in Fig. 1 but for C-densities. The line in panel (b) corresponds to Eq. (20) (color online)

Fig. 5. Same as in Fig. 2 but for C-densities. The line in panel (a) corresponds to Eq. (21) (color online)

Fig. 6. Same as in Fig. 3 but for C-densities. The line in panel (b) corresponds to Eq. (22) (color online)

Fig. 7. Comparison between the approximations for the barrier characteristics obtained with V- and C-densities: (a) barrier energies B0 (see Eqs. (16) and (20)),

(b) barrier curvatures CB0 (see Eqs. (18) and (21)), and

(c) barrier radii RB0 (see Eqs. (19) and (22)) (color

online)

6. Conclusions

In the literature, every now and again, the spherical heavy-ion capture cross sections are evaluated using the characteristics of the s-wave barrier: its energy, radius, and stiffness. In the present work, we have calculated these quantities systematically within the framework of the double-folding (DF) model. In these calculations, for the effective nucleon-nucleon forces the M3Y Paris forces with zero-range exchange part have been used. The amplitude of this part has been modified to reproduce the barrier energy obtained with the density-dependent finite-range exchange part. For the nucleon density, two options have been used. The first one (V-option) is based on the experimental charge densities. The second one (C-option) has come from the IAEA data base.

For both options, the analytical approximations have been obtained for three quantities required for evaluation of the capture cross sections within the barrier penetration model (parabolic barrier approximation). The comparison of the V- and C-approximations demonstrates that those are not very different. The proposed approximations can be used by everyone for fast estimation of the capture cross sections in the collision of two spherical complex nuclei.

We would like to stress that in the literature there are many recipes for crucial ingredients of the DF model, namely the effective NN-forces and nucleon densities. For instance, in the literature sometimes the Reid M3Y forces [36] are used although in Ref. [37] it is clearly stated that "The Reid soft-core potential is based on earlier and partially erroneous phase-shift data". The Migdal forces were used successfully in quantum diffusion model [41, 50, 51], however for this aim very special nucleon densities were employed. Application of the Migdal forces with densities coming from the Hartree-Fock SKX calculations [52] results in cross sections which do not leave any room for dissipation of collective energy [53]. We believe that the versions of NN-forces and densities used in the present work are the best which are available in the literature for systematic calculations. At the time being, we do not see any arguments allowing to prefer Cor V-option of the densities.

Of course, one would like to see an application of the proposed approximation for the analysis of experimental cross sections as well as a numerical analysis of the accuracy of approximate formulas (3) and (4). However, this would make the present

paper unjustifiably long and would distract the attention of the reader. We hope to complete such study in near future.

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Поступила в редакцию 02.01.2023; одобрена после рецензирования 26.01.2023; принята к публикации 03.02.2023 The article was submitted 02.01.2023; approved after reviewing 26.01.2023; accepted for publication 03.02.2023