Impact parameter and energy dependence of observables in ...PHYSICAL REVIEW C VOLUME 40, NUMBER 4...

PHYSICAL REVIEW C VOLUME 40, NUMBER 4 OCTOBER 1989

Impact parameter and energy dependence of observablesin intermediate energy heavy-ion reactions

M. Betty Tsang, George F. Bertsch, William G. Lynch, and Mitsuru Tohyama

(Received 24 March 1989)

The dependence of various measurable properties of heavy-ion collisions is calculated with theBoltzmann equation, in an attempt to find the best observables to determine the impact parameterin a collision. The observables considered -are nucleon multiplicity, longitudinal momentumtransfer, angular momentum transfer, projectile mass loss, and projectile energy loss. Systems stud-ied in detail include the ' N+" Sm reaction at E/A=35 MeV, the Ar+' Au reaction atE/A =60 MeV, and the ' 0+ ' Au reaction at E/A =90 MeV where some experiments have beenperformed with reaction filters. The energy and impact parameter dependence of nucleon multipli-city distributions have also been studied for the mass symmetric Ar+ Ar and asymmetric' 0+ ' Au systems at energies ranging from E/A = 35 to 400 MeV. These calculations suggest thatthe mean multiplicity of fast nucleons and the linear momentum transferred to the target residue arerelatively insensitive to the impact parameter at small impact parameters and incident energiesbelow E/A 60 MeV. These nucleon multiplicity distributions become more sensitive to impactparameters with increasing incident energy and may provide rather accurate impact parameter in-formation for incident energies as large as E/A =90 MeV.

I. INTRODUCTION

To interpret experimental data, it is often important toknow the impact parameter of the collision, and the exci-tation energy and angular momentum transferred to aheavy reaction residue, should one survive the collision.Information about these quantities is obtained most fre-quently from experimental observables, such as the multi-plicities of charged particles, ' y rays, ' or neutrons, '

or the velocity of heavy reaction residues. ' Totalcharged particle multiplicities have been used extensivelyat high energies E/A ~ 200 MeV to indicate the centrali-ty of the collision. ' For low incident energies,E/A ~20 MeV, the multiplicities of neutrons ' and yrays ' from the reaction residue can supply similar infor-mation. The optimum choice of observables for inter-mediate energy reactions, 20 MeV ~ E/ A ~ 200 MeV, isnot clear. Some guide to the properties of such reactionfilters, however, may be obtained from calculations usingpresently available reaction models.

I

To explore the properties of reaction filters in this ener-

gy domain, we have performed calculations with theBoltzmann equation ' for several systems. We havecalculated experimental observables such as the linearand angular momentum transferred to the target residue,the multiplicities and momentum distributions of emittednucleons, and properties of the projectile-like residuespredicted in collisions at large impact parameters. Somenumerical details of the calculations are described in Sec.II. The results of calculations for individual systems aredescribed in Sec. III. A summary of the results is provid-ed in Sec. IV.

II. DETAILS OF THE CALCULATIONS

The calculations were performed by numerically solv-ing the Boltzmann equation, given below, which de-scribes the time evolution of the Wigner functionf (r, k, t) in phase space:

+v V„fi —V„U V~f, =3 fd kzd k3dQo„„(Q)v, z(2~)'

X[fzf~(1 f, )(1 fz) —f,fz(1 —f—3)(1 f4)]5 (k—, +kz ——k3 —k4) .

U(p)= —356p/po+303(p/po) (MeV) . (2)

The nuclear mean-field defined by Eq. (2) reproducesnuclear matter saturation properties and gives a compres-sibility coeScient of K =200 MeV. For most of the cal-

Here, o „„(Q) and viz are the differential cross sectionand relative velocity for the colliding nucleons, and U isthe mean-field potential approximated by

U, =32 ~, (MeV)Pn Pp

po(3)

culations presented here, 4mcr„„(Q) is taken to be 41 mband is isotropic. ' To explore the inhuence of theCoulomb repulsion between protons, we also used amodified Boltzmann code which distinguishes protonsfrom neutrons. The mean field in this modified code in-cludes the Coulomb potential and a symmetry potentialof the form

40 1685 1989 The American Physical Society

1686 TSANG, BERTSCH, LYNCH, AND TOHYAMA

in addition to the mean field defined by Eq. (2). Here, p„and p are the neutron and proton densities and w, is theisospin operator with eigenvalues +1 and —1 for neu-trons and protons, respectively. Numerical solutions toEq. (1) were obtained by propagating test particles ac-cording to Newtonian mechanics. The mean-field andthe Pauli blocking factors in the collision integral werecalculated with distribution functions which were ensem-ble averaged over 100 parallel simulations. "

Calculations were initiated with the surfaces of theprojectile and target separated by 2 fm. The test particlesare distributed spatially within the projectile and targetaccording to spherical distributions with a radius param-eter ro = 1.12 fm. The momenta are obtained within a lo-cal Thomas-Fermi approximation using a mean-field po-tential defined on a grid with a cell size of 1 fm ." Thedensity distributions provided by this procedure for a

Sm nucleus are compared in Fig. 1 to the predictionsof a spherical Hartree-Fock calculation. ' The solid linedenotes the density provided by the Hartree-Fock calcu-lation; the dashed line in the figure designates the densityprovided by the Thomas-Fermi approximation, if onedoes not distinguish between protons and neutrons anddoes not include the Coulomb repulsion between protons;the open points are obtained when the Coulomb meanfield and the symmetry potential are included in the cal-culation. The densities predicted by the Thomas-Fermiprescription and the spherical Hartree-Pock calculationare similar at high density p ~ 0.4po. In the nuclear sur-face, however, the local Thomas-Fermi method providesdensities which are considerably less than the results of

I I I I I I I I I

5r (f~)

FIG. 1. Density distribution for a " Srn nucleus. The openpoints designate the density provided by the Boltzmann calcula-tion, which includes the Coulomb mean-field potential. Thedashed line corresponds to the density from the Boltzmann cal-culation without the Coulomb mean field. The density providedby a spherical Hartree-Fock calculation {Ref. 14) is indicated bythe solid line.

the Hartree-Fock calculation, partly due to the absenceof barrier penetration effects in the Thomas-Fermimethod. Both the sharp surfaces and the lack of defor-mation effects cause interactions at large impact parame-ters, (e.g. , b ~ 9 fm for ' N+' Sm at E/A =35 MeV) tobe somewhat underestimated by the present calculations.These surface effects should be much less important forsmall impact-parameter collisions.

For each simulation, we examined the phase-space dis-tribution of nucleons at various time intervals after thestart of the collision. In the analysis of the simulations,individual nucleons are assumed to be contained in afragment if they have a potential energy less than —6MeV; otherwise they are assumed to propagate freely.(This prescription provides observables which agree, towithin a few percent, with those obtained by simpler pro-cedure' such as drawing a sphere of radius R =1.4A '

about the target residue. ) The observables in most calcu-lations were evaluated at a time interval long enough forthe fast unbound nucleons and the bound residues to beclearly separated. In collisions where more than twobound fragments were produced, however, the observ-ables were occasionally evaluated before the target-likeresidue was fully isolated so that all the slow bound frag-ments would be kept within the grid upon which themean-field potential was evaluated. In such cases, the ve-locity, mass, and angular momentum of the target-likeresidue presented here are less accurate.

In an effort to identify accurate parameter filters, wehave calculated the impact-parameter dependence of themultiplicities of nucleons with energies greater than 5MeV, and the mass, the velocity parallel to the beam axis,and the total angular momentum of heavy target-likeresidues. In our calculations, such residues survive col-lisions between asymmetric reaction partners for energiesbelow E/A =100 MeV. We have also calculated theimpact-parameter dependence of the mean transversemomentum and the enhancement of nucleon emissionpatterns in the reaction plane for the ' N+ ' Sm systemat E/A =35 MeV, the Ar+' Au system at E/A =60MeV, and the ' 0+' Au system at E/A =90 MeV.While neither of these latter two observables would besuitable for impact-parameter selection, each may pro-vide useful information about the importance ofnucleon-nucleon collisions at intermediate energies.

Within the test-particle approach, the observables areevaluated by treating the test particles as classica1 parti-cles; quantities like the multiplicity, the residue mass, orthe angular momentum of the residue, are normalized bydividing by the number of parallel ensembles. Statisticaluncertainties for observables associated with unboundtest particles are obtained by assuming statistical fluctua-tions in the Wigner function to be proportiona1 to thesquare root of the number of test particles. To describeour results, we choose a Cartesian coordinate system inwhich the total momentum lies along the positive z axisand the initial orbital angular momentum is parallel tothe y axis. In this convention, nucleons, which are scat-tered to forward angles and experience predominantly at-tractive (repulsive) momentum transfers, emerge withnegative (positive) values of the transverse momentum

IMPACT PARAMETER AND ENERGY DEPENDENCE OF. . . 1687

P . To indicate the extent that nucleon emission isenhanced in the reaction plane, we compute the ratio ofthe yield of nucleons emitted out of the reaction plane tothe yield of nucleons emitted in the reaction plane. Forthis comparison, both in-plane and out-of-plane yields areintegrated over polar angles from 40 to 70'; the out-of-plane yield is integrated over azimuthal angles60'&4&120 and 240 &&&300, and the in-plane yieldis integrated over azimuthal angles, —30'&@&30 and150' & 4 & 210'.

0.020

~ 0.010

0.000

4020

I I I I

/

I

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ t ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

y. —~ —y —~ —~~ ~ ~ ~ ~ ~ ~ ~ ~

0

III. RESULTS OF THE SIMULATIONS

We have simulated collisions between the followingsystems:

200180160140

N+ SrnE/A= 35

0 0 0 0 ~ ~I I I I I I

MeV =

' N+' Sm at E/A =35 MeV,

Ar+' Au at E/A =60 MeV,' 0+' Au at E/A =60, 90, 200, and 400 MeV,

Ar+ Ar at E/A =35, 60, 100, 200, and 400 MeV .

The choice of systems was partly motivated by the availa-bility of experimental data. For example, experimentshave been performed for ' N-induced reactions on heavytargets at incident energies near E/A =35 MeV, wherethe velocity of the heavy reaction residue parallel to thebeam axis has been used as an impact-parameterfilter. ' ' The transverse momentum distributions ofunbound nucleons and projectile-like fragments, emittedin '"N+' Sm collisions at E/A =35 MeV, have beencalculated previously with the Boltzmann equation tocompare them with measurements of the circular polar-ization of coincident y rays from the target residue. ' '

In this work, we examine the impact-parameter depen-dence of observables to search for an appropriateimpact-parameter filter for this reaction. We also explorein detail the impact-parameter dependence of observablesfor Ar-induced reactions on ' Au at E/A =60 MeV(Ref. 21) and ' 0 induced reactions on ' Au atE/A =94 MeV (Ref. 22) because measurements havebeen performed for these systems in which the multiplici-ties of coincident light charged particles were used as areaction filter. To understand how impact parameterfilters based on multiplicity measurements evolve with in-cident energy, we also performed calculations for themass symmetric Ar+ Ar system and the mass asyrn-metric ' 0+' Au system for a range of bombarding en-ergies.

A. '4N+ "4Sm at E/A =35 MeV

We begin our discussions with the ' N+ ' Sm reactionat E/A =35 MeV, where our most extensive set of calcu-lations were performed. Observables for this system werecalculated at an elapsed time of 200 fm/c. At this point,a well-defined target-like residue with an approximatemass of 150 can be identified at each impact parameter.The velocity in the beam direction V„, the angularmomentum I, and the mass A„, of the target residue areshown in Fig. 2 as a function of the impact parameter.

10

FIG. 2. Predictions for the parallel component of the veloci-ty (upper part), the angular momentum {middle part), and themass (lower part) of the target-like residue for the ' N+' Smreaction at E/2 =35 MeV. The dotted line in the upper part ofthe figure corresponds to the residue velocity consistent withfull linear momentum transfer to the target-like residue. Thedashed line is drawn to guide the eye.

(Note that the residue mass and angular momentum willcontinue to decrease with increasing elapsed time of thecalculation rejecting the mass and angular momentumloss due to compound evaporation processes. ) The dottedline in the uppermost part of the figure denotes the targetresidue velocity expected for full linear momentumtransfer, i.e., complete fusion. The target residue velocityV„ is surprisingly insensitive to impact parameters forb &7 fm. At large impact parameters, b ~7 fm, whereprojectile-like residues can be identified in the calcula-tion, V„drops ofF sharply with increasing impact parame-ters. At smaller impact parameters, V, assumes valuesthat correspond to about 85 —90% linear momentumtransfer; these values are larger than the most probablelinear momentum transfer ( =80%) established from sys-tematic measurements on heavy targets, andsignificantly larger than the values ( =60%) reported forthis system. ' Values more comparable to the experimen-tal results are obtained by averaging the target velocityover impact parameter (64%). For b ~ 7 fm, the calcula-tions predict that the impact-parameter dependence isrejected most clearly in the angular momentum of the re-action residue; this increases monotonically with b untilb =6.5 fm then falls off rapidly at larger impact parame-ters. The maximum transferred angular momentum isabout 60fi, much less than the maximum value (l & 90k)predicted by the liquid-drop model for nuclei in this massdomain.

Projectile-like fragments are observed at impact pa-rameters larger than 7 fm. In Fig. 3, the mass, defiectionangle, energy per nucleon of the projectile-like residue,and the mean transverse momentum P I/AI, carried bya nucleon in the projectile-like residue are plotted as solid


i I I

I

I I I i

I

I I I iI

I I I I

I

I i

0

N+ SrnO

I I I I I I

I I I I I Ii i I I I iI i I I I i

0.00

—0.05 &

+E/A=35O

. . . I. . . , I,

8 10

MeV- —0.1012 8 10 12

b (Sm)

I I I I

ooo 0

i I I I

I

I I I I

100

Q

" 100o

~-10—3

&10—6

o o

0 0- i50.

6o.

90'

+ Sm E/A = 35 MeV

I i i

0 50 100E, (Mev)

~ ~ ~

~ 3oe

0 II0 0

150

FIG. 3. Predictions for the mass (upper left), angle (upperright), energy per nucleon (lower left), and transverse momen-tum per nucleon (lower right) of the projectile-like residue forthe ' N+' Sm reaction at E/A =35 MeV. Calculations whichinclude the Coulomb mean field are designated by the openpoints. Calculations which exclude the Coulomb mean field aredesignated by the solid points. The solid lines are drawn toguide the eye.

points for values of the impact parameters greater than 7frn. The open points are results calculated with themodified Boltzmann code, which includes the Coulombfield for proton test particles. (All the other results inthis paper were calculated with the code of Ref. 11,which does not include the Coulomb mean field. ) Themass and velocity of the target-like residue parallel to thebeam axis (not shown) are only slightly modified by theinclusion of the Coulomb mean field.

For calculations with and without the Coulomb meanfield, all four properties of the projectile shown in Fig. 3increase with impact parameters. Inclusion of theCoulomb mean field leads to a systematic decrease in theprojectile-like residue mass, an increase in the deflectionangle that is especially pronounced at impact parametersaround 8 fm, and positive scattering angles at larger im-pact parameters, b ~ 10 fm. Since Auctuations are impor-tant to intermediate mass fragment emission' ' but arestrongly suppressed by the ensemble averaging used inthe computation of the mean field, the connection be-tween the properties of the calculated fragments and theproperties of experimentally observed fragments is some-what tenuous. For this reason, it is questionable whethera comparison of the calculated and measured propertiesof projectile-like fragments can provide accurate impact-parameter information at any but the most peripheral im-pact parameters.

Nucleon differential cross sections, shown in Fig. 4,were obtained by summing the nucleon multiplicity dis-tributions over impact parameter. The differential rnulti-

FIG. 4. Nucleon di8'erential cross sections are shown as solidand open points for the ' N+ ""Sm reaction at E/A =35 MeV.The normalizations of the cross sections at 10' are correct; thecross sections at successively larger angles are suppressed bysuccessively larger powers of 10.

tiplicities for nucleons with laboratory energies greaterthan 5 MeV were also computed at each impact parame-ter and then integrated over laboratory angles for threeangular ranges 0 —30, 30 —90, and 90'—180. The re-sulting nucleon multiplicities are shown in Fig. 5. The

I I I I

I

I I I I

N + Srn E/A = 35 MeV

30 —90

90' — 180'

0

b (fm)10

FIG. 5. Nucleon multiplicities for the ' N+' Sm reaction atE/A =35 MeV are given by the solid and open points in thefigure. These multiplicity distributions were integrated over theangular ranges indicated in the figure. Smaller uncertaintiesoccur for points at impact parameters for which more than onecalculation was performed. The lines are drawn to guide theeye.


~ ~ ~ ~ ~ ~

I

N +E/A =

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( ~ ~ ~ ~ ~ ~ ~ ~ ~

154S

35 MeV

impact-parameter dependence of the total multiplicity issimilar to that previously shown for the residue velocity.At large impact parameters, the nucleon multiplicitiesdrop steeply with increasing impact parameter reAectingthe decreased overlap of projectile and target. At smallerimpact parameter, the nucleon multiplicities are remark-ably insensitive to the impact parameter. When one alsoconsiders that statistical fluctuations will smear out thecorrelation between multiplicity and impact parameter, itis clear that the residue velocity provides a better indica-tion of the impact parameter for this reaction. Experi-mental multiplicity distributions will include particlesevaporated from the highly excited target residue. Suchevaporation processes are not taken into account by ourpresent calculations. Evaporation calculations are need-ed to determine whether the inclusion of the yield of suchevaporated particles significantly improves the impact-parameter sensitivity of the multiplicity distributions.An even better measure of the impact parameter mightbe obtained if one should combine multiplicity or residuevelocity measurement with a probe, like the y ray multi-plicity, which is sensitive to the angular momentumtransfer to the target-like residue.

The shape of the nucleon emission pattern displayssome interesting features. In Fig. 6, the ratio of the out-of-plane nucleon cross section to the in-plane nucleoncross section is plotted as a function of the impact param-eter. The solid and open points represent nucleon-energy

thresholds of Eo ) 5 MeV and Eo )20 MeV, respectively.The calculation indicates that nucleons are emitted pref-erentially in the reaction plane. Similar results have beenreported in Ref. 25. The preference for in-plane emissionincreases with the energy of the outgoing particle. Suchan energy-dependent in-plane enhancement has been ob-served experimentally for the reaction of ' N+' Au atE / A =30 MeV. ' Experimentally, this in-planeenhancement is also observed to increase strongly withthe mass of the detected particle. ' Unfortunately, anymass dependence of the emitted particles cannot be stud-ied in the present model without invoking additional as-sumptions about the fragment production mechanism.Since these uncertainties in the fragment productionmechanism have a significant inhuence on the in-planeenhancement of particle emission patterns, we have notattempted to determine the sensitivity of the different ob-servables to the azimuthal orientation of the reactionplane.

Information about the reaction dynamics is also con-tained in the mean transverse momentum (P ) of un-bound nucleons. ' ' Solutions of the Boltzmann equa-tion indicate that the mean transverse momentum in in-termediate energy heavy-ion collisions is very sensitive tothe interplay between the attractive nuclear mean-fieldand nucleon-nucleon collisions. ' ' The mean transversemomentum of unbound nucleons, shown in Fig. 7, is neg-ative at all impact parameters for this reaction. The neg-ative values for (P ) are consistent with experimentalmeasurements of the circular polarization of y rays emit-ted by the target residue; they suggest the dominance ofthe attractive nuclear mean field. The mean transversemomentum decreases with increasing impact parametersuntil a minimum is reached at about 6 fm, then it in-creases and vanishes at much larger impact parameters.Such a strong impact-parameter dependence clearly illus-trates the need to develop a good experimental measureof the impact parameters. At large impact parameters,

p 0.50

DI O. Z5

0.00

/

Q/

b (fm)10

O

—10—

I I

N + Srn

E/A = 35 MeV

Ik

Ik

I I I

FIG. 6. Ratios of the out-of-plane to in-plane nucleon crosssections are plotted here as functions of the impact parameterfor ' N+' Sm reactions at E/A =35 MeV. These ratios wereobtained by integrating the nucleon cross sections over angularranges specified in Sec. II, and over nucleon energies above en-ergy thresholds of 5 MeV (solid points) and 20 MeV (openpoints). Smaller uncertainties are assigned to points at impactparameters for which more than one calculation was performed.Statistical uncertainties for the points calculated with the 20MeV threshold are 30—40% larger than the corresponding un-certainties for the 5-MeV thresholds. The solid and dashedlines are drawn to guide the eye. The dotted line corresponds tothe expectation for isotropic emission.

—20—

Ik-

1IIN

II

—300 5

b (f~)10

FIG. 7. Values for the mean transverse momentum of un-bound nucleons calculated with the Boltzmann equation for' N+" Sm reactions at E/A =35 MeV. The line is drawn toguide the eye.


the calculations predict that negative transverse momen-ta are carried by the projectile-like residues. Due to thesubsequent evaporation by projectile-like residues, thereis an ambiguity in the magnitude of (I' ) one should as-sign to nucleons or light particles at large impact parame-ters. To avoid this ambiguity, the mean transverse mo-menta of Ref. 15 include both the momenta of unboundnucleons and the momenta of nucleons contained in fastbound residues.

While the predictions of the Boltzmann equation aresensitive to the nucleon-nucleon cross section containedin the collision integral, there is presently little experi-mental information to constrain this nucleon-nucleoncross section. To explore the sensitivity of these predic-tions to the nucleon-nucleon cross section, additional cal-culations were performed with a reduced nucleon-nucleon cross section of 4mo „„=20mb. Comparisons ofthe nucleon-energy spectra, the nucleon multiplicities,the ratios of out-of-plane to in-plane nucleon cross sec-tions, and the mean transverse momenta are shown inFigs. 8 —11 for 4mo. „„=20mb and 4'„„=41mb. Nu-cleon distributions are slightly more forward peaked, andmore enhanced in the reaction plane in calculationswhere the nucleon-nucleon cross section is reduced. Themean transverse momenta are significantly more nega-tive. ' ' These differences are not unexpected sincenucleon-nucleon collisions make the nucleon velocity dis-tributions for this bombarding energy domain more ran-dom and isotropic. The overall nucleon multiplicity isalso somewhat lower with the reduced nucleon-nucleoncross section, and is more sensitive to impact parameters.The velocity of the heavy residue (not shown) is alsosomewhat less. In principle, these differences may enable

I

' ' ' 'I

N+ Sm E/A = 35MeV

30' —90'

Q

90' — 180-s

0 0 Cj 0

0'

0 I I I I I I I I

b (frn)10

I

' ' ' 'I

14N- + 154S

FIG. 9. Nucleon multiplicities for the ' N+" Sm reaction atE/A =35 MeV calculated with the Boltzmann equation and areduced nucleon-nucleon cross section of 20 mb are given by thesolid and open points in the figure. The corresponding calcula-tions with a nucleon-nucleon cross section of 41 mb are depictedby the solid and dashed lines. These multiplicity distributionswere integrated o'ver the angular ranges indicated in the figure.The statistical uncertainties for the calculations with the 41-mbnucleon-nucleon cross section are shown in Fig. 5.

102

10

100

I I I I I I I

~1.000

I

~Q. 75

~0.50

CO

~ 0.25

0.00

4L

E/A = 35 MeV

5 Ib (f~)

50

E, (MeV)100

FIG. 8. Nucleon-energy spectra calculated with theBoltzmann equation and a reduced nucleon-nucleon cross sec-tion of 20 mb are shown as solid and open points for the' N+ ' Sm reaction at E / A =35 MeV. The corresponding en-

ergy spectra calculated with a nucleon-nucleon cross section of41 mb are depicted by the solid and dashed lines. The statisticaluncertainties for the calculations with the 41-mb nucleon-nucleon cross section are shown in Fig. 4.

FIG. 10. Ratios of the out-of-plane to in-plane nucleon crosssections calculated with the Boltzmann equation and anucleon-nucleon cross section of 41 mb are plotted as the solidpoints for ' N+" Sm reactions at E/A =35 MeV. Ratios cal-culated with a reduced nucleon-nucleon cross section of 20 mbare depicted by the open points. These ratios were obtained byintegrating the nucleon cross sections over angles indicated inSec. II and over nucleon energies above an energy threshold of 5MeV. The solid and dashed lines are drawn to guide the eye.The dotted line corresponds to the expectation for isotropicemission.


0 N + Srn

I I I I

10

FIG. 11. Values for the mean transverse momentum of un-

bound nucleons calculated with the Boltzmann equation and anucleon-nucleon cross section of 41 mb are plotted here as solidpoints for ' N+ ' Sm reactions at E /3 =35 MeV. The corre-sponding calculations with a reduced nucleon-nucleon cross sec-tion of 20 rnb are given by the open points. Statistical uncer-tainties for both calculations are about the same. The lines aredrawn to guide the eye. 30 — Ar + Au

E/A=60 MeV

SIo ~ ~~ ~~ ~~0

from fission folding angle distributions, however, are in-consistent with the existence of such high-velocity resi-dues. ' These calculations could still be reconciled tothe experimental data if these highly excited residues donot decay by fission and either decay to heavy residues ordecay by unconventional, perhaps multifragment decaymodes. The angular momentum of the residue, shown inFig. 13, has a maximum value of 100% at about b =4 fm.This value is larger than the maximum value (I ) 85fi)predicted by the liquid-drop model. As mentioned previ-ously, between b =4 and 6 fm more than two fragmentsare found in the final state and are not separated in thesecalculations by an elapsed time of 150 fm/c. The proper-ties of the heavy residue between 4 and 6 fm are thereforesomewhat inaccurate.

Excluding the target residue, we have calculated theproperties of other bound fragments produced in thesimulations. The total mass, average deAection angle, en-ergy per nucleon, and mean transverse momentum pernucleon of these fast bound fragments is shown in Fig.14. The solid points for b ~7 fm indicate the values atimpact parameters where a single fast fragment is pre-

one to determine empirically the most appropriatenucleon-nucleon cross section for these simulations. Thestrongest sensitivities to the nucleon-nucleon cross sec-tion are displayed by the mean transverse momenta andthe in-plane enhancement of the nucleon emission pat-terns. These two observables, however, are stronglyimpact-parameter dependent, illustrating the importanceof establishing reliable measures of the impact parame-ters.

t=90 fm, /c20 b=—10 fm sw0

10--~ s~ 0

5

—10—~ ~

40A 197A I ~ ' @a~e P ~ ' o

E/A=60 MeV " '~Lg fl' ~g. :b=5 fm ~ ~

t= 150 fm/c +I'

I.l ~. . .l.l' I,

Lf ~

B. Ar+' Au at E/A =60 MeV

Calculations were performed for this system for anelapsed time of 150 fm/c for all impact parameters small-er than 10 fm. At 10 fm, the calculation was stopped at90 fm/c. As shown in the upper half of Fig. 12, well-defined projectile- and target-like residues could be clear-ly identified in collisions at large impact parameters. Atimpact parameters 4~b ~6 fm, as shown in the lowerhalf of the figure, more than two bound fragments werefrequently observed at the end of the calculation. Forthese cases, the fragments in the multifragment final statewere sometimes not completely separated at an elapsedtime of 150 fm jc.

The velocity, angular momentum, and mass of the larg-est (target-like) residue are shown in Fig. 13 as a functionof the impact parameters. The residue velocity V„de-creases smoothly with the impact parameters. At smallimpact parameters, b (3 fm, the residue velocity is atleast 80% of the velocity of a residue from a completefusion reaction. Determinations of the residue velocity

10—

—10—~ ~

0x (fm)

20

FIG. 12. The locations of bound test particles calculatedwith the Boltzmann equation Upper half: projectile- andtarget-like residues can be clearly identified for collisions at animpact parameter of 10 fm and an elapsed time of 90 fm/c.Lower half: more than two bound residues are observed for thiscollision at an impact parameter of 5 fm and an elapsed time of150 fm/c. The location of unbound test particles are not plottedin these figures.


0.060.040.020.00

100806020

I~ I ~ ~ ~ o ~ ~ o

I I I

~ ~ ~ ~ ~ ~ ~ o o ~ ~ o ~ ~ ~ ~ ~ ~ ~ o ~ ~ Io ~ ~ ~

o o ~

~" 20—

0 I I I I II I I I I I

40A + 197A

— E/A=60 MeV-

g ~ ~(3

I I I I I I I I I I I

I I I I I I I I I I I

200180160140

ArI

0 0 00

+ Au E/AI I I I

b (f

~ ~

60 MeV

10

~ 20—W

——O. OP p

—0.04I I I I I I I I I I I I ! I I I I I I I I I I I

6 8 10 6 8 10b (fm)

FIG. 13. Predictions for the parallel component of the veloc-ity (upper part), the angular momentum (middle part), and themass (lower part) of the target-like residue for the Ar+ ' Aureaction at E/A =60 MeV. The open points correspond to im-

pact parameters where more than one fast bound fragment wasproduced in the calculations. The solid points denote calcula-tions for which no more than one fast fragment was produced.The dotted line in the upper part of the figure corresponds tothe residue velocity which is consistent with the full linearmomentum transfer to the target-like residue.

dieted. The open points for b ~6 fm indicate the valuesobtained when more than one fast fragment is predicted;these points were calculated by treating all the nucleonsin fast fragments as if they were contained in a singlefragment. As shown by the figure, the mean scatteringangle of fast bound fragments, —10', is nearly indepen-dent of impact parameter. The projectile-like fragmentproduced at b ~7 fm is relatively energetic, Ef/Af )20

I

FIG. 14. Predictions for the mass (upper left), angle (upperright), energy per nucleon (lower left), and transverse mornen-tum per nucleon (lower right) of the projectile-like residue forthe Ar+' Au reaction at E/3 =60 MeV. The open pointscorrespond to impact parameters where more than one fastbound fragment was produced in the calculations. The solidpoints denote calculations for which no more than one fast frag-ment was produced.

MeV. Slower fragments are produced at smaller impactparameters, where the calculation produces multifrag-rnent final states.

The calculated nucleon differential cross sections areshown as the open and solid points in Fig. 15. To com-pare these calculations to experimental data, we have as-surned for simplicity that light particle production can betreated by the coalescence approximation and havecombined the measured light particle spectra using therelationship

on d op=2dEod 0 dE d 0 E:Eo+vc

d20. 0-+4 +9

dE dQ E 2 o+~coU& dE dQ += +o+~cou

03 d 20-

+9 +16dE d 0 = o+ coUi dE d 0 =4 o+2~coU

(4)

This relationship assumes the equality of the neutron andproton production cross sections. The acceleration ofprotons and light composite nuclei in the Coulomb fieldof the target residue was approximated by a Coulombshift AE =ZVc, „& with Vc,„&

= 10 MeV. The factor 3that precedes each term (except the first term) in Eq. (4)rejects both the number of primary nucleons in eachfragment and the conversion from the fragment energy tothe energy of the constituent nucleons. The resultingeffective nucleon cross sections are indicated by the solidand dashed lines in Fig. 15. The shape of the energy

I

spectra are well reproduced at all angles. The magnitudeof the calculations reproduce well the experimental dataat forward angles. At backward angles greater than 70,however, the calculated spectra exceed the experimentalresults, with the discrepancy increasing with scatteringangle. It is not clear at present whether this indicates adeficiency in the coalescence prescription, a need for asmaller nucleon-nucleon cross section, or a shortcomingin the Boltzmann approach.

The nucleon multiplicity distributions are shown inFig. 16 as a function of impact parameters. These calcu-


I I

I

I I I I

Ar + ' Au E/A = 60 MeU

10

w10

b 1

po

0 100E, (MeV)

200

FIG. 15. Nucleon differential cross sections calculated withthe Boltzmann equation are shown as solid and open points forthe Ar+' 'Au reaction at E/A =60 MeV. The normaliza-tions of the cross sections at 22.7' are correct; the cross sectionsat successively larger angles are suppressed by successivelylarger powers of 10. The solid and dashed lines correspond toeffective nucleon cross sections derived from the experimentaldata of Ref. 30 according to Eq. (4).

Ar + Au E/A = 60 MeU

lations suggest that misleading conclusions may bereached if one uses devices which measure the multiplici-ty exclusively in the forward direction. ' ' Similar con-clusions have been reached from measurements at lowerincident energies. Indeed, for this reaction the max-imum multiplicities for nucleons emitted to 0'—30', are ~ 1.00

~0.75

r+ Au~ ~ ~ ~ ~ ~ ~ ~ 4 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 ~ ~

60 MeV

obtained for intermediate impact parameters b =4—5 fm.The calculated multiplicities are remarkably independentof impact parameters at b & 5 fm. Thus, while more ac-curate information may be obtained with experimentaldevices which have a more complete angular coverage,the present calculations suggest that it may not be possi-ble to differentiate between impact parameters less thanb =5 fm using the multiplicity. Even at larger impact pa-rameters, the relationship between multiplicity and im-

pact parameters can become indefinite due to fluctuationsin the multiplicity. The effect of these Auctuations will bereduced, however, if one includes the yields of particlesevaporated from the target residue.

We have also computed the ratio of out-of-plane to in-plane differential multiplicities, shown in Fig. 17. Thecalculations also predict that a larger fraction of the nu-cleons are emitted in the reaction plane at larger impactparameters. The mean transverse momentum, shown inFig. 18, is similar in magnitude to that calculated for the' N+' Sm system. The minimum occurs at an impactparameter of about 5 fm. The emission patterns for themost energetic nucleons are slightly more isotropic forthis reaction than for the ' N+ ' Sm reaction atE/A =35 MeV. Two particle correlation measurementsfor this reaction suggest a much greater isotropy in thenucleon emission patterns. ' This apparent isotropyhas been partly attributed to the effects of averaging overthe azimuthal angle of the reaction plane, since the az-imuthal orientation of the reaction plane is not welldefined by such measurements.

30' —90H.~ -H —-Ej

20—0' —30'

g 0.50

C'I 0.Z5

0.00

b (f~)10

90' — 180'I I I

FIG. 16. Nucleon multiplicities for the Ar+ ' Au reactionat F. /A =60 MeV are given by the solid and open points in thefigure. These multiplicity distributions were integrated over theangular ranges indicated in the figure. The lines are drawn toguide the eye.

FIG. 17. Ratios of the out-of-plane to in-plane nucleon crosssections are plotted here as functions of the impact parameterfor Ar+ ' Au reactions at E/A =60 MeV. These ratios wereobtained by integrating the nucleon cross sections over angles asdiscussed in Sec. II, and over nucleon energy subject to energythresholds of 5 MeV (solid points) and 20 MeV (open points).Statistical uncertainties for the points calculated with the 20-MeV threshold are 30—40 /o larger than the corresponding un-certainties for the 5-MeV thresholds. The solid and dashedlines are drawn to guide the eye. The dotted line corresponds tothe expectation for isotropic emission.


I

40A + 197' Ej'A=60 MeV

QU —1

II/ /

/

II

II

II/

/--/

/

5b (fm}

10

FIG. 18. Values for the mean transverse momentum of un-bound nucleons calculated with the Boltzmann equation for"Ar+' Au reactions at E/A =60 MeV. The line is drawn toguide the eye.

0+ '9 Au at E / A =60 and 90 MeU

We first consider the reaction at E//1 =60 MeV. Theobservables were calculated for this reaction at an elapsedtime of 150 fm/c, by which time a well-defined targetresidue can be identified at each impact parameter. Weconfine our discussion of this reaction to the properties ofthe target residue and the nucleon multiplicities. Theproperties of the target residue are shown in Fig. 19.

Qualitatively, the trends are similar to those calculatedfor the other systems. The residue velocity begins to de-crease smoothly at about 5 fm, slightly smaller than forthe '"N+ ' Sm system at E/3 =35 MeV, and displays atrend that is also similar to that observed for the

Ar+ ' Au reaction at E/3 = 60 MeV. The mostsignificant dNerences with the Ar+ ' Au calculationoccur principally at impact parameters near 5 fm, wheremore than two bound fragments are found in the simula-tions of the Ar+ Au system. The maximumtransferred angular momentum is about 60k for the' 0+' Au system at E/A =60 MeV, much less than themaximum values (I ) 80iri) predicted by the liquid-dropmodel for nuclei in this mass domain. For b &5 fm, theresidue velocity is insensitive to the impact parameter;only the angular momentum transfer is sensitive to theimpact parameter for such small impact parameters.

The impact-parameter dependence of the nucleon mul-tiplicity is shown in Fig. 20. Like the other low-energysystems, the total multiplicity is rather insensitive to im-pact parameters for impact parameters less than 5 fm,where it saturates at values comparable to the mass nurn-ber of the projectile. The multiplicity at forward angles(O~,b~30') is essentially (lat for impact parameters lessthan about 8 or 9 fm.

We turn now to the reaction at E/A =90 MeV. Theobservables for this system were evaluated at an elapsedtime of 100 fm/c. A well-defined target-like residue canbe identified at each impact parameter. Properties of thetarget-like residue are shown in Fig. 21. These results arequalitatively similar to those obtained for the ' N+ ' "Smreaction at E/A =35 MeV and the ' 0+ ' Au reactionat E/A =60 MeV. Both the linear and angular momen-tum transfer begin to decrease at smaller values (b=5fm) of the impact parameters than that observed at

0.03

0.02

0.01

0.00

60:—40:20:0—

200:—190 .:—180:—170:—160:—150

0

~. —~ —-O- W -g.

r ~

b (Zm)10

0 0 ~ ~ ~ ~ y g ~

0 + Au E/A = 60 MeV

~ ~ ~ ~ 20

10—D

0+ AuE/A= 60MeV

i I I I

FIG. 19. Predictions for the parallel component of the veloc-ity (upper part), the angular momentum (middle part), and themass {lower part) of the target-like residue for the ' 0+ ' Aureaction at E/A =60 MeV. The dotted line in the upper part ofthe figure corresponds to the residue velocity consistent withfull linear momentum transfer to the target-like residue. Thedashed line is drawn to guide the eye.

5b (fm)

FIT+. 20. Nucleon multiplicities for the ' 0+' Au reactionat E/A =60 MeV are given by the solid and open points in thefigure. These multiplicity distributions were integrated over theangular ranges indicated in the figure. The lines are drawn toguide the eye.

1695

0.03 projectile-like fragment increase with increasing impactparameters. Interestingly, the absolute value of the cal-culated deflection angle does not decrease beyond —10nor does the energy per nucleon decrease below about 50MeV, regardless of the impact parameter.

Nucleon-energy spectra predicted by the Boltzmannequation are shown as the solid and open points in Fig.23. The solid and dashed lines correspond to effective nu-cleon cross sections obtained by combining data for the' 0+' Au system at E/A =94 MeV (Ref. 22) accordingto Eq. (4). The agreement between the calculated andmeasured spectra is very good. Unfortunately, measure-ments of the energy spectra are not available at largerscattering angles where the discrepancies between theeffective nucleon cross sections and the calculated nu-cleon spectra could be larger.

The dependence of nucleon multiplicities on impact pa-rameters, is shown in Fig. 24 for different angular ranges.It is noteworthy that the multiplicity at forward (0 —30')and backward (30'—180') angular ranges dependdifferently upon impact parameters. In particular, multi-plicities greater than 6 in the forward angular range(0' —30 ) occur preferentially at intermediate impact pa-rameters 3 & b & 8 fm. On the other hand, a multiplicitygreater than 10 in the backward angular range (30 —180')occurs primarily for more central co11isions, b &6 fm.The calculations suggest that suitable combinations offorward, backward, and total multiplicities could permitone to bias experimental data alternatively towards small,intermediate, or large impact parameters.

The ratio of the out-of-plane to the in-plane nucleon

I

I I

- —~00.02

0.01 '0

II0.0060:40:20:0—200—

180:160:140:

O ~ O ~ ~ I ~

16p + 197A E/AI

5

90 MeV

10

FIG. 21. Predictions for the parallel component of the veloc-ity (upper part), the angular momentum (middle part), and themass (lower part) of the target-like residue for the ' 0+' Aureaction at E/A =90 MeV. The residue velocity consistentwith full linear momentum transfer to the target-like residue liesoutside the range of the figure and therefore is not shown.

E/A =60 MeV. Of the properties of the target-like resi-due, only the angular momentum transfer is impact-parameter dependent for impact parameters less than 4fm. The maximum transferred angular momentum isabout 57fi, much less than the maximum values (l ) 80fi)predicted by the liquid-drop model. Projectile-like frag-ments are observed for this reaction at b ~6 fm. Theproperties of the projectile-like residual are shown in Fig.22. Both the energy per nucleon and the mass of the

I I 1 I

"O + '"AI I f I

E/A = 90 Mev10~ ~ Oy ~ 0 ~ y 10'0 ~ ~ 0op pop opp

1OO0

~ ~p 0 ~0 60o ~

0QQ

10

—, 1O—6

50Q10—

Og 0 ~ II I& II j)0 g ~

'5O &~ iaO.

I I I I I I I I I I I I I I0 I I I I I I I I I I I I I I 0.00 90

850~

16O

—0.02 a

0 100E. (MeV)

200197A

E/A 90 MeV

0I I I I I I I I I I I

—0.04FIG. 23. Nucleon di6'erential cross sections calculated with

the Boltzmann equation are shown as solid and open points forthe ' 0+' Au reaction at E/A =90 MeV. The normalizationsof the cross sections at 10' are correct; the cross sections at suc-cessively larger angles are suppressed by successively largerpowers of 10. The solid and dashed lines correspond to effectivenucleon cross sections at 9=40 and 50', respectively, derivedfrom the experimental data of Ref. 22, according to Eq. (4).

6 8 10 6 8 10b (f~)

FIG. 22. Predictions for the mass (upper left), angle (upperright), energy per nucleon (lower left), and transverse momen-tum per nucleon (lower right) of the projectile-like residue forthe ' O+ ' Au reaction at E/A =90 MeV.

IMPACT PARAMETER AND ENERGY DEPENDENCE OF. . .


r

I I I I

0 + Au E/A = 90 MeV

I I I I I I

16O + 197A

15—

U

~-1

E/A=90 MeV

10—

E/A=60 MeV

10 10

FIG. 24. Nucleon multiplicities for the ' 0+' Au reactionat E/A =90 MeV are given by the solid and open points in thefigure. These multiphcity distributions were integrated over the

angular ranges indicated in the figure. The lines are drawn toguide the eye.

FIG. 26. Values for the mean transverse momentum of un-

bound nucleons calculated with the Boltzmann equation for' 0+' Au reactions at E/A =60 and 90 MeV. The lines aredrawn to guide the eye. The statistical uncertainties for thelatter points are about 15%%uo smaller than the uncertainties atE/A =60 MeV.

1.25

~~1.00

N0 P5

~050I

~ 0.25

I I I

16O + 197A

E/A = 90 MeV

cross sections is shown in Fig. 25. Particularly for themost energetic nucleons, the out-of-plane nucleon crosssections are slightly enhanced with respect to those inplane at small impact parameters, b ~2. The ratio ofout-of-plane to in-plane nucleon yields decreases with theimpact parameter. The mean transverse momenta ob-tained for collisions at E/A =90 MeV, shown in Fig. 26,are significantly less negative at b ~ 5 fm than the trans-verse momenta obtained for the same system atE/3 =60 MeV; they are also less negative than for the' N+' Sm system at E/A =35 MeV. This increase of(P„) with incident energy is a consequence of the in-

creased importance of nucleon-nucleon collisions athigher energies where Pauli blocking effects are less im-portant.

D. The bombarding energy dependenceof the nucleon multiplicities

0.00 I

5b (rm)

10

FIG. 25. Ratios of the out-of-plane to in-plane nucleon crosssections are plotted here as functions of the impact parameterfor ' 0+' Au reactions at E/A =90 MeV. These ratios wereobtained by integrating the nucleon cross sections over the an-gles as specified in Sec. II and over the nucleon energy subject toenergy thresholds of 5 MeV (solid points} and 20 MeV (openpoints). Statistical uncertainties for the points calculated withthe 20-MeV threshold are 30—40% larger than the correspond-ing uncertainties for the 5-MeV thresholds. The solid anddashed lines are drawn to guide the eye. The dotted line corre-sponds to the expectation for isotropic emission.

As the bombarding energy is increased, it becomes in-creasingly advantageous to be able to select the impactparameter using the properties of the unbound nucleonsor light particles. To illustrate the evolution of the nu-cleon multiplicity distributions with the impact parame-ters, we have calculated the nucleon multiplicity distribu-tions for the symmetric Ar+ Ar system and the asym-metric ' 0+' Au system for E/3 ranging between 35and 400 MeV.

We consider first calculations for the Ar+ Ar sys-tem. In these calculations, nucleon multiplicities wereevaluated after an elapsed time of 200, 150, 100, 75, and50 fm/c for calculations performed at the incident ener-gies E/A =35, 60, 100, 200, and 400 MeV, respectively.Fusion is predicted to occur only at b =1 fm for col-


lisions at E/A =35 MeV. At this energy and larger im-pact parameters, well-formed projectile- and target-likeresidues can be identified at the end of each calculation.At higher energies, two bound residues occur mainly atlarge impact parameters; at small impact parameters,many fragments are frequently produced. As an exam-ple, Fig. 27 shows a calculation for the Ar+ Ar sys-tem at E/A =60 MeV and b =0 frn after an elapsed timeof 250 fm/c. In this calculation, six separate fragmentscan clearly be seen. About 30% of the calculations atthis impact parameter result in more than two boundfragments.

The total multiplicity and the multiplicity of nucleonsemitted at 0„„~30 are shown in Fig. 28 for the

Ar+ Ar system at four bombarding energies. Like the' N+' Sm system at E/A =35 MeV, the total multipli-city for the Ar+ Ar system at E/A =35 MeV is in-sensitive to impact parameters for small impact parame-ters. At higher energies, however, the total multiplicitydisplays a monotonic, almost linear, dependence on im-pact parameters. The multiplicity at any given parametergrows with incident energy and reaches limiting valuesconsistent with the participant spectator model byE/A =200 MeV. Multiplicities calculated at E/A =400MeV (not shown) are about 10% larger and have essen-tially the same dependence on impact parameters as thosecalculated at E/A =200 MeV.

The multiplicity also shows energy-dependent trendsfor the ' 0+' Au reaction. In addition to the calcula-tions at E/A =60 and 90 MeV presented previously, cal-culations were performed at incident energies ofE/A =200 and 400 MeV For th. ese reactions, nucleonmultiplicity distributions were evaluated after elapsedtimes of 75 and 50 fm/c, respectively. After this time in-

Ar + Ar

-E/A=35 MeV- -- 60 MeV

20 —~=--

0

50—

'G~--I I I I i iI I I I i I I

100 MeV-- V-

0 5 0 5 10b (rm)

terval, well-formed projectile- and target-like residueswere observed mainly in collisions at large impact param-eters. At small impact parameters, multifragment finalstates were occasionally observed.

The energy dependence of the multiplicity distributionsfor ' 0+' Au reactions is shown in Fig. 29. It is in-

FIG. 28. Total nucleon multiplicities are given by the solidpoints and the multiplicities of nucleons at 8&30' by the openpoints for the Ar+ Ar reaction at bombarding energiesE/A =35, 60, 100, and 200 MeV. The lines are drawn to guidethe eye.

20

10

i

]i I I I

(

I i

A + A

E/A=60 MeV ..

16O + l SV~

p0 E/A=60 MeV ~ 90 MeV

30

0

b=O fmt =250 frn/c

«Q

~ ~

$~ « I ~« ~~ I«

~ ~ ~

~ ~0

. . . 4) II I I I i i I' i

400 MeV

10—-note~

o.40 200

—200~ i

—30 —20 0 20 0 5 10 5b (f~)

10

FIG. 27. The locations of bound test particles calculatedwith the Boltzmann equation are shown after an elapsed time of250 fm/c for the Ar+ Ar reaction at E/A =60 MeV andzero impact parameter. The locations of unbound test particlesare not plotted in these figures.

FIG. 29. Total nucleon multiplicities are given by the solidpoints and the multiplicity of nucleons at 0&30' by the openpoints for the ' Q+' Au reaction at bombarding energiesE/A =60, 90, 200, and 400 MeV. The lines are drawn to guidethe eye.


teresting to note that the impact-parameter dependencesof both the calculations at E/A =200 and 400 MeV arequalitatively similar to those displayed for the' 0+' Au reaction at E/3 =90 MeV. Such depen-dences are different from those predicted for collisionsbetween mass-asymmetric systems at lower energies.These calculations suggest that multiplicity measure-ments may provide more quantitative impact-parameterinformation at small impact parameters for collisions atand above E/A =90 MeV than below.

IV. SUMMARY

Guided by solutions of the Boltzmann equation, wehave explored the sensitivity of different experimental ob-servables to the impact parameter. These calculationswere motivated by the knowledge that many of the mostinteresting observables, like the transverse momentum orAow, are strongly impact-parameter dependent and there-fore ambiguities in the impact parameters can prevent usfrom fully utilizing the experimental measurements. Al-though the present calculations indicate that nearly allobservables are sensitive to the impact parameter, a sin-gle reaction variable does not suSce to provide impact-parameter selection in general. Instead, we can identifyreaction variables which provide impact-parameter selec-tion for the different domains of incident energy and im-pact parameters.

For mass-asymmetric entrance channels like the' 0+' Au reaction, the average total nucleon multiplici-ty and the velocity of the reaction residue display a verysimilar dependence on the impact parameter. At low en-ergies E/A ~60 MeV where the multiplicity is low andthe existence of a reaction residue is assured, the residuevelocity provides more accurate impact-parameter infor-mation because it is not subject to the statistical Auctua-tions of multiplicity measurements when the mean multi-plicities are small. Unfortunately, for low-energy col-lisions, both residue velocity and multiplicity are insensi-tive to impact parameters for b ~ 5 fm. (Additional cal-culations are needed to determine whether the inclusionof the yields of particles evaporated from the highly ex-cited target residue, not taken into account by the presentcalculations, can significantly improve the impact-parameter sensitivity of the multiplicity distributions. ) At

these smaller impact parameters, only the angularmomentum of the reaction residue is impact-parameterdependent, suggesting that conventional techniques likethe measurement of the y ray multiplicity should be ex-plored to see if they could provide additional informationabout the impact parameters.

For the Ar+ Ar system at E//l =35 MeV, the cal-culations suggest that compound nuclei are not formed atimpact parameters greater than or equal to 2 fm; the nu-cleon multiplicity is sensitive to the impact parameteronly for large impact parameters, b ) 5 fm. It is likelythat similar results would be obtained for calculationswith other symmetric systems.

At higher energies, where the existence of a reactionresidue is questionable and needs experimentalverification, the calculations suggest that the total nu-cleon multiplicity and the angular distribution of emittednucleons may be combined to provide impact-parameterselectivity. This selectivity exists for symmetric systemsat E/A ~ 60 MeV and for asymmetric systems atE/A ~90 MeV, and extends to rather small impact pa-rameters. The distinction between different impact pa-rameters will be somewhat blurred, however, by theeffects of multiplicity fIuctuations.

The values of some of the important parameters of theBoltzmann equation, like the velocity and density depen-dence of the nucleon-nucleon cross section and the nu-clear mean field are not very well established. For inter-mediate energy collisions, uncertainties in the nucleon-nucleon cross section are probably more important. Forthe ' X+ ' Sm reaction we have shown that these uncer-tainties can strongly affect the calculated observables. Inparticular, the mean transverse momentum and theenhancement of emission patterns in the reaction planeare rather sensitive to the importance of nucleon-nucleoncollisions. Uncertainties in the cluster production mech-anism presently make the comparison of these observ-ables to experimental data somewhat tenuous. These as-sumptions, however, clearly can and will be refined asmore experimental data become available.

This work was supported by the National ScienceFoundation under Grant Nos. PHY-86-11210 and PHY-87-14432, and a U.S. Presidential Young InvestigatorsAward.

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Impact parameter and energy dependence of observables in ...PHYSICAL REVIEW C VOLUME 40, NUMBER 4...

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