Momentum Dependence of Nuclear Mean Field and multifragmentation in Heavy-Ion Collisions
aa r X i v : . [ nu c l - t h ] D ec Momentum Dependence of Nuclear Mean Fieldand multifragmentation in Heavy-Ion Collisions
Yogesh K. Vermani, Supriya Goyal and Rajeev K. Puri ∗ Department of Physics, Panjab University,Chandigarh-160014, India.October 22, 2018
Abstract
We report the consequences of implementing momentum dependent in-teractions (MDI) on multifragmentation in heavy-ion reactions over entirecollision geometry. The evolution of a single cold nucleus using static softequation of state and soft momentum dependent equation of state demon-strates that inclusion of momentum dependence increases the emission offree nucleons. However, no heavier fragments are emitted artificially. Thecalculations performed within the framework of quantum molecular dy-namics approach suggest that MDI strongly influence the system sizedependence of fragment production. A comparison with ALADiN ex-perimental data justifies the use of momentum dependent interactions inheavy-ion collisions.
The heavy-ion collisions have always played a fascinating role in exploring vari-ous aspects of nuclear dynamics such as fission-fusion [1, 2, 3], multifragmenta-tion [4, 5, 6, 7, 8, 9, 10, 11, 12], collective flow [6, 9, 13, 14, 15, 16, 17] as wellas particle production [5, 18, 19] etc. The wider energy spectrum available dueto modern accelerator technologies has become a powerful tool in experimental ∗ [email protected] n-n ) potential is found to affect drastically the collectiveflow observables [21, 22] and particle production [4, 18, 19]. It has been shownthat observables related to the particle production e.g. π, κ, λ yields, n d /n p ratios etc. are strongly influenced by the momentum dependence of the n-n interaction [18, 19]. A strong influence of momentum dependent interactionswas also observed on fragment flow for E lab ≥ M eV /nucleon . In higher in-cident energy regime, momentum dependent interactions (MDI) cause strongerreduction in the number of n-n collisions leading to more pronounced transverseflow. The momentum dependent interactions, therefore, increase the mean freepath of nucleons, and consequently affect the stopping and thermalization ofnuclear matter [21, 23].On the contrary, very few attempts exist in the literature, that shed light onthe consequences of implementing momentum dependent interactions in frag-mentation [10, 22]. One of the basic problem with MDI is the strong repulsioncreated in the nuclear environment. As a result, nuclei propagating with MDItend to be destabilized and start decaying via emission of free nucleons and clus-ters quite early during the reaction. Interestingly, the role of momentum depen-dent interactions depends crucially on the impact parameter of the reaction. Itis found to enhance the energy of disappearance of flow in central collisions [15],whereas it reduces the energy of disappearance of flow in peripheral collisions[16, 17]. Similarly, MDI reduce the production of fragments in central collisionswhereas it enhances the same in peripheral collisions [22]. However, one alwaysremained concerned about the stability of nuclei propagating with momentumdependent interactions. Even a use of cooling procedure via Pauli potential isalso reported in the literature [7]. Our present aim, therefore, is to investigatethe stability of cold nuclei propagating under the influence of momentum de-pendent interactions and to see whether one can study fragmentation with MDIor not. An attempt to study the system size effects in the presence of momen-2um dependent forces will also be made. We shall also confront our calculationswith multifragmentation data of ALADiN group [11] which has a rise and fall variation with impact parameter. This study is carried within the framework ofquantum molecular dynamics (QMD) transport model [4, 5]. The QMD modeland implementation of momentum dependent potential are described in section2. Our results are presented in section 3 and summarized in section 4
The quantum molecular dynamics (QMD) model [4, 19] is an n-body transporttheory that simulates the heavy-ion (HI) reactions between 30 MeV/nucleonand 1 GeV/nucleon on event by event basis. It includes quantum features likePauli blocking, stochastic scattering and particle production. Here each nucleonis represented by Wigner distribution function of the form: f i ( r , p , t ) = 1( π ¯ h ) e − ( r − r i ( t )) / L .e − ( p − p i ( t )) L/ ¯ h , (1)where r i ( t ) and p i ( t ) define the classical orbit, the center of i th Gaussian wavepacket in phase space which evolves in time. The centers of these Gaussian wavepackets propagate according to the classical equations of motion [4, 6]. Theinteraction part used in the QMD model consists of local Skyrme interaction, afinite range Yukawa term and an effective Coulomb interaction among protons i.e. V tot = V Sk + V Y uk + V Coul , (2)with local Skyrme interaction consisting of two- and three-body interactions: V Sk = t δ ( r i − r j ) + t δ ( r i − r j ) δ ( r i − r k ) . (3)Since QMD model is a n -body theory, therefore, Eq. (3) can be reduced to adensity dependent potential in the limit of infinite nuclear matter limit as: U Sk = α ( ρρ o ) + β (cid:18) ρρ o (cid:19) . (4)The momentum dependence of nuclear mean field is included in Eq. (2) viamomentum dependent interaction as: V MDI = t ln h t ( p i − p j ) + 1 i δ ( r i − r j ) , (5)3ith parameters t = 1 . M eV and t = 5 . × − ( M eV /c ) − . This param-eterization is deduced from the real part of the proton-nucleus optical potentialwhich reproduces the experimental data upto 1 GeV/nucleon [4, 24]. In an in-finite nuclear matter limit, generalized n-n potential (Eq.(2) and (5)) leads tofollowing density and momentum dependent potential (without Coulomb andYukawa terms): U ( ρ, p ) = α (cid:18) ρρ (cid:19) + β (cid:18) ρρ (cid:19) γ + t ln [ t ( p + 1)] (cid:18) ρρ o (cid:19) . (6)The parameters α , β and γ in Eq.(6) have to be re-adjusted in the presence ofmomentum dependent interactions so as to reproduce the ground state proper-ties of nuclear matter. The parameters corresponding to soft, hard and theirmomentum dependent versions are labelled as S, H, SM and HM, respectively.The constants α and β give the proper rms radii and binding energies of nucleiacross the periodic table. Parameter γ gives us possibility to examine the com-pressibility and hence equation of state. The set of parameters correspondingto soft (S), hard (H) and their momentum dependent versions SM and HM,respectively, can be found in Ref. [5]. To address the question of stability of computational nucleus in the presenceof momentum dependent interactions (MDI), we initialize a single cold projec-tile using soft (S) equation of state and soft momentum dependent equationof state (SM). Earlier theoretical attempts ranging from giant monopole reso-nances [25] to nucleosynthesis of heavy elements in mergers of neutron stars [26]could be explained if the equation of state (EoS) is relatively soft than whenit is stiff . Another study concerning the linear momentum transfer occuring incentral HI collisions also showed that a soft compressibility modulus is neededto explain the experimental data [27, 28]. These observations motivated us forthe choice of comparatively softer EoS. We follow the cluster emission patternand rms radii of few computational nuclei. Figure 1 shows the time evolutionof cold QMD nuclei, namely N i and Au initialized with S and SM inter-actions. The cluster emission is followed for the time span of 200 fm/c. Here,4 Ni (cid:1) A m ax (cid:2) t (fm/c) Au F ree - N u c l . L C P s (cid:0) ** MM F s I M F s Figure 1: The time evolution of heaviest fragment h A max i , free nucleons, LCPs[2 ≤ A ≤ M M F s ∗ [5 ≤ A ≤
9] and
IM F s ∗ [5 ≤ A ≤ A P /
3] ( A P being themass of projectile nucleus) emitted from a single cold nucleus of N i (left panel)and Au (right panel). Results obtained with soft (S) equation of state arerepresented by solid lines whereas dashed lines show results with soft momentumdependent (SM) equation of state. ‘*’ indicates that heaviest fragment has beenexcluded. A max denotes the size of residual nucleus. This should be close to that of par-ent nucleus if there is no destabilization of the nucleus. The sizes of parent5 N i and Au nuclei reduce with the inclusion of momentum dependent in-teractions compared to nuclei propagating with static soft interactions alone.The SM interactions caused an enhanced emission of free nucleons and lightcharged particles LCPs [2 ≤ A ≤ ≤ A ≤
9] and IMFs { [5 ≤ A ≤ A P / A P being the mass of projectile } are almost insensitive towards momentum dependent interactions. Superscript(*) indicates that heaviest fragment has been excluded from the multiplicitiesof MMFs and IMFs. Only a small fraction is emitted as intermediate massfragments (IMFs). It seems that nucleons close in space are emitted in bulk,therefore, leading to an enhanced emission of light clusters. On the contrary,very few nucleons, LCPs and heavier clusters are emitted when propagatingwith soft EoS. The enhanced evaporation with MDI is also due to repulsive na-ture of these interactions. Does this enhanced emission prohibit one to use MDIfor fragmentation ? If one sees carefully, majority of mass that leaves the goldnucleus (for example, with MDI about 19 units are emitted and h A max i is closeto 177) is in the form of free nucleons. In the above gold nucleus, out of 19 unitsabout 15 are in terms of free nucleons. In other words, we see that nucleonsfrom the surface are emitted and there is no contribution towards the emissionof intermediate mass fragments. One sees that even with MDI, only 0.15 IMFsare emitted on the average. Realizing that as many as 10-12 IMFs can be seenemitted in Au+Au reaction [32], this number with MDI is negligible.A survey of the time evolution of rms radii of a single QMD nucleus alsodepicts the same picture. We show in Fig. 2, the time evolution of rms radiiof N i and Au nuclei till 200 fm/c. The rms radius of nucleus with SMinteractions increases gradually compared to that initialized with static softinteractions. This behavior reflects that MDI create additional repulsions amongnucleons which leads to enhanced emission of free nucleons. The rms radii ofgold and nickel nuclei in soft case shows negligible deviation for the characteristictime of HI collision. As discussed above, this enhanced radius is due to theemission of free nucleons and not due to the IMFs. Therefore, one can studythe fragmentation with MDI since the structure of IMFs is not altered by theinclusion of MDI. 6
50 100 150 20002468 Au t (fm/c) Ni SM S r m s r a d i u s (f m ) Figure 2: The time variation of rms radii of single cold nuclei of N i (top panel)and Au (bottom panel) using soft (S) equation of state (solid lines) and softmomentum dependent (SM) interactions (dashed lines). After investigating the behavior of cold nuclei initialized with momentum de-pendent interactions, let us study the effect of momentum dependent forces inheavy-ion reactions. One of the observables linked with the compression andexpansion of nuclear matter is the density of fragmenting system. The totalnuclear matter density is obtained as : ρ ( r , t ) = A T + A P X j = ( π L ) / e − ( r − r j ( t )) / . (7)Here A T and A P stand for the target and projectile masses, respectively. In ourapproach, average nuclear matter density h ρ/ρ o i is calculated in a sphere of 27 .00.51.01.52.0 (cid:3) r / r o (cid:4) t (fm/c) Ni + Ni E=50 AMeV
SM S
E= 400 AMeV
Au + Au Figure 3: The average nucleonic density h ρ/ρ o i calculated in a central sphere of2 fm radius versus reaction time for the central collisions of N i + N i (toppanel) and Au + Au (bottom panel). The results obtained with soft (S)and soft momentum dependent (SM) interactions are compared at 50 AMeV(left) and 400 AMeV (right).fm radius.In Fig. 3, we display the time evolution of average nucleon density h ρ/ρ o i reached in the central region for the head-on collisions of N i + N i and Au + Au at incident energies of 50 and 400 AMeV. The maximal averagedensity tends to reduce with inclusion of momentum dependent interactions.This happens due to additional n-n repulsions created in the system that pro-hibits compression of nuclear matter to a significant level. This difference in thebehavior of h ρ/ρ o i calculated using S and SM interactions diminishes at higher8 Soft EoS (S)
Soft EoS + MDI (SM) F ree N u c l . -2 -1 0 1 2010203040 Ca+Ca Ni+Ni Nb+Nb
Xe+Xe Er+Er Au+Au L C P s d N f r ag / d y ( a r b . un i t s ) E=400 AMeV b= 0 fm Y c.m. /Y proj. Figure 4: The rapidity distribution dN frag /dy of free nucleons and LCPs [2 ≤ A ≤
4] as a function of scaled rapidity Y c.m. /Y proj. ; Y proj. being the rapidity ofprojectile for the head-on collisions at 400 AMeV.incident energies (400 AMeV). This is due to the fact that in central collisionsat 400 AMeV, most of the initial n-n correlations are already destroyed andmatter is already scattered, and therefore, repulsion generated due to MDI doesnot play any significant role. As a result, we do not see much difference inaverage central density reached at higher incident energy. Another importantquantity related with the initial compression of nuclear matter is the rate ofbinary collisions. We have also checked the collision rate for these reactionsand its behavior is found to be consistent with the density profile. The rapiditydistribution of nucleons is another useful tool to characterize the stopping andthermalization of the nuclear matter. We have displayed in Fig. 4, the fragment9apidity distribution dN frag /dy of free nucleons and LCPs for central collisionsof six different symmetric systems at 400 AMeV. The results are displayed hereusing soft EoS (left panel) and soft EoS including MDI (right panel). The ra-pidity distribution is more ‘isotropic’ and nearly full stopping is achieved inheavier systems like Au+Au and Er+Er. In lighter systems, on other hand,a larger fraction of particles is concentrated near target and projectile rapidi-ties resulting into broad Gaussian shape. This feature can be seen in both Sand SM cases. The lighter systems, therefore, exhibit larger transparency effect i.e. less stopping. Such features are also observed in the experimental dataof FOPI-group [29]. Based on the experimental observations and theoreticaltrends, one can say that smaller the system, lesser is the stopping. With MDI,a slight increase in transparency effect is seen due to lesser stopping of parti-cles in longitudinal direction. This happens due to reduction in n-n collisionswhich deflect the fragments in transverse direction. As a results, one obtainsless particles being stopped in longitudinal direction. The system size effects inthe production probability of different kinds of fragments has been studied andpredicted by our group [8]. Here we extend the same study with reference tomomentum dependent interactions. For this analysis, we simulated the centralcollisions of six symmetric systems Ca + Ca , N i + N i , N b + N b , Xe + Xe , Er + Er and Au + Au at incident energies of 50and 400 AMeV. We also parameterized the multiplicities as a function of to-tal mass of the composite system using a power law of the form: cA τtot ; A tot being the total mass of the system. Figure 5 displays the ‘reduced multiplic-ity’ i.e. multiplicity per nucleon of various kinds of fragments. It is clear thatthe system size effects are more visible in soft equation of state compared tosoft momentum dependent case. A negative slope obtained for the multiplic-ity of free nucleons, fragments of mass A=2, and LCPs at 50 AMeV indicatestheir origin from the surface of interacting nuclei. As we move to momentumdependent version, additional break up of n-n correlations leads to enhancedemission of free nucleons and light charged particles. As a result, multiplicity of M M F s ∗ [5 ≤ A ≤
9] and
IM F s ∗ [5 ≤ A ≤ min { A P / , } ] (excluding largestfragment A max ) gets reduced at 400 AMeV, indicating the vanishing of systemsize effect with MDI. In higher energy regime, cluster production via emission of M M F s ∗ and IM F s ∗ is strongly suppressed in the presence of MDI. It is worthmentioning that earlier studies e.g. see Ref. [30], also reported the momentum10 .00.20.40.60.8 * b=0 fmE=50 AMeV I M F s M u l t i p li c i t y p er nu c l e o n A tot * E=400 AMeV MM F s L C P s A = F ree - N u c l .
60 120 180 240 300 360 4200.000.010.020.03
60 120 180 240 300 360 4200.0000.0020.0040.006
Figure 5: The final state scaled multiplicity (calculated at 200 fm/c) of freenucleons, fragments with mass A=2, LCPs [2 ≤ A ≤ M M F s ∗ [5 ≤ A ≤ IM F s ∗ [5 ≤ A ≤ min { A P / , } ] as a function of total mass of the system A tot . Results shown here are at incident energies of 50 AMeV (l.h.s) and 400AMeV (r.h.s). Open circles depict the calculations with soft (S) interactionwhile solid circles are for soft momentum dependent (SM) interactions. ‘*’means that heaviest fragment has been excluded.11 ALADiN SM SE=400 AMeV
Au + Au b (fm) (cid:5) N I M F (cid:6) Figure 6: The mean IMF multiplicity h N IMF i vs the impact parameter b forthe reaction of Au + Au at 400 AMeV. The QMD calculations (at 300fm/c) using soft EoS (open circles) and soft momentum dependent EoS (solidcircles) are compared with ALADiN experimental data (filled stars).dependent potential to be more repulsive for high momentum nucleons. Thisleads to enhanced emission of free nucleons and LCPs. A similar enhancementof the nucleons emission and light cluster production was predicted on inclusionof momentum dependent effective interactions in the isoscalar nuclear potentialand symmetry potential [30, 31]. Contrary to this, with static soft equation ofstate, the production probability of MMFs and IMFs scale with the system sizeas power law: cA τtot with power factor τ close to 3/2. Let us now try to confrontour calculations with experimental data of ALADiN group [11]. The experi-mental data is very fascinating because it has been shown that there is a riseand fall of multiplicity of intermediate mass fragments with impact parameter1211]. However universality is observed with mass of the system and with incidentenergies exceeding 400 AMeV. In Fig. 6, we display the multiplicity of interme-diate mass fragments as a function of impact parameter using soft (S) and softmomentum dependent (SM) equations of state. We see that entire spectrumis very well reproduced by the momentum dependent interactions. One shouldalso keep in the mind that for central impact parameters, different experimentalgroups like FOPI [9], ALADiN [10, 11] and Miniball [32] differ significantly inthe multiplicities of IMFs. Overall, we see a clear need of momentum dependentinteractions in heavy-ion collisions. Summarizing these findings, we here presented a detailed study on the conse-quences of employing momentum dependent potential in multifragment-emission.Investigation of a single cold nucleus initialized with soft (S) and soft momentumdependent (SM) equations of state reveals that momentum dependent interac-tions act as destabilizing factor, which results into enhanced emission of freenucleons only. However, no change is seen towards artificial emission of IMFs.Further, momentum dependent interactions are observed to weaken the systemsize effects studied at 50 and 400 AMeV. A comparison of model calculationswith ALADiN data on Au+Au reactions favor strongly the use of momentumdependent equation of state in heavy-ion collisions.This work was supported by a research grant from Department of Scienceand Technology, Government of India vide grant no. SR/S2/HEP-28/2006.
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