directed transverse flow and its disappearance for asymmetric reactions

ISSN 1547�4771, Physics of Particles and Nuclei Letters, 2014, Vol. 11, No. 3, pp. 232–237. © Pleiades Publishing, Ltd., 2014.

232

1 1. INTRODUCTION

It is now well accepted that collective transverse in�plane flow serves as one of the best candidates to studythe properties of hot and dense nuclear matter. It isalso known that collective flow is sensitive to equationof state and the measurements on flow has proved to beuseful in constraining equation of state of symmetricnuclear matter [1]. In past three decades, extensiveefforts have been carried out in both theoretical andexperimental fronts regarding the behavior of collec�tive flow with various entrance channel parameterssuch as incident energy [2, 3], colliding geometry[4, 5], system size [6, 7], mass [8] and isospin symme�try [9–11]. At low incident energies (i.e. fewMeV/nucleon) dynamics is governed by the attractivemean field and flow is negative. At higher incidentenergies, flow is positive because of the dominance ofrepulsive nucleon nucleon scattering. Therefore, whilegoing from the low to higher incident energies, flowdisappears at a particular incident energy. This isbecause of the counterbalancing of the attractive andrepulsive interactions. This energy at which disappear�ance of flow takes place is termed as the energy of van�ishing flow (EVF) [12]. A large number of theoreticaland experimental efforts have been carried out to studythe energy of vanishing flow. Its dependence on themass of colliding nuclei [13, 14], colliding geometry[15], mass asymmetry of the reaction [8] and on isos�pin degree of freedom [10, 16] has been studied exten�sively. Similar studies have been carried out regardingentrance channel dependence on elliptical flow [17]and multifragmentation [18].

Colliding geometry, on the other hand, also plays asignificant role in directed flow and its disappearance.

1 The article is published in the original.

The directed flow first increases when we go from per�fectly central to semi�central collisions, reaches amaximum at a particular value of the impact parame�ter, and again decreases as one moves to peripheralcollisions. This is because of near absence of binarynucleon nucleon collisions. The value of the impactparameter at which directed flow crosses zero is calledgeometry of vanishing flow (GVF) [19]. The GVF hasbeen found to be dependent on the mass of the collid�ing nuclei. The mass dependence of GVF is found tobe sensitive towards binary nucleon�nucleon crosssection, thus pointing that GVF can be used as a probeto study in�medium nucleon�nucleon cross section[19]. On the other hand, the mass dependence of GVFhas been found to be insensitive towards the nuclearmatter equation of state. All the above mentionedstudies were restricted to symmetric reactions (sameprojectile and target nuclei).

Reaction dynamics also depends on the asymmetrybetween the projectile and target nuclei [8]. The exci�tation energy will be stored in the form of compres�sional energy in case of symmetric reactions whilemost of the excitation energy will be in the form ofthermal excitation energy for reactions involvingasymmetric colliding nuclei. So we aim to study thebehavior of GVF by simulating the asymmetric reac�tions, i.e., for different projectile and target nuclei.This is done by keeping the target fixed and varying theprojectile from lighter to heavier masses. So the focusof the present work is

(i) to see how GVF behaves for the asymmetricreactions.

(ii) to see whether GVF still scales with the totalmass of the colliding pair or with the mass of the pro�jectile nuclei only (as the target nuclei is kept fixed).On the experimental front, such reaction studies can

Directed Transverse Flow and Its Disappearance for Asymmetric Reactions1

Lovejota and Sakshi Gautamb

aDepartment of Physics, Punjab University, Chandigarh, 160014 IndiabDev Samaj College for Women, Sector 45 B Chandigarh, 160047 India

e�mail: [email protected]

Abstract—We study the directed transverse flow for mass asymmetry reactions. This is done by keeping thetarget fixed and varying the projectile mass from 4He to 131Xe. We find that directed transverse flow is sensitiveto the mass of the projectile. We also study the disappearance of flow at a particular impact parameter calledGeometry of Vanishing Flow (GVF) for such mass asymmetry reactions. Our results indicate that GVF is sen�sitive to the beam energy as well as to the mass of the projectile.

DOI: 10.1134/S1547477114030121

PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY

PHYSICS OF PARTICLES AND NUCLEI LETTERS Vol. 11 No. 3 2014

DIRECTED TRANSVERSE FLOW AND ITS DISAPPEARANCE 233

also be confirmed at Super Conducting Cyclotron atVECC, Kolkata, where we have the possibility oflighter beams. This study is carried out within theframework of quantum molecular dynamics (QMD)model which is discussed in the following section.

2. QUANTUM MOLECULAR DYNAMICS (QMD) MODEL

In QMD model, each nucleon propagates underthe influence of mutual two and three�body interac�tions. The propagation is governed by the classicalequations of motion:

(1)

where H stands for the Hamiltonian which is given by:

(2)

Here and are, respectively,the Skyrme, Yukawa, Coulomb and momentumdependent potentials. The momentum dependentinteractions are obtained by parameterizing themomentum dependence of the real part of the opticalpotential. The final form of the potential reads as [20]

(3)

Here t4 = 1.57 MeV and t5 = 5 × 10–4 MeV–2. A param�eterized form of the local plus mdi potential is given by

(4)

The parameters α, β, γ are –124, 70.5 and 2 forhard equation of state and –356, 303 and 1.17 for softequation of state.

The phase space of the nucleons is stored at severaltime steps and this is clustered using minimum span�ning tree method that binds the nucleons if they arecloser than 4 fm in coordinate space.

3. RESULTS AND DISCUSSION

We simulated the reactions of 4He + 197Au, 12C +197Au, 58Ni + 197Au, 92Zr + 197Au, and 131Xe + 197Au atincident energies of 100 and 400 MeV/nucleon

throughout the range of colliding geometry (b/bmax( ) =0, 0.2, 0.4, 0.6 and 0.8). For the present study, we useda soft equation of state along with energy�dependentCugnon nucleon�nucleon cross section [21].

The transverse flow has been studied using couplesof variables such as by investigating the slope of flow atmid�rapidity and also by investigating the product ofrapidity and transverse momentum. It has been shownin [14] that both the methods yield are equivalent and

r· i∂H∂pi

��; p· i –∂H∂ri

��,= =

Hpi

2

2mi

��i

A

∑ ViSkyrme Vi

Yuk ViCoul Vi

mdi+ + +( ).

i

A

∑+=

ViSkyrme

, ViYuk

, ViCoul Vi

mdi

Umdi t4 t5 p1 p2–( )2 1+[ ]δ r1 r2–( ).2

ln≈

U α ρρ0

��⎝ ⎠⎛ ⎞ β ρ

ρ0

��⎝ ⎠⎛ ⎞ δ � ρ/ρ0( )2/3 1+[ ]

2ρ/ρ0.ln+ +=

b̂

yield same energy of vanishing flow. The transverse in�plane flow is calculated using directed transverse

momentum “which is defined as [14, 22]

(5)

where y(i) and px(i) are, respectively, the rapidity andthe momentum of the ith particle. The rapidity isdefined as

(6)

where E(i) and pz(i) are, respectively, the energy andlongitudinal momentum of the ith particle. In this def�inition, all the rapidity bins are taken into account.

In Fig. 1, we display the time evolution of for

the collision of 4He + 197Au, 12C + 197Au, 58Ni + 197Au,92Zr + 197Au, and 131Xe + 197Au at different impactparameters of b/bmax = 0.0 (solid line), 0.2 (dashedline), 0.4 (dotted line), 0.6 (dash dotted), 0.8 (dashdouble dotted) at beam energy of 100 MeV/nucleon.

From the figure, we see that remains zero atcentral collisions for all the reactions. The directed

flow is negative initially which signifies the roleof attractive mean field interactions during the initialphase of the reaction. These interactions remain eitherattractive or may turn repulsive, depending on theincident energy. At energy of 100 MeV/nucleon, theyremain negative throughout the time evolution for allthe reacting pairs at all the colliding geometries. Thisis because of the less beam energy, which in turn, yieldless binary nucleon�nucleon (nn) collisions.

In Fig. 2, we display the time evolution of fordifferent colliding pairs as in Fig. 1 but at an incidentenergy of 400 MeV/nucleon. From the figure, we see

that remains negative for the reactions of lighter

projectiles on 197Au target at almost all the collidinggeometries. This implies that the interactions remainattractive throughout the reaction. On the other hand,for heavier projectiles like 58Ni, 92Zr and 131Xe, the nncollisions will dominate the dynamics at final phase of

the reaction at 400 MeV/nucleon and turnspositive.

In Fig. 3, we display the impact parameter depen�

dence of for various reactions at100 MeV/nucleon (upper panel) and400 MeV/nucleon (bottom panel). Squares, circles,triangles, pentagons and diamonds represent the colli�sions of 4He, 12C, 58Ni, 92Zr, and 131Xe respectively on

pxdir⟨ ⟩

pxdir⟨ ⟩ 1

A�� sign y i( ){ }px i( ),

i 1=

A

∑=

y i( ) 12��

E i( ) pz i( )+E i( ) px i( )–��,ln=

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

234


LOVEJOT, SAKSHI GAUTAM

197Au target. From the figure, we see that at

100 MeV/nucleon (upper panel) remains nega�

tive for the reactions of lighter projectiles like 4He, 12Cand 58Ni on 197Au throughout the range of collidinggeometry and decreases when we go from central toperipheral collisions. This is because particles mainlyemit from the body of much more heavier (than pro�jectile) target. As we move to the reaction of heavierprojectiles like 92Zr and 131Xe with 197Au target, we see

that increases when we move from perfectlycentral collisions to semi�central collisions, reaches amaximum and finally decreases and become negativefor peripheral collisions. This is because of the realabsence of nucleon�nucleon collisions at peripheralgeometry as explained earlier also. Also at 400

MeV/nucleon (bottom panel), we see that the is

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

more than that at 100 MeV/nucleon at all the collidinggeometries for all the colliding pairs. This can be inter�preted as the effect of increasing pressure gradient withincrease in incident energy. From the figure, we also

see that for all the reactions, first increase when

we go from perfectly central collision to semi�centralcollision, reaches a maximum value at a particularvalue of impact parameter and then again decreaseexcept for the reaction of very light projectile of 4He on197Au. For 4He + 197Au collisions, is almost zero

through the range of colliding geometry because theprojectile nucleus is too light and so it cannot affectthe heavy target 197Au and flow remains zero. The

value of impact parameter where crosses zero is

called geometry of vanishing flow (GVF).

pxdir⟨ ⟩

pxdir⟨ ⟩

pxdir⟨ ⟩

5

0

–5

–10

–15

–202001000 15050

Time, fm/c

E = 100 MeV/nucleon

p xdir

⟨⟩

MeV

/c,

58Ni + 197Au

5

0

–5

–10

–15

–202001000 15050

131Xe + 197Au

5

0

–5

–10

12C + 197Au

5

0

–5

–10

–15

–20

92Zr + 197Au

5

0

–5

4He + 197Au b/bmax = 0.0

b/bmax = 0.2

b/bmax = 0.4

b/bmax = 0.6

b/bmax = 0.8

Fig. 1. The time evolution of the directed transverse flow for the reactions of 4He + 197Au, 12C + 197Au, 58Ni + 197Au,92Zr + 197Au, and 131Xe + 197Au at different impact parameters of b/bmax = 0.0, 0.2, 0.4, 0.6, 0.8 at beam energy of100 MeV/nucleon. Explanations are given in text.

pxdir

⟨ ⟩



In Fig. 4, we display the GVF as a function of thetotal mass of the colliding nuclei (AP + AT) as well aswith the mass of the projectile nucleus, where AP andAT are the mass of the projectile and target nucleirespectively. From the figure, we see that the GVFincreases with total mass of the colliding system (A =AP + AT) (upper left panel) as well as with the mass ofthe projectile nucleus (AP) (upper right panel). This isbecause of the increasing pressure gradient for heavierprojectiles. So it takes longer (more towards peripheralgeometry) for the flow to vanish. Mass dependence ofGVF follows a power law behavior (α Aτ) where τ =0.9 ± 0.3 for A = AP + AT and τ = 0.2 ± 0.04 for A = AP.In the middle panel, we show the maximum value of

20

15

10

5

0

–52001000 15050

Time, fm/c

E = 400 MeV/nucleon

p xdir

⟨⟩

MeV

/c,

58Ni + 197Au30

20

10

0

–102001000 15050

131Xe + 197Au

12C + 197Au92Zr + 197Au

4He + 197Au b/bmax = 0.0

b/bmax = 0.2

b/bmax = 0.4

b/bmax = 0.6

b/bmax = 0.8

5

0

–5

–10

30

20

10

0

–10

5

3

–3

–5

1

–1

Fig. 2. Same as Fig. 2, but at beam energy of 400 MeV/nucleon.

–100 0.1

⟨b/bmax⟩

p xdir

⟨⟩

MeV

/c,

E = 400 MeV/nucleon

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

10

20

304He + 197Au12C + 197Au

92Zr + 197Au

58Ni + 197Au

131Xe + 197Au

–20E = 100 MeV/nucleon

0

–5

5

10

–10

–15

Fig. 3. Impact parameter dependence of transverse in�

plane�flow at energies of 100 MeV/nucleon (upper

panel) and 400 MeV/nucleon (lower panel) for various sys�tems. Explanations are given in text.

pxdir

⟨ ⟩

236


LOVEJOT, SAKSHI GAUTAM

(labeled as ) as a function of total massof the colliding nuclei (AP + AT) (middle left panel)and mass of the projectile (AP) (middle right panel).

also increases with total mass as well as the

projectile mass. The increase in is alsobecause of increasing pressure gradient for heavierprojectiles. The power law factor τ is 3.5 ± 1.2 and0.9 ± 0.2 for A = AP + AT and AP, respectively. In thebottom panels, we display the system size dependence

of the impact parameter at which becomes

maximum (labeled as b/bmax @ ) as a functionof the total mass (A = AP + AT) and the projectile mass(A = AP). From the figure, we see that b/bmax decreasesas the total system mass and the projectile massincreases and follows a power law behavior. When wego from central to peripheral collisions for heaviermasses, the flow achieves maximum value earlier(towards central geometry). The power law factor τ =–0.5 ± 0.1 for A = AP + AT and –0.1 ± 0.01 for A = AP).From the above discussions, it is clear that GVF, max�imum value of flow and the geometry at which thetransverse momentum becomes maximum haveshown a strong dependence on total system mass andrelatively weak dependence on projectile nuclei.

pxdir⟨ ⟩ px

dir⟨ ⟩max

pxdir⟨ ⟩max

pxdir⟨ ⟩max

pxdir⟨ ⟩

pxdir⟨ ⟩max

4. SUMMARY

Using the quantum molecular dynamics model, weexplored the dynamics of collective transverse flow inasymmetric reactions. We simulated the reactions byvarying the projectile on fixed target Au. Our studyreveals clear dependence of collective flow dynamics onthe mass of the projectile and beam energy. Also, thegeometry of vanishing flow and maximum value oftransverse momentum is found to scale with mass of theprojectile as well as total mass of the reacting system.

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p xdir

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pxdir

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⟨ ⟩



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directed transverse flow and its disappearance for asymmetric reactions

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