Variation of nuclear matter properties in fusion reaction of the 64Ni+64Ni

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.1 (1-13)

Available online at www.sciencedirect.com

ScienceDirect 1

1 2

2

Nuclear Physics A ••• (••••) ••••••

3

www.elsevier.com/locate/nuclphysa

3

4

4

5

5

6

6 7

7 8 9 10

Variation of nuclear matter properties in fusion reaction of the 64 Ni+64Ni

8 9 10 11

11

T. Ghasemi, O.N. Ghodsi

12

12 13

13 14

Department of Physics, Faculty of Science, University of Mazandaran, P.O. Box 47415-416, Babolsar, Iran

14

15

Received 10 December 2019; received in revised form 28 December 2019; accepted 31 December 2019

15

16

16

17

17

18

18

19

Abstract

21 22 23 24 25 26 27 28 29

In this paper, the variation of the nuclear matter properties during the fusion reaction and also its relationship with the hindrance phenomenon are examined in the 64 Ni+64 Ni reaction. For this purpose, the inter-nuclear potential is calculated by using the Skyrme energy density functional formalism in which the used forces are in a wide range of the incompressibility values. The obtained results indicate that by increasing bombarding energy the nuclear matter incompressibility is increasing. Also, this variation shows that nuclear matter exhibits a very soft behavior when moving from the sub-barrier to the deep sub-barrier region in this reaction, which can cause a large overlapping between the interacting nuclei. Since the repulsion arising from the Pauli exclusion principle affects this large overlapping, so it can lead to the fusion hindrance and the fall-off of the cross-sections in the 64 Ni+64 Ni reaction. © 2020 Published by Elsevier B.V.

32

21 22 23 24 25 26 27 28 29 30

30 31

19 20

20

Keywords: Fusion reactions; Nuclear matter properties; Hindrance phenomenon

31 32

33

33

34

34 35

35 36

1. Introduction

38 39 40 41

In the last decades, one of the important topics in nuclear physics has been the study of the heavy-ion reactions both experimentally and theoretically [1–16]. To analyze these reactions, it is essential to know the interaction potential between the projectile and target nuclei. In general, the ion-ion interaction potential consists of the long-range repulsive Coulomb part and short-range

46 47

39 40 41

43

43

45

38

42

42

44

36 37

37

E-mail addresses: [email protected] (T. Ghasemi), [email protected] (O.N. Ghodsi). https://doi.org/10.1016/j.nuclphysa.2020.121691 0375-9474/© 2020 Published by Elsevier B.V.

44 45 46 47

JID:NUPHA AID:121691 /FLA

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:7/01/2020; 16:11] P.2 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

attractive nuclear part. There is an explicit expression to calculate the Coulomb part, whereas based on the statics and dynamics approaches, various models have been introduced to calculate the nuclear part [17–25]. The simplest model to calculate the interacting potential is the onedimensional potential model [26], which well describes the fusion excitation functions above the Coulomb barrier, but it is unable to describe the cross-sections below the barrier and underestimates them compared with the experimental data. This indicates that in analyzing the sub-barrier fusion process, it is essential to take into account the couplings between the relative motion of the interacting nuclei with their internal degrees of freedom [12,27,28]. Therefore, the coupledchannels approach is more applicable to study the heavy-ion reactions at the sub-barrier energies. In recent years, an interesting phenomenon in the interaction between the two nuclei has been the focus of many studies, which showed that at a given threshold energy gradually by decreasing bombarding energy, the measured fusion cross-sections fall off rapidly in the deep sub-barrier region [29–33]. In addition, the conventional coupled-channels approach cannot also predict this steep fall-off. The first time, this phenomenon was observed in the 60 Ni+89 Y system [29] and was named as the fusion hindrance. Two quantities of the S(E) factor and logarithmic slope L(E) can be useful to analyze the fusion hindrance phenomenon. The onset of the hindrance, especially in the fusion systems including medium-heavy nuclei is attributed to the energy at which the S-factor is maximum. A number of the reactions including the hindrance phenomenon were reported in Ref. [34]. One of them is the 64 Ni+64 Ni system, which shows the interaction between two open-shell nuclei. Its fusion cross-sections were measured down to the σ ∼10 nb [35]. So far, this reaction has been investigated by different methods [36–42]. In Ref. [43], by considering the 64 Ni nucleus as deformed, within the density-constrained time-dependent Hartree-Fock formalism, it was shown that taking into account the core polarization effects leads to better agreement of the calculations with the experimental data. As a result, the core nucleons can play an important role at the lowest energies. In Ref. [44], by considering the sudden approximation within the double folding model, a repulsive core was added to the interacting potential that this can somehow simulate the effect of the Pauli repulsion at a short distance. As a result, a shallow pocket was obtained that can, in turn, describe the fusion hindrance. In that study for the 64 Ni+64 Ni reaction, it was shown that the incompressibility K∼228 MeV can reproduce the experimental data in all regions of the energy. In addition to the mentioned study, a performed study on the other reactions [45], which indicate the hindrance phenomenon, such as 40 Ca+40 Ca, 16 O+208 Pb, and 48 Ca+48 Ca [46] using the density-constrained frozen Hartree-Fock method [47] showed that the repulsion arising from the Pauli exclusion principle reduces the probability of tunneling at energies below the Coulomb barrier and taking it into account in the interacting potential can lead to better agreement between the theoretical calculations and the experimental data. In another study [48] performed on the 64 Ni+64 Ni system, the interacting potential was determined by employing the Skyrme energy density formalism within the proximity potential and the fusion cross-sections were calculated by using the Wong model. The obtained results for this reaction indicated that some forces describing near and above the Coulomb barrier can no longer describe the energies below the barrier. Therefore, it was necessary to modify the barrier at lower energies, which was done by choosing a more suitable force. Since in the mentioned studies, the effect of the core nucleons and the addition of a repulsive core arising from the Pauli exclusion principle to the interacting potential were discussed, it may be possible that in the 64 Ni+64 Ni reaction, a large overlapping of the density distributions of the colliding nuclei is occurring in the deep sub-barrier region. This requires that the nuclear matter exhibits very soft behavior when moving from the sub-barrier to the deep sub-barrier region. Thus, it appears that there may be an energy dependence to the incompressibility when applied to fusion reactions of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.3 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3

finite size nuclei such as the 64 Ni+64 Ni reaction. This dependence is somehow seen in the study of Ref. [48], although there the Skyrme forces were selected, which each of them could alone describe all energy regions for this reaction. Recently, in a performed study on the 16 O+208 Pb reaction [49], different energy regions were described by various Skyrme forces. In that study by using the Skyrme energy density functional formalism, it was shown that the nuclear matter property can change during the reaction process so that by increasing bombarding energy, the fusion is better described by interactions that correspond to larger nuclear matter incompressibility. In the present study on the basis of the used method in Ref. [49], we are motivated to examine the nuclear matter incompressibility during the fusion reaction of the 64Ni+64 Ni and also its relationship with the hindrance phenomenon in this reaction. To this end, the interacting potentials were determined by the Skyrme energy density functional formalism and then using them the fusion cross-sections were computed. In what follows, the details of the calculations are explained. This paper is organized as: The Skyrme energy density functional formalism is introduced in section 2. The calculations of the fusion cross-sections, S(E) factor and logarithmic slope L(E) are analyzed in section 3. The conclusions are presented in section 4.

21 22

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

4 5 6 7 8 9 10 11 12 13 14 15 16

19

A static approach to study the heavy-ion fusion reactions is the Skyrme energy density functional formalism [50–52]. In this formalism, the nuclear part of the interacting potential, VN (R), is determined as:

20 21 22 23

23 24

3

18

2. Skyrme energy density functional formalism

19 20

2

17

17 18

1

VN (R) = ET (R) − (E1 + E2 ), where R denotes the distance between the centers of the colliding nuclei. In Eq. (1), ET (R), E1 and E2 are determined as:     ρ1n (r ) + ρ2n (r − R)  d3 r. ET (R) = E ρ1p (r ) + ρ2p (r − R),    E1 = E ρ1p (r ), ρ1n (r ) d3 r.    E2 = E ρ2p (r ), ρ2n (r ) d3 r.

(1)

24 25 26 27 28

(2)

29 30

(3)

31 32

(4)

ET (R) is the total energy of the combined system and E1 and E2 are the total energy of each of the projectile and target nuclei when they are far away from each other. The notation E(r ) is the Skyrme energy density, which is written as: 

 h¯ 2 1 1 1 2 2 2 E(r ) = τ + t0 1 + x0 ρ − x0 + (ρn + ρp ) 2m 2 2 2 

 1 1 1 α 2 2 2 + t3 ρ (ρn + ρp ) 1 + x 3 ρ − x3 + 12 2 2   

1 1 1 + t1 1 + x1 + t2 1 + x2 (ρτ ) 4 2 2  

 1 1 1 (5) t1 x1 + − t2 x2 + (ρn τn + ρp τp ) − 4 2 2

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[m1+; v1.304; Prn:7/01/2020; 16:11] P.4 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

4

  

1 1 1  2 3t1 1 + x1 − t2 1 + x2 (∇ρ) 16 2 2  

 1 1 1  p )2 )  n )2 + (∇ρ − 3t1 x1 + + t2 x2 + ((∇ρ 16 2 2   1  n + Jp .∇ρ  p .  + Jn .∇ρ + W0 J.∇ρ 2 In this equation, m denotes the nucleon mass, as well as t0 , t1 , t2 , t3 , x0 , x1 , x2 , x3 , α, and W0 are the parameters of the Skyrme interaction. Moreover, ρ = ρn + ρp , τ = τn + τp and J = Jn + Jp indicate nuclear, kinetic energy and spin-orbit densities, respectively. The kinetic energy density τq by considering the h¯ 2 correction within the semi-classical extended Thomas-Fermi model [51,53] is given as (q = n or p)  2  q  q  q )2 1  q .∇f 5 ∇f 3 1 (∇ρ 1 ∇ρ 1 fq 1 2 23 τq (r ) = (3π ) ρq 3 + + ρq + + ρq − ρq 5 36 ρq 3 6 fq 6 fq 12 fq 2  2   + ρq ) 1 2m W0 ∇(ρ + ρq . (6) 2 2 2 fq h¯ +

In the above expression, fq (r ) is the effective form factor, which is written as:

2m 1  x1  x2  fq (r ) = 1 + 2 t1 1 + + t2 1 + ρ(r ) 2 2 h¯ 4  

 1 1 2m 1 (7) t1 x1 + − t2 x2 + ρq (r ). − 2 2 2 h¯ 4 Because of the spin has no classical equivalent and is a quantum-mechanical property, the semi-classical extended Thomas-Fermi model to calculate the spin-orbit density Jq is started from order h¯ 2 . Therefore, Jq is written as: 1 2m 1  + ρq ). (8) Jq (r ) = − 2 W0 ρq ∇(ρ fq h¯ 2 To calculate the ion-ion interaction potential, in addition to calculate the nuclear part, the coulomb part must also be calculated. Therefore, the coulomb part, VC (r ), is written as:  ρp (r´) e2 3e2 3 1/3 VC (r ) = ρp (r ) (9) ( ) (ρp (r ))4/3 . d r´ −  2 4 π |r − r´ | The important inputs of Eq. (5) are the parameters of the Skyrme interaction and the proton and neutron densities of the projectile and target nuclei. There are different sets of the Skyrme parameters that were obtained from fitting to the ground-state properties of a variety of nuclei. These sets indicate the nuclear mean-fields that can be used to extract the density distributions of the colliding nuclei using the HFB method [54]. Here, the density distributions were obtained by employing the HFBRAD code [55]. To calculate the nuclear potential in the 64Ni+64 Ni system, the mean-fields are chosen that can better describe the properties of the 64 Ni nucleus. To this end, the percentage relative deviations (|(T heo. − Exp.)/Exp.| × 100) of the root-mean-square charge radii and binding energies were calculated from their corresponding experimental data. Accordingly, the Skyrme forces that show better results for the 64 Ni+64 Ni reaction are the SKRA [56], SLy230b [57], SK272 [58], and SIII [59] forces whose incompressibility values are 217, 230, 271, and 356 MeV, respectively. Fig. 1 shows the calculated deviations for these forces.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.5 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

5

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24

25

25

26

26 27

27 28 29

Fig. 1. The percentage relative deviations for (a) the root-mean-square charge radii and (b) the binding energies from their corresponding experimental data for the 64 Ni nucleus. The obtained errors are less than 1.41% and 2.90%, respectively.

3. Calculations analysis

34 35 36 37 38 39 40 41 42 43 44 45 46 47

31 32

32 33

29 30

30 31

28

To calculate the inter-nuclear potential within the Skyrme energy-density formalism, the densities obtained from the HFBRAD code were fitted to the two-parameter Fermi density distribution. Fig. 2(a) shows the diffuseness parameters of the proton and neutron density distributions of the 64 Ni nucleus for the selected Skyrme forces. As it is clear from this figure, by increasing the incompressibility values, the diffuseness parameters are decreasing. By using the obtained densities and the parameters of the chosen forces, the interaction potential of the 64 Ni+64 Ni system was computed. The potential barrier characteristics, i.e., the barrier height and position, arising from the selected forces are displayed in Figs. 2(b) and (c), respectively. As it is obvious from Fig. 2(a)(b)(c) for the 64 Ni+64 Ni reaction, from the incompressibility K=230 MeV toward higher values, the barrier heights are increasing and the barrier positions are decreasing. This demonstrates the important role of the surface nucleons in the heavy-ion fusion reactions so that the decrease in the surface diffuseness results in a decrease in the attraction energy between the two nuclei that this causes the barrier height to increase. While from incompressibility K=230 MeV toward lower values, despite increasing surface diffuseness, the barrier heights again increase and the barrier positions decrease. Based on Fig. 2(a), it is observed that in this incompress-

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

6

[m1+; v1.304; Prn:7/01/2020; 16:11] P.6 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24

25

25

26

26

27

27

28

28

29

29

30

30 31

31 32 33

Fig. 2. (a) The diffuseness parameters of the proton and neutron density distributions of the 64 Ni nucleus (b) the calculated barrier heights and (c) the calculated barrier positions in the 64 Ni+64 Ni system.

35

35

37 38 39 40 41 42 43 44 45 46 47

33 34

34

36

32

ibility range, the nuclear matter in the 64 Ni+64 Ni system tends to behave very softly around K=217 MeV. Therefore, a large overlapping can be created between the density distributions of the colliding nuclei, but due to the repulsion arising from the Pauli exclusion principle, which can strongly affect the overlapping of the densities of two nuclei, it is expected that the barrier heights increase in the mentioned incompressibility range. By using the potentials arising from the selected forces, the fusion cross-sections of the 64 Ni+64 Ni system were computed by the CCFULL code [60] by considering the excitations of the low-lying states 2+ and 3− for the 64 Ni nucleus. The used values are listed in Table 1. The fusion cross-sections in the logarithmic and linear scale are indicated in Fig. 3. In this figure, the fusion cross-sections arising from the SKRA, SLy230b, SK272, and SIII Skyrme forces were displayed by the squares, stars, inverted triangles, and triangles symbols, respectively. Moreover, the percentage relative deviations of the cross-sections were calculated from the experimental

36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.7 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

1

7

1

3

Table 1 The excitations of the low-lying states related to the 64 Ni nucleus [35].

4

λπ

Ex (MeV)

βλ coulomb

βλ nuclear

4

2+ 3−

1.346 3.560

0.165 0.193

0.185 0.200

5

2

5 6

2 3

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24

25

25

26

26

27

27

28

28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36

37

37

38

38

39

39 40

40 41 42 43 44

Fig. 3. The left panel shows the fusion cross-sections in the logarithmic scale and the right panel shows the fusion cross-sections in the linear scale for the 64 Ni+64 Ni reaction that the cross-sections were obtained by using the potentials arising from various Skyrme forces. The solid symbols indicate better agreement of each force than the other three forces in describing a given energy region.

47

42 43 44 45

45 46

41

data for each of the selected forces. The obtained values are listed in Table 2. According to Fig. 3, the fusion cross-sections obtained from the selected forces can describe the experimental

46 47

JID:NUPHA AID:121691 /FLA

1

[m1+; v1.304; Prn:7/01/2020; 16:11] P.8 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

8

1

4

Table 2 The percentage relative deviations of the fusion cross-sections arising from the SKRA, SLy230b, SK272 and SIII forces from the experimental data in the 64 Ni+64 Ni system. The experimental value of the barrier height is VB Exp = 93.5 MeV [61].

5

Ecm (MeV)

Ecm /VB Exp

SKRA

SLy230b

SK272

SIII

5

6

86.0 86.7 87.0 87.9 88.9 89.0 89.6 90.8 91.85 92.85 94.95 97.4 99.65 102.15 104.7 108.35

0.920 0.927 0.930 0.940 0.951 0.952 0.958 0.971 0.982 0.993 1.016 1.042 1.066 1.093 1.120 1.159

1.8 17.2 15.0 39.3 70.8 78.0 79.4 62.3 33.4 23.0 18.1 12.8 10.6 5.6 5.7 18.1

1494.7 505.3 154.9 15.0 10.0 10.5 27.2 33.2 11.8 17.7 2.6 8.7 6.2 17.7 6.8 10.1

474.4 172.8 28.6 29.0 41.8 54.5 52.0 54.6 47.0 27.6 19.9 8.3 2.5 14.0 8.8 13.1

162.5 33.4 36.1 65.8 76.5 82.0 83.1 84.6 78.1 61.9 41.2 29.0 17.7 0.1 1.3 9.3

6

2 3

7 8 9 10 11 12 13 14 15 16 17 18

2 3 4

7 8 9 10 11 12 13 14 15 16 17 18

19

19

20

20

21 22 23 24 25 26 27 28 29 30 31

data error bar at most energies. On the other hand, according to Table 2, each force can better describe a given energy region than the other three forces that this was displayed by the solid symbols in Fig. 3. Accordingly in the 64 Ni+64 Ni reaction, the cross-sections arising from the SKRA Skyrme force (K=217 MeV) reproduce the experimental data at the deep sub-barrier energies well, then by increasing bombarding energy the cross-sections arising from the SLy230b force (K=230 MeV) are in a good agreement with the data related to energies below and near the barrier. At energies above the barrier and at higher energies, the cross-sections arising from the SK272 (K=271 MeV) and SIII (K=356 MeV) forces can better describe the experimental data. By using the obtained fusion cross-sections, one can calculate the astrophysical S(E) factor as: S(E) = Eσf us exp(2π(η − η0 )).

(10)

36 37 38 39 40

Consequently, its logarithmic derivative is written as: 1 dS(E) πη d = Ln(Eσf us (E)) − . S(E) dE dE E

43 44 45

L(E) =

d(Eσf us (E)) . Eσf us (E) dE

25 26 27 28 29 30 31

33

36

(11)

1

LCS (E) =

πη . E

37 38 39 40 41

(12)

42 43

On the basis of Eq. (11), where dS(E)/dE = 0 is, the S(E) factor function is constant and shows a maximum, as a result, the logarithmic slope L(E) is equal to:

46 47

24

35

In the above expression, the first term on the right represents the logarithmic slope L(E) that is written as:

41 42

23

34

34 35

22

32

32 33

21

44 45 46

(13)

47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.9 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

9

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23 24

24 25 26

Fig. 4. The astrophysical S(E) factor calculated by the fusion cross-sections arising from various Skyrme forces in the 64 Ni+64 Ni system. The contribution of each force was specified in describing different energy regions. The figure inset was plotted for a clearer display of the S-factor maximum.

28

28

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

26 27

27

29

25

In the above expressions, E and η are the center of mass energy and the Sommerfeld parameter, respectively. In this paper to calculate S(E) and L(E) in the 64 Ni+64 Ni system, the fusion cross-sections are employed, which are in better agreement with the experimental data at each energy (see Table 2). The obtained results for these two quantities are presented in Figs. 4 and 5. In these two figures, the contribution of each force was displayed at different energy regions. According to Fig. 4, the S-factor obtained from the SLy230b force is consistent with the data related to region near and below the barrier, but by decreasing energy and approaching the deep sub-barrier region, it deviates. Instead, the S-factor arising from the SKRA force describes the experimental data in this region well and reproduces the S-factor maximum at the energy E=87.0 MeV, which is close to its experimental value, i.e., E=87.3±0.9 MeV [35]. These agreements are also observed on the logarithmic slope L(E). According to Fig. 5, the logarithmic slope arising from the SLy230b force describes the experimental data at energies near and below the barrier, whereas the logarithmic slope arising from the SKRA force intersects the LCS , and well reproduces the increasing trend of the data in the deep sub-barrier region. Based on the results obtained from the fusion cross-sections, S-factor and logarithmic slope L(E), each energy region in the 64 Ni+64 Ni reaction can be described by an incompressibility value whose corresponding force shows the best agreement between the theoretical calculations and the experimental data. Accordingly, the incompressibility values are plotted as a function

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.10 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

10

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23 24

24 25 26

Fig. 5. The logarithmic slope L(E) calculated by the fusion cross-sections arising from various Skyrme forces in the 64 Ni+64 Ni system. The contribution of each force was specified in describing different energy regions.

29 30 31 32 33 34 35

Exp

Ecm /VB in Fig. 6. Based on this figure, the nuclear matter in the 64 Ni+64 Ni reaction is approaching a very soft matter around K=217 MeV when moving from the sub-barrier to the deep sub-barrier energies, thus many nucleons contribute in the overlapping region. As already mentioned, the Pauli repulsion can strongly affect a large overlapping region, it is therefore anticipated that this repulsion reduces the probability of tunneling at the energies below the Coulomb barrier and subsequently increases the barrier height in the transition from below the barrier to the deep sub-barrier energies (see Fig. 2(b)). Accordingly, the occurrence of the hindrance phenomenon in the 64 Ni+64 Ni reaction can be attributed to the Pauli repulsion. 4. Conclusions

40 41 42 43 44 45 46 47

29 30 31 32 33 34 35

37 38

38 39

28

36

36 37

26 27

27 28

25

In the present study, we examined the energy dependence of the incompressibility during the fusion process and its effect on the hindrance phenomenon in the 64 Ni+64 Ni system. To this end, the inter-nuclear potential was calculated by using the Skyrme energy density formalism in which the used forces are in the incompressibility values range from 217 to 356 MeV. The obtained potentials for the 64 Ni+64 Ni reaction indicated that from the incompressibility K=230 MeV toward higher values, by decreasing the surface diffuseness the barrier heights increase and their corresponding positions decrease. While from incompressibility K=230 MeV toward lower values as the surface diffuseness increases, the barrier heights are again increasing and their corresponding positions are decreasing. Since in the mentioned incompressibility range, the nuclear

39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.11 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

11

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23 24

24 25

Fig. 6. The predicted incompressibility values for the 64 Ni+64 Ni system in different energy regions.

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

25 26

26

64 Ni+64 Ni

matter in the reaction shows a tendency toward a very soft behavior around K=217 MeV, a large overlapping can occur between the densities of the colliding nuclei. Consequently, the repulsion arising from the Pauli exclusion principle, which strongly affects large nuclear overlapping causes the barrier heights to increase. The fusion cross-sections obtained by using the potentials arising from different Skyrme forces can show the variation in the nuclear matter property during fusion reaction of the 64 Ni+64 Ni so that by increasing bombarding energy the forces with higher incompressibility values are in better agreement with the experimental data. In addition, it was shown that the SKRA force with the incompressibility K=217 MeV can describe the fall-off of the fusion cross-sections in the 64 Ni+64 Ni system also, it is able to reproduce the S-factor maximum and the increasing trend of the L(E) in the deep sub-barrier region. It is therefore predicted when moving from the sub-barrier to the deep sub-barrier in the 64 Ni+64 Ni system, because of the variation of the incompressibility and the approaching a very soft nuclear matter, the repulsion arising from the Pauli exclusion principle can lead to the hindrance phenomenon occurrence in this reaction. It should be noted that in the performed calculations in this paper, no adjustable parameters were used. References [1] R.G. Stokstad, Z.E. Switkowski, R.A. Dayras, R.M. Wieland, Phys. Rev. Lett. 37 (1976) 888. [2] J.R. Birkelund, L.E. Tubbs, J.R. Huizenga, J.N. De, D. Sperber, Phys. Rep. 56 (1979) 107. [3] J.R. Birkelund, J.R. Huizenga, Annu. Rev. Nucl. Part. Sci. 33 (1983) 265.

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

18 19 20 21

[24] [25] [26] [27]

22 23 24 25 26 27

[28] [29] [30] [31] [32]

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

[m1+; v1.304; Prn:7/01/2020; 16:11] P.12 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

R.A. Broglia, C.H. Dasso, S. Landowne, A. Winther, Phys. Rev. C 27 (1983) 2433. C.H. Dasso, S. Landowne, A. Winther, Nucl. Phys. A 405 (1983) 381. M. Beckerman, Phys. Rep. 129 (1985) 145. T. Udagawa, B.T. Kim, T. Tamura, Phys. Rev. C 32 (1985) 124. S.G. Steadman, M.J. Rhoades-Brown, Annu. Rev. Nucl. Part. Sci. 36 (1986) 649. M. Beckerman, Rep. Prog. Phys. 51 (1988) 1047. N. Rowley, G.R. Satchler, P.H. Stelson, Phys. Lett. B 254 (1991) 25. W. Reisdorf, J. Phys. G 20 (1994) 1297. A.B. Balantekin, N. Takigawa, Rev. Mod. Phys. 70 (1998) 77. Z.Q. Feng, G.M. Jin, F.S. Zhang, Nucl. Phys. A 802 (2008) 91. S. Ayik, B. Yilmaz, D. Lacroix, Phys. Rev. C 81 (2010) 034605. K. Hagino, N. Takigawa, Prog. Theor. Phys. 128 (2012) 1001. B.B. Back, H. Esbensen, C.L. Jiang, K.E. Rehm, Rev. Mod. Phys. 86 (2014) 317. ´ atecki, C.F. Tsang, Ann. Phys. 105 (1977) 427. ˛ J. Błocki, J. Randrup, W.J. Swi G.R. Satchler, W.G. Love, Phys. Rep. 55 (1979) 183. J. Aichelin, H. Stöcker, Phys. Lett. B 176 (1986) 14. D.T. Khoa, G.R. Satchler, W.V. Oertzen, Phys. Rev. C 56 (1997) 954. ´ atecki, Phys. Rev. C 62 (2000) 044610. ˛ W.D. Myers, W.J. Swi N. Wang, Z. Li, X. Wu, Phys. Rev. C 65 (2002) 064608. L.C. Chamon, G.P.A. Nobre, D. Pereira, E.S. Rossi Jr., C.P. Silva, L.R. Gasques, B.V. Carlson, Phys. Rev. C 70 (2004) 014604. I.I. Gontchar, D.J. Hinde, M. Dasgupta, J.O. Newton, Phys. Rev. C 69 (2004) 024610. A.S. Umar, V.E. Oberacker, Phys. Rev. C 74 (2006) 021601. C.Y. Wong, Phys. Rev. Lett. 31 (1973) 766. A.M. Stefanini, D. Ackermann, L. Corradi, D.R. Napoli, C. Petrache, P. Spolaore, P. Bednarczyk, H.Q. Zhang, S. Beghini, G. Montagnoli, L. Mueller, F. Scarlassara, G.F. Segato, F. Soramel, N. Rowley, Phys. Rev. Lett. 74 (1995) 864. M. Dasgupta, D.J. Hinde, N. Rowley, A.M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48 (1998) 401. C.L. Jiang, et al., Phys. Rev. Lett. 89 (2002) 052701. C.L. Jiang, H. Esbensen, B.B. Back, R.V.F. Janssens, K.E. Rehm, Phys. Rev. C 69 (2004) 014604. C.L. Jiang, et al., Phys. Rev. C 71 (2005) 044613. A.M. Stefanini, G. Montagnoli, L. Corradi, S. Courtin, E. Fioretto, A. Goasduff, F. Haas, P. Mason, R. Silvestri, Pushpendra P. Singh, F. Scarlassara, S. Szilner, Phys. Rev. C 82 (2010) 014614. ¸ Mi¸sicu, Phys. Rev. C 76 (2007) 054609. H. Esbensen, S. K. Cheng, C. Xu, Nucl. Phys. A 992 (2019) 121642. C.L. Jiang, et al., Phys. Rev. Lett. 93 (2004) 012701. S. ¸ Mi¸sicu, H. Esbensen, Phys. Rev. Lett. 96 (2006) 112701. H. Esbensen, Phys. Rev. C 72 (2005) 054607. C.L. Jiang, B.B. Back, H. Esbensen, R.V.F. Janssens, K.E. Rehm, Phys. Rev. C 73 (2006) 014613. T. Ichikawa, K. Hagino, A. Iwamoto, Phys. Rev. C 75 (2007) 057603. T. Ichikawa, K. Hagino, A. Iwamoto, Phys. Rev. Lett. 103 (2009) 202701. K. Hagino, A.B. Balantekin, N.W. Lwin, E.S.Z. Thein, Phys. Rev. C 97 (2018) 034623. T. Ichikawa, Phys. Rev. C 92 (2015) 064604. A.S. Umar, V.E. Oberacker, Phys. Rev. C 77 (2008) 064605. S. ¸ Mi¸sicu, H. Esbensen, Phys. Rev. C 75 (2007) 034606. C. Simenel, A.S. Umar, K. Godbey, M. Dasgupta, D.J. Hinde, Phys. Rev. C 95 (2017) 031601(R). P.G. Reinhard, A.S. Umar, P.D. Stevenson, J. Piekarewicz, V.E. Oberacker, J.A. Maruhn, Phys. Rev. C 93 (2016) 044618. C. Simenel, A.S. Umar, Prog. Part. Nucl. Phys. 103 (2018) 19. R. Kumar, M.K. Sharma, R.K. Gupta, Nucl. Phys. A 870 (2011) 42. O.N. Ghodsi, F. Torabi, Phys. Rev. C 93 (2016) 064611. T.H.R. Skyrme, Nucl. Phys. 9 (1958–1959) 615. M. Brack, C. Guet, H.B. Hakansson, Phys. Rep. 123 (1985) 275. R.K. Puri, R.K. Gupta, Phys. Rev. C 45 (1992) 1837. J. Bartel, K. Bencheikh, Eur. Phys. J. A 14 (2002) 179. D. Vautherin, D.M. Brink, Phys. Rev. C 5 (1972) 626.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHA AID:121691 /FLA

[m1+; v1.304; Prn:7/01/2020; 16:11] P.13 (1-13)

T. Ghasemi, O.N. Ghodsi / Nuclear Physics A ••• (••••) ••••••

1 2 3 4 5 6

[55] [56] [57] [58] [59] [60] [61]

K. Bennaceur, J. Dobaczewski, Comput. Phys. Commun. 168 (2005) 96. M. Rashdan, Mod. Phys. Lett. A 15 (2000) 1287. E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 627 (1997) 710. B.K. Agrawal, S. Shlomo, V.K. Au, Phys. Rev. C 68 (2003) 031304. M. Beiner, H. Flocard, N. Van Giai, P. Quentin, Nucl. Phys. A 238 (1975) 29. K. Hagino, N. Rowley, A.T. Kruppa, Comput. Phys. Commun. 123 (1999) 143. M. Beckerman, M. Salomaa, A. Sperduto, J.D. Molitoris, A. DiRienzo, Phys. Rev. C 25 (1982) 837.

13

1 2 3 4 5 6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

16

16

17

17

18

18

19

19

20

20

21

21

22

22

23

23

24

24

25

25

26

26

27

27

28

28

29

29

30

30

31

31

32

32

33

33

34

34

35

35

36

36

37

37

38

38

39

39

40

40

41

41

42

42

43

43

44

44

45

45

46

46

47

47