A dynamical description of 136Xe + p spallation at 1000 MeV/nucleon

  • This work proposes a dynamical description of the 136Xe + p spallation at 1000 MeV/nucleon with the aim of probing the mechanism which rules the IMFs production. The isospin-dependent quantum molecular dynamics (IQMD) model is applied to describe the dynamical process of the spallation until the hot fragments with excitation energy less than a special value Estop are formed. The statistical code GEMINI is applied to simulate the light-particle evaporation of the hot fragments. It is found that the IMFs production is well described by the model when the Estop = 2 MeV/nucleon is used. But comparison of the mean neutron-to-proton ratios between the data and calculations indicates the value of Estop = 3 MeV/nucleon.
  • 加载中
  • [1] R. Serber, Physical Review, 72: 1114 (1947) doi: 10.1103/PhysRev.72.1114
    [2] J. Wei, H. Chen, Y. Chen, Y. Chen, Y. Chi, C. Deng, H. Dong, L. Dong, S. Fang, J. Feng, S. Fu, L. He, W. He, Y. Heng, K. Huang, X. Jia, W. Kang, X. Kong, J. Li, T. Liang, G. Lin, Z. Liu, H. Ouyang, Q. Qin, H. Qu, C. Shi, H. Sun, J. Tang, J. Tao, C. Wang, F. Wang, D. Wang, Q. Wang, S. Wang, T. Wei, J. Xi, T. Xu, Z. Xu, W. Yin, X. Yin, J. Zhang, Z. Zhang, Z. Zhang, M. Zhou, and T. Zhu, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 600: 10 (2009)
    [3] J. Yang, J. Xia, G. Xiao, H. Xu, H. Zhao, X. Zhou, X. Ma, Y. He, L. Ma, D. Gao, J. Meng, Z. Xu, R. Mao, W. Zhang, Y. Wang, L. Sun, Y. Yuan, P. Yuan, W. Zhan, J. Shi, W. Chai, D. Yin, P. Li, J. Li, L. Mao, J. Zhang, and L. Sheng, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 317: 263 (2013) doi: 10.1016/j.nimb.2013.08.046
    [4] T. Kubo, M. Ishihara, N. Inabe, H. Kumagai, I. Tanihata, K. Yoshida, T. Nakamura, H. Okuno, S. Shimoura, and K. Asahi, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 70: 309 (1992) doi: 10.1016/0168-583X(92)95947-P
    [5] H. Abderrahim, P. Kupschus, E. Malambu, P. Benoit, K. V. Tichelen, B. Arien, F. Vermeersch, P. D’hondt, Y. Jongen, S. Ternier, and D. Vandeplassche, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 463: 487 (2001)
    [6] L. Yang and W. Zhan, Science China Technological Sciences, 58: 1705 (2015) doi: 10.1007/s11431-015-5894-0
    [7] Z.-Q. Chen, Nuclear Science and Techniques, 28: 184 (2017) doi: 10.1007/s41365-017-0335-3
    [8] M. Casolino, V. Bidoli, A. Morselli, L. Narici, M. P. D. Pascale, P. Picozza, E. Reali, R. Sparvoli, G. Mazzenga, M. Ricci, P. Spillantini, M. Boezio, V. Bonvicini, A. Vacchi, N. Zampa, G. Castellini, W. G. Sannita, P. Carlson, A. Galper, M. Korotkov, A. Popov, N. Vavilov, S. Avdeev, and C. Fuglesang, Nature, 422: 680 (2003) doi: 10.1038/422680a
    [9] J. C. David, The European Physical Journal A, 51: 68 (2015) doi: 10.1140/epja/i2015-15068-1
    [10] J. R. Grover, Physical Review, 126: 1540 (1962) doi: 10.1103/PhysRev.126.1540
    [11] A. I. Warwick, A. Baden, H. H. Gutbrod, M. R. Maier, J. P′eter, H. G. Ritter, H. Stelzer, H. H. Wieman, F. Weik, M. Freedman, D. J. Henderson, S. B. Kaufman, E. P. Steinberg, and B. D. Wilkins, Physical Review Letters, 48: 1719 (1982) doi: 10.1103/PhysRevLett.48.1719
    [12] L. Andronenko, A. Kotov, L. Vaishnene, W. Neubert, H. Barz, J. Bondorf, R. Donangelo, and H. Schulz, Physics Letters B, 174: 18 (1986) doi: 10.1016/0370-2693(86)91120-2
    [13] W. c. Hsi, K. Kwiatkowski, G. Wang, D. S. Bracken, E. Cornell, D. S. Ginger, V. E. Viola, N. R. Yoder, R. G. Korteling, F. Gimeno-Nogures, E. Ramakrishnan, D. Rowland, S. J. Yennello, M. J. Huang, W. G. Lynch, M. B. Tsang, H. Xi, Y. Y. Chu, S. Gushue, L. P. Remsberg, K. B. Morley, and H. Breuer, Physical Review Letters, 79: 817 (1997) doi: 10.1103/PhysRevLett.79.817
    [14] P. Napolitani, K.-H. Schmidt, A. S. Botvina, F. Rejmund, L. Tassan-Got, and C. Villagrasa, Physical Review C, 70: 054607 (2004) doi: 10.1103/PhysRevC.70.054607
    [15] M. V. Ricciardi, P. Armbruster, J. Benlliure, M. Bernas, A. Boudard, S. Czajkowski, T. Enqvist, A. Keli′c, S. Leray, R. Legrain, B. Mustapha, J. Pereira, F. Rejmund, K.-H. Schmidt, C. St′ephan, L. Tassan-Got, C. Volant, and O. Yordanov, Physical Review C, 73: 014607 (2006) doi: 10.1103/PhysRevC.73.014607
    [16] P. Napolitani, K.-H. Schmidt, L. Tassan-Got, P. Armbruster, T. Enqvist, A. Heinz, V. Henzl, D. Henzlova, A. Keli′c, R. Pleskaˇc, M. V. Ricciardi, C. Schmitt, O. Yordanov, L. Audouin, M. Bernas, A. Lafriaskh, F. Rejmund, C. St′ephan, J. Benlliure, E. Casarejos, M. F. Ordonez, J. Pereira, A. Boudard, B. Fernandez, S. Leray, C. Villagrasa, and C. Volant, Physical Review C, 76: 064609 (2007) doi: 10.1103/PhysRevC.76.064609
    [17] P. Napolitani, K.-H. Schmidt, and L. Tassan-Got, Journal of Physics G: Nuclear and Particle Physics, 38: 115006 (2011) doi: 10.1088/0954-3899/38/11/115006
    [18] E. L. Gentil, T. Aumann, C. O. Bacri, J. Benlliure, S. Bianchin, M. Bohmer, A. Boudard, J. Brzychczyk, E. Casarejos, M. Combet, L. Donadille, J. E. Ducret, M. Fernandez-Ordo~nez, R. Gernhouser, H. Johansson, K. Kezzar, T. Kurtukian-Nieto, A. Lafriakh, F. Lavaud, A. L. F`evre, S. Leray, J. Luhning, J. Lukasik, U. Lynen, W. F. J. Muller, P. Pawlowski, S. Pietri, F. Rejmund, C. Schwarz, C. Sfienti, H. Simon, W. Trautmann, C. Volant, and O. Yordanov, Physical Review Letters, 100: 022701 (2008) doi: 10.1103/PhysRevLett.100.022701
    [19] P. Napolitani and M. Colonna, Physical Review C, 92: 034607 (2015) doi: 10.1103/PhysRevC.92.034607
    [20] F.-F. Duan, X.-Q. Liu, W.-P. Lin, R. Wada, J.-S. Wang, M.-R. Huang, P.-P. Ren, Y.-Y. Yang, P. Ma, J.-B. Ma, S.-L. Jin, Z. Bai, and Q. Wang, Nuclear Science and Techniques, 27: 131 (2016) doi: 10.1007/s41365-016-0138-y
    [21] C.-W. Ma, C.-Y. Qiao, T.-T. Ding, and Y.-D. Song, Nuclear Science and Techniques, 27: 111 (2016) doi: 10.1007/s41365-016-0112-8
    [22] J. Cugnon, D. L. Hote, and J. Vandermeulen, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 111: 215 (1996) doi: 10.1016/0168-583X(95)01384-9
    [23] D. D. S. Coupland, W. G. Lynch, M. B. Tsang, P. Danielewicz, and Y. Zhang, Physical Review C, 84: 054603 (2011) doi: 10.1103/PhysRevC.84.054603
    [24] M. Papa, T. Maruyama, and A. Bonasera, Physical Review C, 64: 024612 (2001) doi: 10.1103/PhysRevC.64.024612
    [25] J. Su, W. Trautmann, L. Zhu, W.-J. Xie, and F.-S. Zhang, Physical Review C, 98: 014610 (2018) doi: 10.1103/PhysRevC.98.014610
    [26] R. Charity, M. McMahan, G. Wozniak, R. McDonald, L. Moretto, D. Sarantites, L. Sobotka, G. Guarino, A. Pantaleo, L. Fiore, A. Gobbi, and K. Hildenbrand, Nuclear Physics A, 483: 371 (1988) doi: 10.1016/0375-9474(88)90542-8
    [27] B. Borderie and M. Rivet, Progress in Particle and Nuclear Physics, 61: 551 (2008) doi: 10.1016/j.ppnp.2008.01.003
  • 加载中

Figures(7)

Get Citation
Fan Zhang and Jun Su. A dynamical description of 136Xe + p spallation at 1000 MeV/nucleon[J]. Chinese Physics C. doi: 10.1088/1674-1137/43/2/024103
Fan Zhang and Jun Su. A dynamical description of 136Xe + p spallation at 1000 MeV/nucleon[J]. Chinese Physics C.  doi: 10.1088/1674-1137/43/2/024103 shu
Milestone
Received: 2018-11-24
Article Metric

Article Views(54)
PDF Downloads(8)
Cited by(0)
Annacment
Reuse Permission or SCOAP3
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

A dynamical description of 136Xe + p spallation at 1000 MeV/nucleon

    Corresponding author: Fan Zhang, zhangfan@mail.bnu.edu.cn
  • 1. Department of Electronic Information and Physics, Changzhi University, Changzhi 046011, China
  • 2. Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China

Abstract: This work proposes a dynamical description of the 136Xe + p spallation at 1000 MeV/nucleon with the aim of probing the mechanism which rules the IMFs production. The isospin-dependent quantum molecular dynamics (IQMD) model is applied to describe the dynamical process of the spallation until the hot fragments with excitation energy less than a special value Estop are formed. The statistical code GEMINI is applied to simulate the light-particle evaporation of the hot fragments. It is found that the IMFs production is well described by the model when the Estop = 2 MeV/nucleon is used. But comparison of the mean neutron-to-proton ratios between the data and calculations indicates the value of Estop = 3 MeV/nucleon.

    HTML

    1.   Introduction
    • The heavy nucleus bombarded by nucleons or light nuclei at relativistic energies, or its inverse reaction, will produce abundant neutrons and isotopes from the hydrogen to the reacting nucleus. This kind of nuclear reaction is early described by Serber several decades ago and called the spallation [1]. The spallation reaction plays an important role in a wide domain of applications, such as neutron sources, production of rare isotope, transmutation of nuclear waste, and astrophysics. The neutrons can be produced through the spallation process of the steel target colliding by the short proton pulses [2]. In the radioactive ion beam facility, the rare isotope generated in the spallation is separated and focussed as the secondary beam [3, 4]. Moreover, the spallation reactions induced by either protons or neutrons are considered as a cornerstone in the accelerator-driven subcritical reactors [5-7]. In the astrophysics, one uses the cross section data of hydrogen induced spallation to evaluate the propagation of cosmicray nuclei in the interstellar medium [8].

      Decade ago, the International Atomic Energy Agency (IAEA) promoted a benchmark of spallation models with the effort to collect the data and assess the prediction capabilities of the spallation models [9]. In the IAEA benchmark, the codes contain the descriptions of the fast excitation stage by the dynamical models and the slow decay process by the statistical-decay models. The above described two-step models have achieved successes to extract precious information about the influence of deexcitation. Recently, one makes an effort to clarify the emission mechanism of the intermediate-mass fragments (IMFs). The production of IMFs in the spallation was already found by the pioneering experiments [10-13]. With the help of the high-resolution detector array, recent experiments tried to track the process of the IMFs production by the velocity distribution [14-17]. From the theoretical point of view, the IMFs could originate from the multifragmentation of the most excited configurations or asymmetric fission. Comparing the data to predictions of several deexcitation models coupled to the intranuclear cascade model, it is indicated that the multifragmentation stage is not crucial for adequate prediction of residue-production cross sections [17, 18]. However, investigation by the dynamics model propose that the unstable modes in spallation should exhibit quite a similar phenomenology as spinodal instability in dissipative central heavy-ion collisions [19-21].

      The present work is an attempt to describe the emission mechanism of the IMFs in the 136Xe + p spallation at 1000 MeV/nucleon. Being different from the two-step models in the IAEA benchmark, the spallation is described dynamically until the hot fragments with excitation energy less than a special value Estop are formed. The statistical code is applied to simulate the light-particle evaporation of the hot fragments. The $E_{\rm stop}$ dependence of the IMF production will be studied. The paper is organized as follows. In Section 2, the method is introduced. In Section 3, the results are presented. In Section 4, the summary is given.

    2.   Theoretical ramework

      2.1.   Isospin-dependent quantum molecular dynamics model

    • In the IQMD model, the system is described by the $N$ -body wave function, which is the direct product of the Gaussian wave packet for each nucleon.

      where ${{r}}_{i}$ and ${{p}}_{i}$ are the central position and momentum of the i-th nucleon. The width of the Gaussian wave packet depends on the parameter $L$ .

      Using the Wigner transformation, the quantum wave function can be transformed to the one-body phase-space density, which is given by

      Then the local density is,

      In this case, the Hamiltonian can be calculated by the integral,

      where, $T$ is the kinetic energy, $U_{\rm Coul}$ is the Coulomb potential energy, and the third term is the nuclear potential energy, which is given by

      where $\rho_0$ is the normal density, $\delta$ is asymmetry. The parameters used in the this work are $\alpha$ = 356.00 MeV, $\beta$ = 303.00 MeV, $\gamma$ = 7/6, $C_{sp}$ = 38.06 MeV, and $\gamma_{i}$ = 0.75.

      The time evolution of the nucleons in the system under the self-consistently generated mean-field are governed by Hamiltonian equations of motion,

      Moreover, the nucleon-nucleon (NN) collisions are included, in order to simulate the short-range residual interaction. In the pertinent energy region, the elastic NN scattering and the inelastic NN collisions which produce the $\Delta$ particle should be considered. The NN collision are performed according to the differential cross sections, which is the product of the cross section in free space $\sigma^{\rm free}$ , the angular distribution $f^{\rm angl}$ , and the in-medium factor $f^{\rm med}$ ,

      The subscript $i$ refer to the channels of the NN collisions. $i$ = pp for elastic proton-proton scatterings, $i$ = nn elastic neutron-neutron scatterings, $i$ = np elastic neutron-proton scatterings, and $i$ = in inelastic NN collisions. We use the parametrization of the cross section, angular distribution, and in-medium factor, which are taken from Ref. [22, 23].

      At last but not the least, the fermionic feature is compensated by the Pauli blocking and the phase space density constraint (PSDC). The PSDC method was introduced by M. Papa et al., who found that the method affects the dynamics of the nucleus-nucleus collision by reducing the low-momentum particles produces obviously on the average a nonlocal repulsion effect [24]. Then J. Su et al. suggested that the PSDC method can increase the production of the IMFs [25]. The phase space occupation probability of each nucleon can be calculated,

      The integration is performed on an hypercube of volume h $^{3}$ in the phase space centered around the point ( ${{r}}_{i}$ , ${{p}}_{i}$ ). The Pauli blocking is performed by accepting the NN collisions which produce the final states with $\overline{f}_{i} <$ 1 and $\overline{f}_{j} <$ 1. Even though the Pauli blocking is considered, the phase space occupation probabilities of some nucleons will increase to the values larger than 1. In each time step, the phase space occupation probabilities are checked. The many-body elastic scattering will be performed for the nucleons with $\overline{f}_{i} >$ 1.

      In the IAEA benchmark, the dynamical model is only applied to describe the excitation stage, while the IMF emission in the decay stage is described statistically. In this work, the dynamical description is performed not only for the excitation stage but also for the IMF emission. After the excitation stage, the time evolution by the IQMD code continues until the excitation energy of the heaviest host fragment has decreased to a specified value $E_{\rm stop}$ in each event. If the excitation energy is already lower than $E_{\rm stop}$ at t=25 fm/c, the IQMD calculation is immediately stopped. Then the charge number, mass number, excitation energy and momentum of each hot fragment are outputted by the IQMD code. The value of the $E_{\rm stop}$ corresponds to the threshold energy of the IMF emission. After the excitation energy of the fragment is smaller than $E_{\rm stop}$ , the nucleon evaporation is dominant. The decay time of the nucleon evaporation is longer than that of the IMF emission. The effect of spurious nucleon emissions in the IQMD model, which means that a few nucleons will be evaporated even if the excitation energy of the fragment is close to zero, becomes stronger as the time proceeds and for lighter nuclei. Thus, we choose to describe the nucleon evaporation statistically rather than dynamically.

    • 2.2.   GEMINI

    • The output of the IQMD code are the hot fragments. In order to obtain the cold fragments, emissions of the light-particles (Z < 3) from the hot fragments are performed by the statistical code GEMINI [ 26]. A Monte Carlo technique is employed to follow the decay chains until the excitation energy of the product is zero. The partial decay widths are taken from the Hauser-Feshbach formalism,

      where $Z_{i}$ is the charge number, $A_{i}$ is the mass number, $J_{i}$ is the spin, $\rho_{i}$ is the level densities. The subscript $i$ has the values of 0, 1, and 2, which refer to the initial fragment, emitted light particle, and the residual fragments. E*, B, $ E_{\rm rot}$ , and $\varepsilon$ are the excitation energy, separation energy, rotation plus deformation energy, and kinetic energy. $T_{l}$ is the transmission coefficient. The separation energy B is calculated from the tabulated masses.

    3.   Results and discussion
    • The excitation stage in the spallation is displayed as the time evolutions of the local density (up panels) and local excitation energy (down panels) in the x-z plane in Fig. 1. In order to calculate the distribution of the density and excitation energy, the ${}^{136}{\rm Xe}$ nucleus is divided into subsystems with Cartesian coordinate grids. The average density $\rho$ of each grids is calculated within 10000 events. The energies per nucleon E in the grids are calculated from the density $\rho$ and the transverse kinetic energy $E_{tr}$ .

      Figure 1.  (color online) Time evolutions of (a-e) density and (f-j) excitation energy in the x-z plane in central 136Xe + p collision at 1000 MeV/nucleon.

      The transverse kinetic energy is used to deduct the translational energy from the total energy. The local excitation energies are obtained by comparing the energies per nucleon of the subsystems to the those of the nuclear matter at zero temperature.

      The initial time t = 0 fm/c is defined as the moment when the proton touches the surface of the 136Xe nucleus. The kinetic energy of the proton dissipates gradually into the thermal energy of the 136Xe nucleus. The 136Xe nucleus is heated from one side. The local excitation energy on the side reaches the maximum up to 20 MeV. Because of the high excitation, some nucleons escape from the hot spot, leaving a region with low density. After 20 fm/c, the hot spot moves to the center of the nucleus, and the density distribution return to being spherical symmetry. At the same time, the average excitation energy reaches to the maximum, see Fig. 2. In the two-step models, this excitation stage is described dynamically, while the deexcitation stage is described statistically with the equilibrium hypothesis. In this work, we describe the reaction dynamically until the IMFs are produced. For this purpose, we need to recognize the fragments during the time evolution, and decide when the dynamical evolution is stopped.

      Figure 2.  (color online) Time evolution of the excitation energy of the largest fragment in central 136Xe + p collision at 1000 MeV/nucleon.The solid curve shows the average value.The colors show the distribution of the excitation energy.

      The minimum spanning tree (MST) algorithm is applied to recognize the fragments at each time step. The nucleons with relative distance of coordinate and momentum of $|r_{i} - r_{j}| \leqslant R_{0}$ and $|p_{i} - p_{j}| \leqslant P_{0}$ belong to a fragment. The parameters $R_{0}$ = 3.5 fm and $P_{0}$ = 250 MeV/c are chosen. The excitation energy of the fragments can be calculated

      where U is the potential energy, T is the internal kinetic energy, B is the binding energy, $Z_{f}$ is the charge number of the fragment, and $A_{f}$ is the mass number of the fragment.

      The excitation energy of the largest fragment as a function of time is shown in Fig. 2. The solid curve shows the average value, and the colors show the event distribution. The average value of the excitation energy raises rapidly in the excitation stage and then falls slowly in the deexcitation stage. Accompanying with the increase of the average value, the distribution of the excitation energy becomes wider. This is the fluctuation-dissipation phenomenon, which indicates that the dissipation of the kinetic energy of the incident proton is responsible for the fluctuation of the excitation energy.

      The dynamical evolution time of the IQMD model is chosen by the excitation energy of the largest fragment, i.e. the value of each event but not the average value. When the excitation energy of the largest fragment is less than a special value $E_{\rm stop}$ , the dynamical evolution is stopped and the GEMINI code is switched on. The choice of the $E_{\rm stop}$ depends on the threshold energy of the IMF emission. It has been proposed that the threshold energy of the multifragmentation in central heavy ion collision is close to 3 MeV/nucleon [27]. However, the IMF emission in spallation may correspond to a different breakup mechanism. Hance, we don't use straightway the value $E_{\rm stop}$ = 3 MeV/nucleon, but study the $E_{\rm stop}$ dependence of the IMF production.

      Figure 3 (a) shows the cross sections $\sigma$ as a function of the charge number Z of the fragments. The solid circles show the experimental data taken from Ref. [16]. The cross sections for 3 < Z < 20 show a power law, which has been considered as the character of the multi-fragmentation. The solid and dashed curves show the calculated values with and without the sequential decays performed by the GEMINI code. In the calculations, $E_{\rm stop}$ = 2 MeV/nucleon is used. Since the fragments outputted from the IQMD model are excited, they are called the hot fragments. In contrast, the fragments outputted from the IQMD+GEMINI model are called the cold fragments. Overall, the model calculations reproduce the U-type-shape of the data. Only the cross sections in the valley (15 < Z < 30) are somewhat underrepresented. The main effect of GEMINI, as observed from the difference between the hot and cold fragments, is the small increment of the light ( Z = 1 and 2) and heavy fragments ( $Z >$ 30), but reduction of the IMFs. It is caused by the evaporations of the protons or $\alpha$ particles in the final deexcitation stage.

      Figure 3 (b) shows the cross sections $\sigma$ as a function of the neutron number N of the fragments in ${}^{136}$ Xe + p at 1000 MeV/nucleon. The data also show the U-type-shape, but with a platform in the region of 60 < N < 80. The isotopes of the I ( Z = 53) and Te (Z = 52), which are produced abundantly in peripheral collision, are responsible for the platform. Without the sequential decays performed by the GEMINI code, the model reproduces the cross sections for N < 20, but underestimates grossly those in the region of 20 < N < 70. With the help of the neutron evaporation performed by the GEMINI code, the calculations for the cold fragments reproduce the data rather satisfactorily. The effect of the GEMINI code is obvious. However, the neutron evaporation causes the overestimation of the odd-even staggering.

      Figure 3.  (color online) Cross sections as a function of (a) charge number and (b) neutron number of the hot and cold fragments produced in the 136Xe + p spallation at 1000 MeV/nucleon. The calculations were performed with Estop = 2 MeV/nucleon. The experimental data (solid circles) are taken from Ref. [16].

      Figure 4 (a) shows the mean neutron-to-proton ratios $\langle N \rangle/Z$ as a function of the charge number of the hot and cold fragments. The calculated $\langle N \rangle/Z$ values of the hot fragments show a uniform increase over Z in the IMFs region, and keep the memory of the initial N/Z = 1.52 for Z > 30. After the second decays, $\langle N \rangle/Z$ values decrease obviously, indicating that the neutron evaporation dominates rather than the proton evaporation. The experimental values are not well described by the model. The experimental data show that the Li and Be production is neutron rich. But the calculation underestimates the $\langle N \rangle/Z$ values of Li and Be. In the IMFs region, the odd-even staggering is weak for the data but obvious for the calculations.

      Figure 4 (b) shows the $E_{\rm stop}$ dependence of the $\langle N \rangle/Z$ of the fragments. Three values of $E_{\rm stop}$ , i.e. 1, 2, and 3 MeV/nucleon, are applied. The isospin memory is kept somewhat in the IQMD evolution. After the second decay, the $\langle N \rangle/Z$ values are attracted by the valley of stability, because the tabulated masses are applied in the GEMINI code. When a smaller $E_{\rm stop}$ value is used to match the IQMD with GEMINI, more isospin memory is kept due to the more evolution time in IQMD and less evaporated nucleons in GEMINI. In this case we see the larger $\langle N \rangle/Z$ values overall for $E_{\rm stop}$ = 1 MeV/nucleon. More remarkably, the calculations for $E_{\rm stop}$ = 3 MeV/nucleon reproduce the data rather satisfactorily, except for the region near Z = 45. The calculation without the PSDC method for $E_{\rm stop}$ = 1 MeV/nucleon is also displayed. The effect of the PSDC on the $\langle N \rangle/Z$ value can be found by comparing the calculations with and without the PSDC method, but using the same $E_{\rm stop}$ value (1 MeV/nucleon). It is found that the PSDC method increase the $\langle N \rangle/Z$ values of the IMFs, but decrease those for the heavy fragments.

      In order to show the isospin effect of the PSDC method, we display the number of the free nucleons as a function of time in Fig. 5. The shapes of the curves are similar for both cases with and without the PSDC method. In the excitation stage of the spallation (before 20 fm/c), the system mainly lose the neutrons rather than the protons due to the neutron richness of the system and the Coulomb barrier for protons. In the decay stage of the pre-fragments (after 30 fm/c), both neutrons and protons are emitted. Comparing to the protons, the neutrons are emitted more quickly. This is why we see smaller $\langle N \rangle/Z$ values for $E_{\rm stop}$ = 3 MeV/nucleon than those for $E_{\rm stop}$ = 1 MeV/nucleon. Comparing the calculations with and without the PSDC method, it is found that the PSDC method increases the stability of the pre-fragments and decreases the numbers of the free nucleons.

      Figure 5.  (color online) Number of the free nucleons as a function of time in central 136Xe + p collision at 1000 MeV/nucleon. Both the cases with and without the PSDC method are displayed. The average excitation energy is also displayed.

      Figure 6 shows the $E_{\rm stop}$ dependence of the cross sections of the fragments. All calculations, with three values of $E_{\rm stop}$ , reproduce U-type-shape of the data and the platform in the region of 60 < N < 80. In the figure, the calculation without the PSDC method for $E_{\rm stop}$ = 1 MeV/nucleon is also displayed. The cross sections of the IMFs and the heavy fragments without PSDC method are much smaller than the corresponding values with the PSDC method. However, the cross sections of the IMFs and the heavy fragments depend strongly on the $E_{\rm stop}$ values. In the deexcitation process from 3 to 1 MeV/nucleon of the hot fragments, the IMFs are produced persistently. For $E_{\rm stop}$ = 1 MeV/nucleon, the system is fragmenting too much. Compering to this phenomenon, the calculation without the PSDC method for $E_{\rm stop}$ = 1 MeV/nucleon substantially underestimate the data. It may be related to the fermionic character introduced by PSDC method. On one hand, the PSDC method compensates the fermionic character and hence increase the productions of the IMFs. On the other hand, it may provide much repulsive force between the fragments and cause excessive fragmentation.

      Figure 6.  (color online) Cross sections as a function of (a) charge number and (b) neutron number of the cold fragments produced in 136Xe + p at 1000 MeV/nucleon. The calculations were performed for three values of Estop as indicated. The experimental data (solid circles) are taken from Ref. [16].

      More interestingly, only the calculation for $E_{\rm stop}$ = 2 MeV/nucleon reproduces the data well. It is indicated that the threshold energy of the IMF emission in the spallation is about 2 MeV/nucleon, which is smaller than the value of 3 MeV/nucleon suggested by B. Borderie et al. for heavy-ion collision [27]. This may relate to the different mechanisms of the IMF emission in heavy-ion collision and spallation. In the heavy-ion collision, the incident energy dissipates into both the potential energy and thermal energy. The system undergoes the compression-expansion phase, and then splits into fragments. The spinodal decomposition is a major mechanism behind IMF emission in heavy-ion collision. While in the spallation, the incident energy mainly dissipates into the thermal energy. Being different to the heavy-ion collision, the compression-expansion phase is absent. It is still open question if the excitation system undergoes the multifragmentation. The model developed in this work can be applied to study further the breakup mechanism in the spallation.

      In Fig. 7 (a), we display the $E_{\rm stop}$ dependence of the IMFs productions. The columns show the calculation, and the dashes show the data. It is shown that the calculated cross sections of the IMFs decrease obviously with increasing $E_{\rm stop}$ value. Figure 7 (b) shows the yield of the IMFs as a function of the excitation energy of the fragmenting source, which is defined as the excitation energy of the largest fragments at 20 fm/c. For all $E_{\rm stop}$ values, the yield of the IMFs decreases with decreasing excitation energy, and reaches zero at the used $E_{\rm stop}$ values. It means that the current version of IQMD model can describe the IMF emission dynamically, but we must choose an appropriate threshold energy to stop the dynamical evolution. Only when the $E_{\rm stop}$ = 2 MeV/nucleon is used, the data the IMFs productions is well described by the model. However, the mean neutron-to-proton ratios $\langle N \rangle/Z$ in Fig. 4 indicates that $E_{\rm stop}$ = 3 MeV/nucleon is the best choice.

      Figure 4.  (color online) Calculated mean neutron-to-proton ratios ${\langle N \rangle}/Z$ as a function of the charge number of fragments produced in 136Xe + p at 1000 MeV/nucleon, in comparison with the experimental data taken from Ref. [16]. The panel (a) shows the cases for the hot and cold fragments calculated with Estop = 2 MeV/nucleon. The panel (b) shows the cases for the cold fragments calculated with three values of Estop as indicated.

      Figure 7.  (color online) $E_{\rm stop}$ dependence of the IMFs productions in the 136Xe + p spallation at 1000 MeV/nucleon. In the calculations, Estop = 1, 2, and 3 MeV/nucleon are used. The panel (a) shows the cross sections of the IMFs as a function of Estop. The panel (b) shows the yield of the IMFs as a function of the excitation energy of the fragmenting source.

    4.   Conclusion
    • In this work, the 136Xe + p spallation at 1000 MeV/nucleon was investigated within the isospin-dependent quantum molecular dynamics (IQMD) model matched with the statistical code GEMINI. The IQMD model is used to describe not only the excitation stage but also the emission of the intermediate-mass fragments (IMFs). In the GEMINI code, only the channel of light-particle evaporation is switched on. The dynamical evolution of the central collision shows that the 136Xe nucleus is heated from one side. Due to the high excitation, some nucleons escape from the hot spot, leaving the region with low density. Accompanying with the dissipation of the kinetic energy of the incident proton, the fluctuation of the excitation energy of the largest fragment become stronger. When the average excitation energy increases to its maximum 2.6 MeV/nucleon, the excitation energies distribute from 0 to 6 MeV/nucleon. In order to describe the IMFs emission dynamically and avoid the spurious emissions of nucleons in the dynamical evolution, the dynamical evolution time of each events is chosen as the moment when the excitation energy of the largest fragment is less than a special value $E_{\rm stop}$ . It is found that the data the IMFs productions are well described by the model when the $E_{\rm stop}$ = 2 MeV/nucleon is used. But comparison of the mean neutron-to-proton ratios between the data and calculations indicates the value of $E_{\rm stop}$ = 3 MeV/nucleon. It seems that the fermionic character is compensated by the PSDC method, and hence the productions of the IMFs is increased. But the model sill can not describe the IMF emission consistently, because we need to use different values of $E_{\rm stop}$ to reproduce the cross sections and neutron-to-proton ratios of the fragments.

Reference (27)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return