- Email: [email protected]
On the multifragmentation around the energy of vanishing flow using isospin-dependent model
Available online at www.sciencedirect.com
Nuclear Physics A 875 (2012) 173–180 www.elsevier.com/locate/nuclphysa
On the multifragmentation around the energy of vanishing flow using isospin-dependent model Rajni a , Suneel Kumar a,∗ , Rajeev K. Puri b a School of Physics and Materials Science, Thapar University, Patiala-147004, Punjab, India b Department of Physics, Panjab University, Chandigarh-160014, India
Received 30 September 2011; received in revised form 10 November 2011; accepted 25 November 2011 Available online 28 November 2011
Abstract We analyze the fragmentation of colliding nuclei around the energy of vanishing flow (EVF) and evaluated its mass dependence throughout the periodic table using isospin-dependent quantum molecular dynamic model. The free nucleons, light mass fragments (LMFs) are emitted from the participant source that scales with the size of the emitting source while medium mass fragments (MMFs) and intermediate mass fragments (IMFs) come from the spectator part showing well known universality in the multiplicity. No particular fragment structure found at the energy of vanishing flow. © 2011 Elsevier B.V. All rights reserved. Keywords: Balance energy; Transition energy; Multifragmentation; IQMD model
1. Introduction The breaking of colliding nuclei into several complex pieces (consisting of light, intermediate as well as heavy masses) has been of core importance both in experimental and theoretical studies. It is now quite clear that the multiplicity of these fragments depend crucially on the bombarding energy, masses of colliding nuclei as well as on the impact parameter of the reaction. The experimental studies can be divided into symmetric and asymmetric reactions. Former category leads to higher compression whereas latter lacks the compression and therefore, large part of excitation energy converts into thermal energy [1,2]. * Corresponding author.
E-mail address: [email protected] (S. Kumar). 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.11.009
174
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Recently, one of us and collaborators [3] carried out systematic calculations of energy of vanishing flow (EVF) over entire periodic table and power law dependence of A−1/3 proposed earlier by [4] was found to be valid. They also explored the fragment’s structure at EVF. These calculations were later on extended by one of us and collaborators for isospin symmetry energies [5]. It still remain to be seen whether the fragmenting matter at the energy of vanishing flow is sensitive towards isospin-symmetry energies and how structure of fragments at EVF differ from the one at different incident energies. Unfortunately, no study exists in the literature so far that sheds light on the fragment’s structure around EVF using isospin-dependent quantum molecular dynamics (IQMD) model. It is worth reminding that the energy of vanishing flow is much higher in lighter colliding nuclei compared to heavier one where matter counterbalances attractive and repulsive force at low incident energies. It is also evident from the literature that the form and strength of symmetry energy depends on the incident energy and corresponding density. Therefore, one would like to investigate how multiplicity of various fragments behaves at energy of vanishing flow and whether the trend is similar to that obtained at low or higher incident energies. Further, how symmetry energy affects the dynamics around the energy of vanishing flow. We plan to address these questions in this paper with the help of isospin quantum molecular dynamic model which is discussed in brief below. 2. Isospin-dependent quantum molecular dynamics (IQMD) model The isospin-dependent quantum molecular dynamics (IQMD) [6] model treats different charge states of nucleons, deltas and pions explicitly, as inherited from the Vlasov–Uehling– Uhlenbeck (VUU) model [6]. The IQMD model has been used successfully for the analysis of large number of observables from low to relativistic energies. The isospin degree of freedom enters into the calculations via symmetry potential, cross-sections and Coulomb interactions. The details about the elastic and inelastic cross-section for proton–proton and neutron–neutron collisions can be found in Ref. [6]. These cross-section are well explained by particle data group. In this model, baryons are represented by Gaussian-shaped density distributions fi (r , p, t) =
1 π 2 h2 ¯
· e−(r −ri (t))
2 1 2L
·e
−(p− pi (t))2 2L2 h¯
.
(1)
Nucleons are initialized in a sphere with radius R = 1.12A1/3 fm, in accordance with the liquid drop model. Each nucleon occupies a volume of h3 , so that phase space is uniformly filled. The initial momenta are randomly chosen between 0 and Fermi momentum (pF ). The nucleons of target and projectile interact via two and three-body Skyrme forces, Yukawa potential and momentum-dependent interactions. These interactions are similar as used in the molecular dynamical models like quantum molecular dynamics (QMD) [7] and relativistic QMD [8]. The isospin degree of freedom is treated explicitly by employing a symmetry potential and explicit Coulomb forces between protons of colliding target and projectile nucleons. This helps in achieving correct distribution of protons and neutrons within the nucleus. The hadrons propagate using Hamilton equations of motion: d ri dH = , dt dpi with
dH d pi =− dt dri
(2)
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
175
H = T + V p2 i = + t)V ij r , r × fj r , p , t d r d r d p d p . fi (r , p, 2mi i
i
(3)
j >i
The baryon–baryon potential V ij , in the above relation, reads as: ij ij ij ij ij V ij r − r = VSkyrme + VYukawa + VCoul + Vmdi + Vsym r + r = t1 δ r − r + t2 δ r − r ρ γ −1 2 exp(|r − r|/μ) Zi Zj e2 + (|r − r|/μ) |r − r| 1 2 j + t4 ln2 t5 pi − p + 1 δ r − r + t6 T3i T3 δ ri − rj . 0
+ t3
(4)
j
Here Zi and Zj denote the charges of ith and j th baryon, and T3i , T3 are their respective T3 components (i.e. 1/2 for protons and −1/2 for neutrons). Meson potential consists of Coulomb interaction only. The parameters μ and t1 , . . . , t6 are adjusted to the real part of the nucleonic optical potential. For the density dependence of the nucleon optical potential, standard Skyrmetype parameterization is employed. As is evident, we choose symmetry energy that depends linearly on the baryon density. The binary nucleon–nucleon collisions are included by employing collision term of well known VUU–BUU equation. The binary collisions are done stochastically, in a similar way as are done in all transport models. During the propagation, two nucleons are supposed to suffer a binary collision if the distance between their centroids
√ σtot , σtot = σ ( s, type) (5) |ri − rj | π “type” denotes the ingoing collision partners (N –N, N –, N –π, . . .). In addition, Pauli blocking (of the final state) of baryons is taken into account by checking the phase space densities in the final states. The final phase space fractions P1 and P2 which are already occupied by other nucleons are determined for each of the scattering baryons. The collision is then blocked with probability Pblock = 1 − (1 − P1 )(1 − P2 ).
(6)
Delta decays are checked in an analogous fashion with respect to the phase space of the resulting nucleons. Recently, several studies have been devoted to pin down the strength of the NN crosssection [7]. 3. Results and discussion The present analysis is made using the reactions of 197 Au79 + 197 Au79 (b = 2.5 fm), 139 La57 + 93 Nb + 93 Nb 86 Kr + 93 Nb 64 Zn + 58 Ni 57 (bˆ = 0.3), 41 41 (bˆ = 0.3), 36 41 (bˆ = 0.4), 30 28 40 45 ˆ ˆ (b = 2 fm), Ar18 + Sc21 (b = 0.4) where b is the scaled impact parameter, defined as 1/3 1/3 b (b is particular impact parameter in Fermi (fm) and bmax = 1.12(AT + AP )), AT bˆ = bmax and AP is the mass of target and projectile respectively. The corresponding energy of vanishing flow (EVF) for these system were reported to be 48, 58, 62, 56, 64, 80 MeV/nucleon 139 La
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Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Fig. 1. Time evolution of largest fragment Amax , free nucleon, LMFs, MMFs, IMFs for the collision of 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc at their corresponding theoretical balance energies of 48, 58, 62, 56, 64 and 80 MeV/nucleon.
respectively [9]. The calculations have been carried out with symmetry energy corresponding to normal nuclear matter density which is Esym = 32 MeV [10]. It is worth mentioning that these EVF is now found to be in closer agreement with experimental energy of vanishing flow. Each above reaction was simulated at their corresponding energy of vanishing flow for 1000–3000 events till reaction saturates. It has been found out by many authors that the reaction time in lighter colliding nuclei is shorter compared to heavy nuclei. For comparison, calculations were also made below and above the energy of vanishing flow. In the present calculations, mass dependence Gaussian width was employed. After a couple of hundreds fm/c, reaction is assumed to be over. The clusterization at first instance is made using the minimum spanning tree (MST) method, that binds two nucleons in a cluster if their distance is less than 4 fm [11]. In Fig. 1, we display the time evolution of different fragments for the above mentioned reactions at their corresponding energy of vanishing flow. The displayed windows are the heaviest fragment Amax , free nucleons (A = 1), light mass fragment (LMF) (2 Af 4), medium mass fragment (MMF) (5 Af 11) as well as intermediate mass fragment (IMF) (5 Af Atot/6 ). The MST method gives one big excited compound cluster at the time of maximum density that decays later into free nucleons and fragments. As a result, free nu-
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
177
Fig. 2. Same as in Fig. 1 for 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc but for above theoretical balance energy i.e. 100 MeV/nucleon.
cleons, LMFs, MMFs, IMFs continue to rise till interaction among them ceases. One also see that lighter colliding nuclei saturate much faster compared to heavier nuclei. In addition, since balance energy in heavier nuclei is much smaller than that of lighter nuclei, it takes longer time for heavier nuclei to saturate. At the start of the reaction, all nucleons constitute spectator matter. Since 40 Ar + 45 Sc reaction happens at relatively higher energy of vanishing flow. The transition from the spectator to participant matter is swift and sudden. In contrast, due to low bombarding energy, such transition is slow and gradual in heavier colliding nuclei. Since energy of vanishing flow represents counterbalancing between the attractive and repulsive forces, this also point towards the formation of lighter fragment that emerge from midrapidity. In Fig. 2, we display the time evolution of different fragments at incident energy of 100 MeV/nucleon. Due to the increase in the incident energy, frequent nucleon–nucleon collisions enhance the emission of free nucleons, LMFs, IMFs and MMFs. In Fig. 3, we display the time evolution of different fragments below the corresponding energy of vanishing flow. The displayed windows are same as reported in Fig. 1, with the decrease in the incident energy, excitation energy of the systems gets depleted severely and as a result, binary collision are reduced drastically that suppresses the emission of free nucleon and LMFs. This point towards the dominance of attractive mean field below the balance energy. Reduced binary
178
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Fig. 3. same as in Fig. 1 for 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc but for below theoretical balance energy i.e. 40 MeV/nucleon.
collisions and stronger attractive forces binds more and more nucleons in various fragments, therefore, reducing the production of medium and heavy mass fragments. In Fig. 4, we display the Amax , free nucleon, LMFs, IMFs and MMFs as a function of the system mass at all three incident energies i.e. at EVF with Esym = 32 MeV, below 40 MeV/nucleon, above 100 MeV/nucleon and at EVF with Esym = 0 MeV. From the figure, it is clear that no typical structure is obtained at the energy of vanishing flow with Esym = 32 MeV and Esym = 0 MeV. As noted in Ref. [12], multiplicity of various fragment show power law dependence without any particular structure at energy of vanishing flow. In the Fig. 5, multiplicity of IMFs in 86 Kr + 93 Nb, 40 Ar + 45 Sc are displayed as a function of beam energy at scaled impact parameter bˆ = 0.3 (semi central collisions) using isospin-dependent reduced cross-section (σ = 0.9σNN ) [9]. Final results are compared with NSCL experimental data [13]. Multiplicity of IMFs increases with increase in beam energy. It is also clear from experimental data that no particular structure is obtained at the energy of vanishing flow (EVF) as shown by arrow in this figure. 4. Conclusion Using the isospin-dependent quantum molecular dynamic model, we studied the fragment structure at, above and below the energy of vanishing flow. Fragmentation produced due to two colliding nuclei does not follow any particular structure at the energy of vanishing flow.
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
179
Fig. 4. Multiplicities of Amax , free nucleon, LMFs, MMFs, IMFs as a function of composite mass of colliding nuclei (AT + AP ) for the reactions of 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc at, above, below and at EVF with Esym = 0 MeV.
Fig. 5. The average multiplicity of IMFs vs beam energy in the reaction 86 Kr + 93 Nb, 40 Ar + 45 Sc. The symbol represent the NSCL experimental results, while lines represent the result obtained using cross-section (σ = 0.9σNN ).
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Rajni et al. / Nuclear Physics A 875 (2012) 173–180
References [1] B. Jakobsson, et al., Nucl. Phys. A 509 (1990) 195; M. Begemann-Blaich, et al., Phys. Rev. C 58 (1998) 1639; M. Begemann-Blaich, et al., Phys. Rev. C 48 (1993) 610; S.C. Jeong, et al., Nucl. Phys. A 604 (1996) 208. [2] J. Aichelin, Phys. Rep. 202 (1991) 233. [3] A.D. Sood, R.K. Puri, Phys. Rev. C 69 (2004) 054612; A.D. Sood, R.K. Puri, Phys. Rev. C 70 (2004) 034611. [4] G.D. Westfall, et al., Phys. Rev. Lett. 71 (1993) 1986. [5] S. Kumar, S. Kumar, R.K. Puri, Phys. Rev. C 81 (2010) 014601. [6] C. Hartnack, et al., Eur. Phys. J. A 1 (1998) 151; V. Kaur, S. Kumar, R.K. Puri, Phys. Lett. B 697 (2011) 512; V. Kaur, S. Kumar, R.K. Puri, Nucl. Phys. A 861 (2011) 37. [7] D.T. Khoa, et al., Nucl. Phys. A 619 (1992) 102. [8] E. Lehmann, et al., Phys. Rev. C 51 (1995) 2113; E. Lehmann, et al., Prog. Part. Nucl. Phys. 30 (1993) 219. [9] S. Kumar, Rajni, S. Kumar, Phys. Rev. C 82 (2010) 024610. [10] B.A. Li, L.W. Chen, C.M. Ko, Phys. Rep. 464 (2008) 113. [11] D.T. Khoa, et al., Nucl. Phys. A 548 (1992) 102. [12] J. Singh, R.K. Puri, Phys. Rev. C 65 (2002) 024602. [13] W.J. Llope, et al., Phys. Rev. C 51 (1995) 1325.
Nuclear Physics A 875 (2012) 173–180 www.elsevier.com/locate/nuclphysa
On the multifragmentation around the energy of vanishing flow using isospin-dependent model Rajni a , Suneel Kumar a,∗ , Rajeev K. Puri b a School of Physics and Materials Science, Thapar University, Patiala-147004, Punjab, India b Department of Physics, Panjab University, Chandigarh-160014, India
Received 30 September 2011; received in revised form 10 November 2011; accepted 25 November 2011 Available online 28 November 2011
Abstract We analyze the fragmentation of colliding nuclei around the energy of vanishing flow (EVF) and evaluated its mass dependence throughout the periodic table using isospin-dependent quantum molecular dynamic model. The free nucleons, light mass fragments (LMFs) are emitted from the participant source that scales with the size of the emitting source while medium mass fragments (MMFs) and intermediate mass fragments (IMFs) come from the spectator part showing well known universality in the multiplicity. No particular fragment structure found at the energy of vanishing flow. © 2011 Elsevier B.V. All rights reserved. Keywords: Balance energy; Transition energy; Multifragmentation; IQMD model
1. Introduction The breaking of colliding nuclei into several complex pieces (consisting of light, intermediate as well as heavy masses) has been of core importance both in experimental and theoretical studies. It is now quite clear that the multiplicity of these fragments depend crucially on the bombarding energy, masses of colliding nuclei as well as on the impact parameter of the reaction. The experimental studies can be divided into symmetric and asymmetric reactions. Former category leads to higher compression whereas latter lacks the compression and therefore, large part of excitation energy converts into thermal energy [1,2]. * Corresponding author.
E-mail address: [email protected] (S. Kumar). 0375-9474/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2011.11.009
174
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Recently, one of us and collaborators [3] carried out systematic calculations of energy of vanishing flow (EVF) over entire periodic table and power law dependence of A−1/3 proposed earlier by [4] was found to be valid. They also explored the fragment’s structure at EVF. These calculations were later on extended by one of us and collaborators for isospin symmetry energies [5]. It still remain to be seen whether the fragmenting matter at the energy of vanishing flow is sensitive towards isospin-symmetry energies and how structure of fragments at EVF differ from the one at different incident energies. Unfortunately, no study exists in the literature so far that sheds light on the fragment’s structure around EVF using isospin-dependent quantum molecular dynamics (IQMD) model. It is worth reminding that the energy of vanishing flow is much higher in lighter colliding nuclei compared to heavier one where matter counterbalances attractive and repulsive force at low incident energies. It is also evident from the literature that the form and strength of symmetry energy depends on the incident energy and corresponding density. Therefore, one would like to investigate how multiplicity of various fragments behaves at energy of vanishing flow and whether the trend is similar to that obtained at low or higher incident energies. Further, how symmetry energy affects the dynamics around the energy of vanishing flow. We plan to address these questions in this paper with the help of isospin quantum molecular dynamic model which is discussed in brief below. 2. Isospin-dependent quantum molecular dynamics (IQMD) model The isospin-dependent quantum molecular dynamics (IQMD) [6] model treats different charge states of nucleons, deltas and pions explicitly, as inherited from the Vlasov–Uehling– Uhlenbeck (VUU) model [6]. The IQMD model has been used successfully for the analysis of large number of observables from low to relativistic energies. The isospin degree of freedom enters into the calculations via symmetry potential, cross-sections and Coulomb interactions. The details about the elastic and inelastic cross-section for proton–proton and neutron–neutron collisions can be found in Ref. [6]. These cross-section are well explained by particle data group. In this model, baryons are represented by Gaussian-shaped density distributions fi (r , p, t) =
1 π 2 h2 ¯
· e−(r −ri (t))
2 1 2L
·e
−(p− pi (t))2 2L2 h¯
.
(1)
Nucleons are initialized in a sphere with radius R = 1.12A1/3 fm, in accordance with the liquid drop model. Each nucleon occupies a volume of h3 , so that phase space is uniformly filled. The initial momenta are randomly chosen between 0 and Fermi momentum (pF ). The nucleons of target and projectile interact via two and three-body Skyrme forces, Yukawa potential and momentum-dependent interactions. These interactions are similar as used in the molecular dynamical models like quantum molecular dynamics (QMD) [7] and relativistic QMD [8]. The isospin degree of freedom is treated explicitly by employing a symmetry potential and explicit Coulomb forces between protons of colliding target and projectile nucleons. This helps in achieving correct distribution of protons and neutrons within the nucleus. The hadrons propagate using Hamilton equations of motion: d ri dH = , dt dpi with
dH d pi =− dt dri
(2)
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
175
H = T + V p2 i = + t)V ij r , r × fj r , p , t d r d r d p d p . fi (r , p, 2mi i
i
(3)
j >i
The baryon–baryon potential V ij , in the above relation, reads as: ij ij ij ij ij V ij r − r = VSkyrme + VYukawa + VCoul + Vmdi + Vsym r + r = t1 δ r − r + t2 δ r − r ρ γ −1 2 exp(|r − r|/μ) Zi Zj e2 + (|r − r|/μ) |r − r| 1 2 j + t4 ln2 t5 pi − p + 1 δ r − r + t6 T3i T3 δ ri − rj . 0
+ t3
(4)
j
Here Zi and Zj denote the charges of ith and j th baryon, and T3i , T3 are their respective T3 components (i.e. 1/2 for protons and −1/2 for neutrons). Meson potential consists of Coulomb interaction only. The parameters μ and t1 , . . . , t6 are adjusted to the real part of the nucleonic optical potential. For the density dependence of the nucleon optical potential, standard Skyrmetype parameterization is employed. As is evident, we choose symmetry energy that depends linearly on the baryon density. The binary nucleon–nucleon collisions are included by employing collision term of well known VUU–BUU equation. The binary collisions are done stochastically, in a similar way as are done in all transport models. During the propagation, two nucleons are supposed to suffer a binary collision if the distance between their centroids
√ σtot , σtot = σ ( s, type) (5) |ri − rj | π “type” denotes the ingoing collision partners (N –N, N –, N –π, . . .). In addition, Pauli blocking (of the final state) of baryons is taken into account by checking the phase space densities in the final states. The final phase space fractions P1 and P2 which are already occupied by other nucleons are determined for each of the scattering baryons. The collision is then blocked with probability Pblock = 1 − (1 − P1 )(1 − P2 ).
(6)
Delta decays are checked in an analogous fashion with respect to the phase space of the resulting nucleons. Recently, several studies have been devoted to pin down the strength of the NN crosssection [7]. 3. Results and discussion The present analysis is made using the reactions of 197 Au79 + 197 Au79 (b = 2.5 fm), 139 La57 + 93 Nb + 93 Nb 86 Kr + 93 Nb 64 Zn + 58 Ni 57 (bˆ = 0.3), 41 41 (bˆ = 0.3), 36 41 (bˆ = 0.4), 30 28 40 45 ˆ ˆ (b = 2 fm), Ar18 + Sc21 (b = 0.4) where b is the scaled impact parameter, defined as 1/3 1/3 b (b is particular impact parameter in Fermi (fm) and bmax = 1.12(AT + AP )), AT bˆ = bmax and AP is the mass of target and projectile respectively. The corresponding energy of vanishing flow (EVF) for these system were reported to be 48, 58, 62, 56, 64, 80 MeV/nucleon 139 La
176
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Fig. 1. Time evolution of largest fragment Amax , free nucleon, LMFs, MMFs, IMFs for the collision of 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc at their corresponding theoretical balance energies of 48, 58, 62, 56, 64 and 80 MeV/nucleon.
respectively [9]. The calculations have been carried out with symmetry energy corresponding to normal nuclear matter density which is Esym = 32 MeV [10]. It is worth mentioning that these EVF is now found to be in closer agreement with experimental energy of vanishing flow. Each above reaction was simulated at their corresponding energy of vanishing flow for 1000–3000 events till reaction saturates. It has been found out by many authors that the reaction time in lighter colliding nuclei is shorter compared to heavy nuclei. For comparison, calculations were also made below and above the energy of vanishing flow. In the present calculations, mass dependence Gaussian width was employed. After a couple of hundreds fm/c, reaction is assumed to be over. The clusterization at first instance is made using the minimum spanning tree (MST) method, that binds two nucleons in a cluster if their distance is less than 4 fm [11]. In Fig. 1, we display the time evolution of different fragments for the above mentioned reactions at their corresponding energy of vanishing flow. The displayed windows are the heaviest fragment Amax , free nucleons (A = 1), light mass fragment (LMF) (2 Af 4), medium mass fragment (MMF) (5 Af 11) as well as intermediate mass fragment (IMF) (5 Af Atot/6 ). The MST method gives one big excited compound cluster at the time of maximum density that decays later into free nucleons and fragments. As a result, free nu-
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
177
Fig. 2. Same as in Fig. 1 for 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc but for above theoretical balance energy i.e. 100 MeV/nucleon.
cleons, LMFs, MMFs, IMFs continue to rise till interaction among them ceases. One also see that lighter colliding nuclei saturate much faster compared to heavier nuclei. In addition, since balance energy in heavier nuclei is much smaller than that of lighter nuclei, it takes longer time for heavier nuclei to saturate. At the start of the reaction, all nucleons constitute spectator matter. Since 40 Ar + 45 Sc reaction happens at relatively higher energy of vanishing flow. The transition from the spectator to participant matter is swift and sudden. In contrast, due to low bombarding energy, such transition is slow and gradual in heavier colliding nuclei. Since energy of vanishing flow represents counterbalancing between the attractive and repulsive forces, this also point towards the formation of lighter fragment that emerge from midrapidity. In Fig. 2, we display the time evolution of different fragments at incident energy of 100 MeV/nucleon. Due to the increase in the incident energy, frequent nucleon–nucleon collisions enhance the emission of free nucleons, LMFs, IMFs and MMFs. In Fig. 3, we display the time evolution of different fragments below the corresponding energy of vanishing flow. The displayed windows are same as reported in Fig. 1, with the decrease in the incident energy, excitation energy of the systems gets depleted severely and as a result, binary collision are reduced drastically that suppresses the emission of free nucleon and LMFs. This point towards the dominance of attractive mean field below the balance energy. Reduced binary
178
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
Fig. 3. same as in Fig. 1 for 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc but for below theoretical balance energy i.e. 40 MeV/nucleon.
collisions and stronger attractive forces binds more and more nucleons in various fragments, therefore, reducing the production of medium and heavy mass fragments. In Fig. 4, we display the Amax , free nucleon, LMFs, IMFs and MMFs as a function of the system mass at all three incident energies i.e. at EVF with Esym = 32 MeV, below 40 MeV/nucleon, above 100 MeV/nucleon and at EVF with Esym = 0 MeV. From the figure, it is clear that no typical structure is obtained at the energy of vanishing flow with Esym = 32 MeV and Esym = 0 MeV. As noted in Ref. [12], multiplicity of various fragment show power law dependence without any particular structure at energy of vanishing flow. In the Fig. 5, multiplicity of IMFs in 86 Kr + 93 Nb, 40 Ar + 45 Sc are displayed as a function of beam energy at scaled impact parameter bˆ = 0.3 (semi central collisions) using isospin-dependent reduced cross-section (σ = 0.9σNN ) [9]. Final results are compared with NSCL experimental data [13]. Multiplicity of IMFs increases with increase in beam energy. It is also clear from experimental data that no particular structure is obtained at the energy of vanishing flow (EVF) as shown by arrow in this figure. 4. Conclusion Using the isospin-dependent quantum molecular dynamic model, we studied the fragment structure at, above and below the energy of vanishing flow. Fragmentation produced due to two colliding nuclei does not follow any particular structure at the energy of vanishing flow.
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
179
Fig. 4. Multiplicities of Amax , free nucleon, LMFs, MMFs, IMFs as a function of composite mass of colliding nuclei (AT + AP ) for the reactions of 197 Au + 197 Au, 139 La + 139 La, 93 Nb + 93 Nb, 86 Kr + 93 Nb, 64 Zn + 58 Ni, 40 Ar + 45 Sc at, above, below and at EVF with Esym = 0 MeV.
Fig. 5. The average multiplicity of IMFs vs beam energy in the reaction 86 Kr + 93 Nb, 40 Ar + 45 Sc. The symbol represent the NSCL experimental results, while lines represent the result obtained using cross-section (σ = 0.9σNN ).
180
Rajni et al. / Nuclear Physics A 875 (2012) 173–180
References [1] B. Jakobsson, et al., Nucl. Phys. A 509 (1990) 195; M. Begemann-Blaich, et al., Phys. Rev. C 58 (1998) 1639; M. Begemann-Blaich, et al., Phys. Rev. C 48 (1993) 610; S.C. Jeong, et al., Nucl. Phys. A 604 (1996) 208. [2] J. Aichelin, Phys. Rep. 202 (1991) 233. [3] A.D. Sood, R.K. Puri, Phys. Rev. C 69 (2004) 054612; A.D. Sood, R.K. Puri, Phys. Rev. C 70 (2004) 034611. [4] G.D. Westfall, et al., Phys. Rev. Lett. 71 (1993) 1986. [5] S. Kumar, S. Kumar, R.K. Puri, Phys. Rev. C 81 (2010) 014601. [6] C. Hartnack, et al., Eur. Phys. J. A 1 (1998) 151; V. Kaur, S. Kumar, R.K. Puri, Phys. Lett. B 697 (2011) 512; V. Kaur, S. Kumar, R.K. Puri, Nucl. Phys. A 861 (2011) 37. [7] D.T. Khoa, et al., Nucl. Phys. A 619 (1992) 102. [8] E. Lehmann, et al., Phys. Rev. C 51 (1995) 2113; E. Lehmann, et al., Prog. Part. Nucl. Phys. 30 (1993) 219. [9] S. Kumar, Rajni, S. Kumar, Phys. Rev. C 82 (2010) 024610. [10] B.A. Li, L.W. Chen, C.M. Ko, Phys. Rep. 464 (2008) 113. [11] D.T. Khoa, et al., Nucl. Phys. A 548 (1992) 102. [12] J. Singh, R.K. Puri, Phys. Rev. C 65 (2002) 024602. [13] W.J. Llope, et al., Phys. Rev. C 51 (1995) 1325.