Collisions between two nuclei have been modeled by numerical solution of classical approximations to the equations of motion of the constituent nucleons. For the reaction Nb(400 MeV/u)+Nb, a correlated sidewards emission of nucleons is observed. This is attributed to the repulsive short-range component of the nucleon-nucleon potential. A strong dependence of the flow angle on the impact parameter is observed, in accord with recent experimental results

Hoffer, James B.

Kruse, Hans

Molitoris, Joseph J.

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLUME  53,  NUMBER  9  PHYSICAL REVIEW LETTERS  27 AUGUST  1984 Microscopic Calculations of Collective Flow Probing the Short-Range Nature of the Nuclear Force J. J. Molitoris, J. B. ~offer,(~)  H. ~ruse,(~)  and H. Stöcker Department of Physics und Astronomy und iVutionul Superconducting Cyclotron Laboratory, Michigan Stute Universiiy, East Lansing, Michigan 48824 (Received 27 April 1984) Collisions  between two  nuclei have been modeled by  numerical solution of  classical ap- proximations  to  the  equations of  motion  of  the constituent  nucleons.  For  the reaction Nb(400  MeV/u)+Nb,  a  correlated  sidewards emission  of nucleons  is  observed.  This is attributed  to  the  repulsive  short-range  component  of  the nucleon-nucleon  potential.  A strong dependence of the flow angle on the impact parameter is observed, in accord with re- Cent experimental results. PACS numbers:  25.70.N~ The  recent  experimental  observation'  of  side- wards  peaks  in the emission Pattern  of  fragments emitted  in collisions of  heavy  nuclear  Systems has stimulated a dispute among theorists about how to interpret  these  data.  It  has  been  shown2 that  the observations are in  agreement  with  the results of macroscopic  nuclear  fluid  dynamical  calculations, but several microscopic  calculations done to repro- duce the sidewards emission  [via the intranuclear cascade (INC) approach3*41  failed-the  angular dis- tributions obtained were always forward peaked.1s2 We  have  recently  developed  a  many-body equations-of-motion  (EOM)  approach  to  study heavy-ion  collisons.  Our approach is analogous to that of Bodmer, Panos, and ~ac~ellar~  and Wilets and  CO-worken6  Hamilton7s equations of  motion are  solved  for  an  ensemble  of  A  nucleons  with simultaneous  mutual  two-body  interactions  be- tween all particles: The Hamiltonian is where  r =  I Ti  -  T'j  1  is  the  distance  between  nu- cleons  i and j. This  classical  nonrelativistic  approach  neglects spin, the Pauli principle, and the uncertainty princi- ple.  On the other hand, this approach allows for a study  of  the  simultaneous  (classical)  interactions between  many  particles  in  contrast  to  the independent-particle  models  such  as  the  intranu- clear  cascade,  which  treat  nuclear  collisions  as  a simple  superposition  of  successive  free-space nucleon-nucleon  collisions.  Also,  the  EOM  ap- proach allows for a systematic study of the repulsive core of the nucleon-nucleon interaction  (because of its  deterministic nature, the EOM calculation  pro- duces an excluded-volume effect), which  is of  in- terest in connection with the possible  study of  nu- clear matter properties at high baryon densities. We also neglect Coulomb forces and pion produc- tion, although the former are trivial to include and we  have  checked  that  one  obtains  identical  flow results when Coulomb interaction  is included  at  a typical  impact  parameter.  At  E,,,,  = 100 MeVIu, the total Coulomb energy is only a small fraction of the incident  kinetic energy.  Pion  production  does not play a significant role for the present investiga- tion;  in fact, less  than  pion  is produced  on the average in the collision studied here.4 The  nucleon-nucleon  potential  consists  of  two terms,  an attractive long-range Yukawa interaction and a repulsive short-range core5s6: where  VR =  2970 MeV-fm,  VA  = 765 MeV-fm, KR =  2.66 fm-',  and KA  = 1.75 fm-l.  The Parameters in  the  potential  were  chosen  in  a  compromise between reproducing in the EOM calculation the n- p differential scattering cross section at large angles B,,,,  =  90"  (which  influences  the  transverse  mo- mentum transfer  the most)  and  at  the same time giving  reasonable  nuclear  radii  and  binding  ener- gies. The differential cross section is hard to obtain in a classical calculation because of  diffraction and ex- change effects.  A meaningful quantity to fit is the viscosity moment of the scattering cross section, which determines the viscosity and thermal conduc- tivity  in  a  Boltzmann-equation  approach.5,6 The present  potential  gives values  of  41.1, 31.9, 27.5, and 24.6 mb for U, at 100, 200, 300, and 400 MeV, respectively.  This compares well  with  the experi- mental values  of  39.2, 26.5,  25.0 mb, which  have @ 1984 The American Physical Society  899 VOLUME  53,  NUMBER  9  PHYSICAL REVIEW LETTERS  27 AUGUST  1984 been calculated by  Bodmer and Pan~s.~  We calcu- late daldR (90")  to be 2.0 mb at 400 MeV, which is quite close to the experimental  value of  1.5 mb. If  we  assume charge exchange  to occur with  50% probability in  n-p reactions, then the measured dif- ferential  Cross  section  at  400  MeV  is  well  repro- duced for large angles 40" S O,,  C 140". In the EOM approach, nuclei are described as an ensemble of Protons and neutrons initially distribut- ed randomly throughout  a sphere with  the nuclear radius  R = 1.2~''~  fm.  However,  the  nuclei  ob- tained  are  not  stable:  They  tend  to  collapse  and evaporate many  nucleons when  the classical equa- tions  of  motion  are  integrated  over  a  typical  col- lision time.  In the present  approach, a metastable ground state has been obtained by  allowing the nu- cleons  to  drift  toward  the  configuration  of minimum  energy  of  the  chosen  nucleon-nucleon potential  which  is  a  crystalline  str~cture.~  The characteristic  interparticle  distance  corresponds  to that of  the potential minimum.  This minimization is  continued  only  until  the  interparticle  distances begin  to show structure and the nuclear density is still constant.  This leads to nuclei sufficiently stable for a collision calculation  to be  meaningful  rather than resulting in a disruption of the nuclei before a collision can actually take place.  Our potential has a minimum  of  -4.67  MeV  at  R =1.85  fm.  The resulting crystalline  ground-state configuration has an average binding energy of -31  MeV/u.  The nu- FIG. 1. Time evolution of a Nb(400 MeV/u) +Nb reaction at b = 3 fm. VOLUME  53,  NUMBER  9  PHYSICAL REVIEW LETTERS  27 AUGUST  1984 cleons then are given random Fermi momenta with an  average  Fermi  energy  of  23  MeV/u  which results in  an average  binding  energy of  7  MeV/u. These  nuclei  are  stable  for  t = O(30 fmlc), i.e., typical collision times. To simulate a collision  process  numerically, the nuclei  are  Galilei-boosted  with  the  respective center-of-mass momenta at given impact parameter. The  equations  of  motion  are  integrated  by  a fourth-order  Adams-Moulton  predictor-corrector method.  Energy conservation to better than 1%  has been  demanded.  A  total  of  535  collisions  of Nb(400 MeV/u) +Nb have been performed, which provides  reliable  statistics  for the global  event-by- event analysis  subsequently performed  on the nu- merical  "data."  The  computations  are  stopped after  t =  30  fmlc, after which  the flow results are found to be constant.  The evolution of  a collision at  b =  3-fm  impact  parameter  is  shown  in  Fig.  1. The resulting  sidewards  flow  can  clearly  be  seen. Note that the average deflection angle of the Center of  mass  of  projectile  and target  nucleons,  respec- tively, is approximately equal to the average calcu- lated flow angle, thus associating a collective repul- sion between the two nuclei. The  individual  collisions  are  analyzed  by  di- agonalizing the kinetic energy flow tensor17 where  the  sum is  over  all  charged  particles  in  a given event.  By  diagonalization of  this tensor, the flow angle OF is obtained for each event. The distri- bution  of  flow  angles  dNldcosOp is presented  in Fig. 2 for two impact-parameter intervals and com- pared  to  the  experimental  data.'  The qualitative and quantitative behavior of the flow Pattern in the EOM  model  is  very  similar  to  the  behavior  ob- served in hydrodynamics2: The flow angle OF rises smoothly from 0"  at large impact parameters to 90' at  b -0.  A  finite  range  of  impact  parameters  is sampled to compute the angular distributions of the flow  angles,  dN/dcosOF.  The distribution of  flow angles is computed by  taking into account the for- mation of  fragments via  a six-dimensional coales- cence model recently deve~o~ed.~  We find roughly the Same flow distribution by  doing the flow anal- ysis with and without clustering. It should be pointed out that the intranuclear cas- cade  m0de1~8~  yields  Zero  flow  angles  even at  the highest  m~lti~licities'~~~~~~  in  contrast  to  the data. The  present  model,  on  the  other  hand,  predicts peaks  in  the angular  distributions of  the flow  an- gles.  The peak  shifts to larger angles with increas- ing  multiplicity,  in  agreement with  the data.  The -  r,,,,,,,  I Y, C + C  EXPERIMENT D 0  0 U \ z 0  30  60  0  30  60  90 FLOW  ANGLE  BF (deg) FIG.  2.  Comparison  of  the experimental  flow-angle distributions  with  the  present  theory,  averaged  over b =  0-3  fm and b =  0-5  fnl, respectively. physical difference between the INC model and the EOM approach, which leads to such distinct predic- tions, can be traced back to the different treatments of  the  nn collision process.  The INC applies  to  a stochastic 4.rr  scattering at the point  of  closest ap- proach  of  straight-line trajectories;  this  allows  for substantial transparency.  In contrast, the repulsive short-range component in the NN potential used for the EOM approach is a hard core and thus effective- ly  results  in  an excluded-volume  effect;  thus the nuclei are not as transparent and easily compressi- ble as in the INC.  This causes incident nucleons to be  deflected away from zones of high density, i.e., small  interparticle  Separations,  towards  sidewards angles.  It is important  to point out the role of the binding potential:  The potential field keeps the nu- clei from expanding before collisions can occur.  If binding  is  neglected  as in  Cugnon's  original  pro- gram,4  finite flow angles occur because of sidewards expansion of  the unbound projectile and target nu- cleons due to Fermi m~tion.~ Much more work remains before one can make a precise connection between the nuclear forces and the  experiments  on  collective  flow.  Both  the kinetic-  and  potential-energy  contributions  to  the pressure  tensor  will  be  important,  as will  also the transport properties.  It is a challenge for the EOM approach to make "realistic"  calculations which ac- curately incorporate these and other requirements, such as Spin and Pauli and uncertainty principles, as well as relativity, into the calculations. (a)~ermanent  address:  Laboratory  for  Atmospheric and  Space  Physics,  University  of  Colorado,  Boulder, Colo. 80309. (b)Permanent address:  TELCO Research Corporation, Nashville, Tenn. 37240. lH.  A.  Gustafsson,  H.  H.  Gutbrod,  B.  Kolb, H. Löhner, B.  Ludewigt,  A. M. Poskanzer,  T. Renner, H. Riedesel,  H. G. Ritter,  A. Warwick,  F. Weik,  and 90 1 VOLUME  53, NUMBER  9  PHYSICAL REVIEW LETTERS  27 AUGUST  1984 H. Wieman, Phys. Rev. Lett. 52, 18 (1984). 2G.  Buchwald,  G.  Graebner,  J.  Theis,  J.  Maruhn, W.  Greiner,  and  H.  Stöcker,  Phys.  Rev. Lett.  52,  18 (1 984). 3Y.  Yariv  and  Z.  Frankel,  Phys.  Rev.  G  20,  2227 (1979). 45.  Cugnon, D. Kinet,  and J. Van der Meulen, Nucl. Phys. A379, 553 (1982). sA. R. Bodmer and C. N. Panos, Phys. Rev. C 15, 1342 (1977).  Transverse peaking  (i.e., the 90" cross section is larger  than the 0"  cross  section)  has  been  observed  in classical  EOM calculations  for central  ( b =  0) collisions of A,=  AT=40 at 400 MeVIu, with lighter Systems being forward peaked:  A. R. Bodmer, C. N. Panos, and A. D. MacKellar, Phys. Rev. C 22, 1025 (1980); A. R. Bodmer and C. N. Panos, Nucl. Phys. A356,  517 (1981).  A  de- tailed discussion of the transverse peaking and the role of the repulsive core and transparency effects can be found in  A. R. Bodrner,  in  Proceedings  of  the Fifth  Heavy-Ion Conference,  University of Califortlia at Berkeley,  Berkeley, California,  1981  (Univ.  of  California  Press,  Berkeley, 19811, p. 648. 6L. Wiiets, E. M. Henley, M. Kraft, and A. D. MacKel- lar, Nucl. Phys. A282, 341  (1977); L. Wilets, Y. Yariv, and R. Chestnut, Nucl. Phys. A301, 359 (1978); D.  J. E. Callaway, L. Wilets, and Y. Yariv, Nucl. Phys. A327, 250 (1979). 7M. Gyulassy, K. Frankel, and H. Stöcker, Phys. Lett. llOB, 185 (1982). *H.  Kruse, B. Jacak, and H. Stöcker, to be published. 

Microscopic calculations of collective flow probing the short-range nature of the nuclear force

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