Comparison of Nuclear Transport Models with 800 A MeV La + La Data

VOLUME 62, NUMBER i3

PHYSICAL REVIEW

LETTERS

27MARCH 1989

Comparison

of

Nuclear

Transport Models

with

SOOA-Mev

La+La

Data

J.

Aichelin, '

J.

Cugnon,

Z.

Fraenkel,

K.

Frankel,

C.

Gale, M. Gyulassy,

D.

Keane,

C.

M. Ko,

J.

Randrup, A.Rosenhauer, H. Stocker, '

G.

Welke,

''

and

J.

Q.Wu

' _Institut

fur Theoretische Physik, Universitat Heidelberg, D 6900-Heidelberg, Federal Republic

of

Germany

Institut de Physique Sart Tilman, Universite deLiege, 8-4000Liege J,Belgium Nuclear Physics Department, 8'eizmann Institute, Rehovot 76100, Israel

Research Medicine and Biophysics Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 Theoretical Physics Branch, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ1JO

Nuclear Science Division, Lawrence Berkeley Laboratory, University

of

California, Berkeley, California 94720 Physics Department, Kent State University, Kent, Ohio 44242

t _{~Cyclotron Institute and Center for Theoretical Physics, Texas Ad'cM University, College Station, Texas 77843}

' _~School

of

Physics and Astronomy, TelAviv University, 69978TelAviv, Israel

Institut fur Theoretische Physik,

J.

W. Goethe Universitat, D 6000Fra-nkfurt, Federal Republic

of

Germany

t"

_~Department

_of

_{Physics, State University}

_of

_{New York, Stony Brook, New York Il 794}

(Received 9January 1989)

Nuclear transport models including density- and momentum-dependent mean-field eA'ects are

com-pared to intranuclear-cascade models and tested on recent data on inclusive p-like cross sections for 800M-MeV La+La. We find a remarkable agreement between most model calculations but a systematic disagreement with the measured yield at

20',

possibly indicating a need for modification of nuclear transport properties athigh densities.

PACSnumbers: 25.70.Np, 24.10.—i

Since the discovery ofcollective-nuclear-flow phenom-ena ' in high-energy nuclear collisions, there has been an intensified eAort to develop microscopic nuclear trans-port models including eff'ects due to nuclear mean fields. Up to that time, intranuclear-cascade models, 3 which include only the effects

of

incoherent nucleon-nucleon scattering, could reproduce most features

of

double-diff'erential inclusive cross sections. While there were earlier hints ofa possible breakdown of cascade models, collective flow could only be confirmed after it became possible to measure triple-diA'erential inclusive cross sec-tions for collisions of heavy nuclei with A &

100.

Such nuclear flow was first predicted in terms of hydrodynami-cal models, but the directed in-plane flow momenta were typically overestimated by a factor of 2. On the other hand, the flow momenta were typically underes-timated by a factor of 2 by cascade modes. ' _{The extra}

"side splash" has been interpreted as evidence for extra nuclear repulsion due tothe stiA'ness

of

nuclear matter at high densities, while the relative smallness of the flow momenta shows the importance ofnonequilibrium trans-port eff'ects in finite nuclei. In terms oftransport theory, these observed flow patterns motivated the addition of a nuclear Vlasov term to the Boltzmann collision term.

Several groups have developed transport models in-cluding such a nuclear Vlasov term. ' The essential new input in this class

of

models is the nucleon optical potential U(p,

p),

which depends not only on density but also on the momentum

of

the nucleon. The goal

of

such approaches is to constrain the possible form

of

U up to several times normal nuclear density by fitting triple-diA'erential data. In this way, it is hoped that

high-energy heavy-ion collisions will eventually lead to reli-able experimental constraints on the nuclear equation of state. In addition, by studying the eAect of varying the eAective nucleon-nucleon cross sections in the Boltzmann term, it is hoped that information on the nuclear trans-port coefficients in dense, highly excited nuclear matter can also beextracted from the data.

While most ofthe new transport models can fit the ob-served in-plane flow momenta by adjusting the nuclear potential U(p,

p),

the form

of

Uthat leads to the best fit of the data diff'ers substantially from one model to the next. Expressed in terms ofthe nuclear incompressibility modulus, the results from the various approaches range between

K=200-400

MeV. These differences are due to diA'erences in the dynamical implementation

of

Pauli blocking and binding effects, the momentum dependence of U, and diA'erences between numerical techniques. At present, considerable controversy still surrounds the va-lidity of particular model assumptions and the correct self-consistent formulation

of

high-energy nuclear trans-port theory remains under active debate. '

_It

_is

there-fore essential that all models be tested on the data other than just the moments

of

the high-multiplicity-selected triple-differential yields.

The purpose of this Letter is to report the results of a new test of competing nuclear transport models. We compare calculated double-diA'erential p-like inclusive cross sections to data on

La+La

at 800M MeV.' Recall that the p-like inclusive cross section is defined as

d'~I

0'july

g

ZfAf

Ef

f

d ky

VOLUME 62, NUMBER 13

PHYSICAL REVIEW

LETTERS

27 MARcH 1989 where the sum extends over all nuclear fragments with

charge and mass number

(Zt,

A~), and

E~d

o /d kt is

the invariant fragment cross section with

_{(Et, kt)}

denot-ing the energy and momentum per nucleon

of

the frag-ment. This reaction was considered because this is the only one involving heavy nuclei with A & 100for which the absolute diA'erential fragment cross sections for

f

=p,

d, t, He, and He have been measured. This rep-resents therefore the most severe absolute test of the models at this time. Since these data are not multiplicity selected, an unrestricted impact-parameter average is in-volved, and possible trigger biases are thereby mini-mized.

Before discussing the results, we first describe briefly each transport model. In the intranuclear-cascade mod-els,

'

nuclear transport is described by straight-line propagation of nucleons to potentia1-scattering points defined by the distance d of closest approach of two nu-cleons.

If

d & (aN&/tr)

',

then a binary scattering is as-sumed to take place. The NN cross section o~N is taken from free-space NN data, and scattering is treated as a stochastic process with final momenta selected randomly according to the measured differential cross sections. DiA'erences between the Fraenkel-Yariv

(FY)

cascade model, the original Cugnon

(CG1)

cascade model, and the latest Cugnon version

(CG2)

' _arise

due to diII'erent prescriptions adopted to simulate Pauli blocking, initial Fermi motion, and nuclear binding eA'ects.

To incorporate nuclear mean-field efIects in addition to Pauli-blocked collision dynamics, several versions of the Vlasov-Uehling-Uhlenbeck (VUU) transport theo-ry' were developed. We consider here two versions, VUU and BUU (Boltzmann, Uehling, and Uhlen-beck). In each event, particles propagate on curved trajectories as determined by the nuclear mean field. In order to reduce fluctuations, the mean field is calculated by averaging over an ensemble

of

synchronously calcu-lated events. Binary collisions between nucleons and

6

resonances are processed as in intranuclear-cascade mod-els, using experimental scattering cross sections and in-cluding Pauli-blocking factors.

In VUU, the isospin of each particle is explicitly in-corporated. The mean field is assumed to be given by a local momentum-independent potential, with a function-al form

U(x)

=ap(x)+bp'(x)

.

The local density of nucleons

p(x)

is determined by an ensemble average, taking a spherical volume

of

radius 2 fm. The parameter y fixes the incompressibility

E

and

the remaining two parameters are constrained by nuclear equilibrium conditions. In this work a "stifI"' nuclear equation of state corresponding to y

=

2 and EC

=

380

MeV was considered. In the special case in which 6U/ Bp

=0

above p=po (equilibrium nuclear density) VUU reduces essentially to

CG2.

In BUU, the momentum dependence of the nuclear

potential is considered explicitly, and each parallel en-semble contains fifty events, as opposed to fifteen in the case

of

VUU. It is important to emphasize that both VUU and BUU are one-body transport theories because the ensemble average washes out many-body correla-tions. While pion production is incorporated, modifica-tions for pion propagation in the nuclear medium are neglected, as in all present nuclear transport models. In this model the nuclear potential is parametrized as

U(p,

p)

=a

+b

Po Po

+2

d '

',

(2)

po"

I+

Np

—

p')/~]'

' where

f(r,

p)

is the one-body phase-space density

of

nu-cleons. The five constants above are fixed by requiring that

E/A

= —

16 MeV,

po=0.

16

fm,

K=215

MeV, U(po,p

=0)

= —

75 MeV, and U(po,p /2m

=300

MeV)

=0.

Their values are then

a

=

—

110.

44 MeV, b

=140.

9 MeV,

c=

—

64.95 MeV, o.

=1.

24, and

A=1.

58pF,

and yield an efTective mass at the Fermi surface of m

=0.

67m. With these parameters, the potential becomes repulsive for cold nuclear matter at normal density for kinetic energy EI, greater than 300 MeV. For much higher kinetic energies, the potential reaches an asymp-totic value of

30.

5 MeV. These features are in accord with optica1-model-potential fits to nucleon-nucleus scattering. Unlike VUU, this BUU calculation assumes isospin degeneracy. The p-like fragments are obtained by summing over all nucleons and scaling by

Z/A.

The relativistic Vlasov-Uehling-Uhlenbeck

(RVU)

model considered here is the one based on Refs. 10and

11. It

follows in the semiclassical and local approxima-tion from the extended quantum hadrodynamics

(QHD)'

with scalar meson self-interaction. The pa-rameters are the same as in

Ref. 11,

corresponding to

%=380

MeV and a nucleon eff'ective mass of

0.

83m at normal nuclear-matter density. The RUU mode1 is solved with the method of test particles'

"

and the re-sults are obtained with fifty test particles for each physi-cal nucleon.

Unlike VUU and BUU which are one-body transport models, the quantum-molecular-dynamics (QMD) mod-el' follows the evolution

of

the 3-body phase-space dis-tribution.

It

goes beyond classical-molecular-dynamics models, which solve the A-body Newtonian equations of motion numerically, by incorporating quantal stochasti-city through random two-body scattering as in intranuclear-cascade models.

It

evolves the particles in a Gaussian-smoothed mean field between two-body col-lisions. The Gaussian smoothing is taken to simulate finite wave-packet eAects with a FWHM taken to be Ar

=1.

8 fm. This smoothing ofthe nuclear field reduces the fluctuations and gradients of the mean field. The present results are not sensitive to the exact value ofh,

r.

In the present calculation, the potential density is taken

VOLUME 62, NUMBER 13

PHYSICAL

REVIEW

LETTERS

27 MARcH 1989 as asum ofa Skyrme-type local two- and three-body

po-tential, an effective Yukawa one-pion exchange poten-tial, and a Coulomb potential. However, the momentum dependence of the optical potential was neglected. The parameters for the potentials were chosen to correspond to the stiff' nuclear equation

of

state with

K=380

MeV.

We now turn to the comparison of the calculated re-sults. In Fig. 1, the inclusive p-like data at laboratory angles

20,

40,

and 60 are compared to the various calculations. In

1(a),

results of cascade models are com-pared. Note that the

FY

cascade model significantly overpredicts the cross sections although the shapes are roughly reproduced. This problem was also observed in earlier comparisons on lighter nuclear reactions such as

Ne+U

at 400M MeV. The dashed curves show that the original Cugnon code,

CGl,

converges to the same re-sults as

FY

at high momentum but differs substantially at low momentum. At low momentum, the difference between

FY

and CG1 is presumably due to the different nuclear-binding prescriptions. The solid curves in Fig.

1(a)

show the effect of an improved Pauli-blocking algo-rithm in

CG2.

The high-momentum yield is reduced by

800 MeV La+La -~p-like at Olab=20,40, 60

Nuclear Cascade Cascade +Mean Field

10 -= 10 V (D I 10 10 '4% ~'i y CG1 a.

„~

_$

~', )'g \ 0.5 ~sq'.

30

'y ~ -e RS 10 10—, F

0

05

1.5

0

0.

5 MOMENTUM (GeV/cj t

15

FIG. 1. Comparison of nuclear transport calculations to data (Ref. 13). (a)Comparison ofCugnon cascade model ver-sions CG1 (Ref. 3) and CG2 (Ref. 14) with the Fraenkel-Yariv cascade model FY (Ref. 2). (b) Comparison of

momentum-independent VUU (Ref. 8) and QMD (Ref. 12)

with lt

=380

MeV, to momentum-dependent BUU (Ref. 9)

with K

=210

MeV, and relativistic RVU (Ref.

11).

(c) Eftects at 20 and 60 of rescaling the free-space NN cross sections in CG1 by factors of 0.5, 1.0, and 3.0. The dotted

curves show results of the FREESCo fireball model FRS (Ref. 17). (d) The contributions to the 20' yield for QMD and CG2

from single-collision (N,

=1)

and multiple-collision (N,

=2-6)

corn ponents.

this effect. The difference between CG1 and CG2 illus-trates the magnitude of uncertainties associated with different Pauli-blocking algorithms.

In Fig.

1(b),

the models incorporating the nuclear mean fields are compared. Recall that the incompressi-bility modulus varies by afactor

of

2 between the various models. We note the remarkable insensitivity

of

the re-sults to variations in the nuclear equation of state and to the details

of

the transport methods. In fact VUU, BUU, QMD, and RVU give results within 20% of CG2

in Fig.

1(a).

This shows that even for very heavy nu-clear collisions, the double-differential cross sections can-not be used to constrain the nuclear equation of state.

On the other hand, Fig.

1(c)

shows that the results are sensitive to variations

of

a factor of 2 in the nucleon-nucleon cross sections. Using the CG1 code with all cross sections scaled by

0.

5, 1.0, and

3.

0, we see that an improved agreement with data at high momentum with a reduced cross section can only be achieved at the expense ofunderpredicting the low-momentum yield at 20

.

The results for the three-times free-space cross sections are obtained with the additional constraint that the scatter-ing is repulsive. From previous studies, ' we know that this case corresponds closely to the predictions

of

ideal hydrodynamics. We see that this simulated hydro-dynamics badly overpredicts the data in this reaction. The same is true for the statistical FREESCO model

FRS,

' _{which considers the microcanonical}

explosion and subsequent evaporation from fully equilibrated par-ticipant and spectator sources.

The important point we emphasize in Fig. 1 is the failure of all models to reproduce the low-cross section yields at 20

.

To provide a better understanding

of

the physics associated with that region of momentum space where the discrepancies between the models and the data are the largest, we show in Fig.

1(d)

a breakdown ofthe QMD and CG2 calculations into components involving nucleons that have suffered a particular range of two-body scattering. The

N,

=l

curve shows the contribu-tion from nucleons suffering only one hard nucleon-nucleon collision. We see that this is a negligible contri-bution to the 20 yield. Even the intermediate com-ponent corresponding to

2-6

collisions only accounts for about half the yield at high momentum. This region of momentum space is then strongly influenced by the reac-tion zone in which the largest number ofbinary interac-tions occurred. The discrepancy is therefore of interest, since the highest nuclear densities are likely to be pro-duced there.

The common feature of all models is the assumption that the NN cross sections can be taken from free-space data. However, many-body eff'ects _{can modify} the in-medium cross sections.

'

The results in Fig.

1(c)

show that no simple rescaling of those cross sections is satis-factory.

It

is possible that momentum-dependent ef-fective cross sections, reducing from free-space values for low-momentum nucleons to about half that value for the 1463

VOLUME 62, NUMBER 13

PHYSICAL REVIEW

LETTERS

27 MARcH 1989 higher-momentum nucleons, could lead to better

agree-ment with the data. However, such corrections for time-dependent in-medium efIIects would require sub-stantial modifications of the present models.

If

the present data are free from additional systematic errors, then a better understanding of nuclear transport at high densities is called for. We note that in a similar study on rapidity distributions, the free-space cross sections gave the best agreement; however, the data in that case were dominated by particles at angles beyond 20

.

We conclude that further tests ofthe nuclear collision term via double-diferential data on heavy nuclear col-lisions are urgently needed. Uncertainties in nuclear transport properties suggested by this study could ob-scure the eAects due to the sought-after equilibrium equation of state. For example, one study ' indicated that the in-plane flow momenta may be just as sensitive to the eff'ective NN cross sections as to the nuclear in-compressibility. Especially important would be a sys-tematic measurement

of

absolute p-like cross sections in

4+A

collisions ranging from

Ne+Ne

to Au+Au in the entire energy range

(0.

2-1.

0)A GeV.

This comparitive study was made possible through a grant from the Lawrence Berkeley Laboratory Director's Program Development Fund. M.

G.

and

J.

R.

acknowl-edge support by the Director, Oftice

of

Energy Research, Division ofNuclear Physics ofthe

0%ce

ofHigh Energy and Nuclear Physics, and

K.

F.

acknowledges support by the

Once of

Health and Environmental Research ofthe U.

S.

Department

of

Energy under Contract No.

DE-AC03-76SF00098.

We are grateful for a

DOE

computer time allocation on the

LLL-NMFECC

Cray system. C.M. K. and

J.

Q.W. acknowledge support in part by the National Science Foundation Grant No.

PHY-8608149

and by the Robert A. Welch Foundation Grant No.

A-1110.

D.

K.

acknowledges facilities provided by the San Diego Supercomputer Center. A.

R.

acknowledges sup-port by the German-Israeli Foundation for Scientific Research and Development.

G.

W. acknowledges support

in part by DOE Grant No.

DE-F602-88ER40388.

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