Nuclear collisions from 0.3 to 2 GeV/nucleon are studied in a microscopic theory based on Vlasov's self-consistent mean field and Uehling-Uhlenbeck's two-body collision term which respects the Pauli principle. The theory explains simultaneously the observed collective flow and the pion multiplicity and gives their dependence on the nuclear equation of state

Jacak, Barbara V.

Kruse, Hans

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLU~IE  54,  NLMHER  4  PHYSICAL REVIEW LETTERS  28 JANCARY  1985 Microscopic Theory of Pion Production and Sidewards Flow in Heavy-Ion Collisions H. ~ruse,(~)  B. V. Jacak, and H. Stöcker Deparrnient of'Ph~,sics  and Astronomy. und .Mational Supercondircfrng Cyclotron Laborafor.y, .'Micl~rgari  Stare Urirversi[)', Easr Lansing, Micliigan 48824 (Received 9 July 1984) Nuclear  collisions  from 0.3 to  2  GeVInucleon  are studied in  a  microscopic  theory  based  on Vlasov's  self-consistent  mean  field  and  Uehling-Uhlenbeck's  two-body  collision  term  which respects the Pauli principle.  The theory explains simultaneously the observed collective flow and the pion multiplicity and gives their dependence on the nuclear equation of state. PACS numbers: 25.70.-z One of  the most intriguing motivations for studying relativistic nucleus-nucleus collisions is the unique op- portunity  to  explore compressed and excited  nuclear matter in  the laboratory.  A signature of  compression is  the collective sidewards flow predicted theoretically by  nuclear  fluid  dynamics'  and  classical  microscopic many-body  simu~ations.~  Recently,  the predicted  col- lective  sidewards  flow  has  been  observed  in  high- multiplicity  selected  collisions  of  heavy  n~clei.~,~ Another  observable compression  effect predicted  by fluid dynamics is the dependence of the pion multipli- city on the nuclear compression energy at high  densi- tie~.~  The pion  multiplicities have been measured for near-central  collisions  of  Ar  (0.3-1.8  GeV/nucleon) + KCI.~ Both  data  sets  present  a  challenge  to  microscopic theories:  The flow calculationsL4 done with the stan- dard  intranuclear-cascade  pr~~rams'.~  result  in  for- ward-peaked  angular  distributions,  in  contrast  to  the data. and the calculated pion m~lti~licities~-~  drastical-  I ly  overestimate the experimental yields.  These large discrepancies are surprising in  view of  the success of the cascade  model  in  describing  inclusive data.'j8  It has  been  conjectured6  that  the  difference  between measured pion yields and cascade predictions is due to the neglect  of  compression energy  in the cascade ap- proach  and thus may  be  used  to extract  the nuclear equation of state at high densities. In this Letter we present a microscopic theory which explains for the first time simultaneously both the ob- served  collective  flow  and  the  pion  multiplicity  and gives  their  dependence  on the  nuclear  equation  of state.  Our approach is based on Vlasov's equation for the evolution of  the single-particle  distribution  func- tion f of  a  collisionless  plasma  in  a  self-consistent mean  potential  field  supplemented  by  Uehling- Uhlenbeck's  quantum  mechanical  extension  of Boltzmann's  two-body  collision  term  which  respects the Pauli principle.  This extended Boltzmann equation can be ~ritten~.'~ The Vlasov equation is  solved by  simultaneous nu- merical integration of the classical equations of motion of  fifteen parallel ensembles of  Ap+ AT test particles, which are initially  assigned  Fermi  momenta and ran- dom  positions  in  a  sphere  of  nuclear  radiiis.  The ensemble-averaged phase-space density is computed at each  synchronization  time  step  in  a  six-dimensional sphere  around  each  test  particle.  This  ensemble a~eraging~,'~  ensures  a  reasonably  smooth  single- particle  distribution function f (r,p,r) which is used to determine the mean field  U(  n) and the Pauli-blocking probability ( 1 -  ,f)  ( 1 -,f). I I  Two test particles from a given ensemble may  un- dergo  s-wave  scattering  if  they  approach  each  other within  a  distance  d2=a/.rr  and  the  resulting  final states are not  Pauli  blocked.  Here a denotes the ex- perimental  scattering  cross  section  of  Protons,  neu- trons, and pions.  The experimental inelastic-scattering cross  sections  are  used  to calculate  pion  production and absorption through the delta resonance. Three different Skyrme parametrizations have been chosen  to represent  the mean  field  U due to a  stiff (compression  constant  K =  375  MeV),  a  medium (K  = 200  MeV), and a supersoft  nuclear equation of state: U(n)  = -  124t7/no+70.5(n/no)2  MeV  (K = 375 MeV),  (2a) Cr(n)  = -  356n/no+ 303(n/n,,)'/~ MeV  (K = 200 M~v).  (2b) VOLL  hf~  54,  NU~IBER  4  PHYSICAL REVIEW LETTERS  28 J.\YLAR) 1985 The supersoft  potential  equals  (2a) at  n < no, and is constant for n > no. It allows the study of  bound nu- clei with  Zero compression energy.  U is directly relat- ed  to  the  nuclear  equation  of  state  via  U(n) =an(E)lan. We have extensively tested the present method and the  newly  developed  Computer  program.''  First,  in- clusive  spectra  of  protons  emitted  from  high-energy heavy-ion reactions have been calculated.  They com- pare well  with recent data.'Oa"  Second, we have stud- ied proton-induced pion production on nuclear targets. Figure 1 shows the target-mass dependence of the ob- served  and  T- yields12 in  comparison  with  the present  theory.  The absolute  yields,  the target-mass dependence,  and  the  large  (factors  of  5)  difference between  the  .rrf  and  x-  yields  as  well  as  the pion spectra (not shown) are well reproduced. The present  approach  has  further  been  tested  by turning off the Pauli blocking and mean potential field. Then  the  parallel  ensembles  decouple,  and  the  test particles  move on straight-line trajectories  until  they scatter.  Thus  the  conventional  intranuclear  cascade model is  recovered.  The thus obtained  "cascade  nu- clei"  are unstable and expand as a result of the Fermi rn~tion.~  It has been shown13 that this artificial expan- sion causes a rapid decrease of the density and drasti- cally  reduces the number of  n-n  collisions if  massive nuclei  (A -  100) are studied.  The artificial expansion of the cascade nuclei can be prevented by  "freezing"13 the  Fermi momenta  of  the nucleons  or-as  done in the present work-by  use of a binding potential.  The supersoft equation of  state can be  used to study bind- ing without introducing compressional energy explicit- ly  into the calculation.  In order to simulate the stan- dard  cascade  calculations  as  closely  as  possible,  we have  used  in  our  "cascade  mode"  Cugnon's  "Pauli principle"  prescription  (n-n collisions with  c.m. ener- gies below  50 MeV/nucleon  are inhibited).  The pion yields calculated with the present program  in the "cas- cade mode"  (i.e., with the simplified Pauli blocker and no  field)  agree  quantitatively  with  the  cascade resu~ts.~,~  Both  results  differ  substantially  from  the results obtained with the present  theory which  uses a nuclear equation of state and phase-space Pauli  block- ing  : Take the 360-MeV/nucleon data, for instance:  The negative pion yield is 1.05 in the "cascade  mode,"  but drops by  a factor  of  2-to  0.56-if  the compression energy (2a) is included-the  suggested large influence of the nuclear matter equation of  state5j6  is observed. The  pion  yield  drops  further  (to  0.46)  when  the Uehling-Uhlenbeck  Pauli  blocking  is  applied.  These numbers differ by  factors of 3  from results published in  Ref. 9, where a method similar to the present one has been used to solve the the Boltzmann equation.  A revised  version  of  that  program  now  reproduces  our results.  We would like to refer to other papers on pion production which use different approaches.'4-'7 The pion multiplicities are shown in Fig. 2 as a func- tion  of  the  bombarding  energy.  The present  theory with  equation  of  state  (2a)  plus  phase-space  Pauli blocker compares well  with  the data.6 while our "cas- cade  mode"  overestimates the pion  yields  by  factors > 2  at  energies  up  to  1  GeV/nucleon, just  like  the cascade ca~culations.~-~  The drop in  the pion  yield  is found to be due to the transformation of  kinetic ener- gy  into potential energy during the high-density phase of  the reaction  as well  as due to  Pauli  blocking.  To Ar + KCI -+ T- $  CASCADE A BOLTZMANN  EQ. 0'  0  L  .,  L  L---L-_  -  ELAB [G~v/NI 0  20  40  60  80  100  120  140  FIG. 2.  Pion  multiplicity for  central  collisions  (h  < 2  3 TARGET  MASS  A  fm) of ArtKC1. The data  (Ref. 6. circles) are con~pared  to FIG. 1.  Pion  yields from  730-MeV protons 3s  a  function  thc present  theory  in the  "cascade  mode"  (crosses) and to of  the target mass  (Ref. 12).  The lines and statistical error  the  same  theory  with  compression  energq. and phase-space bars refer to the present theory.  Pauli blocking included (triangles). VOLLME  54,  NU~IBER  4  PHYSICAL REVIEW LETTERS  28 JAKLARY 1985 check the sensitivity of the pion yields to the equation of  state, the calculations have been repeated with the medium potential  (2b).  At  772 MeVInucleon we find (T-) =  2.45 k0.09 and 2.13 I0.07 with  the medium and the stiff equation of state, respectively.  At  lower energies, statistical error bars of  150h preclude an accu- rate assessment of  the influence of  the potential.  At all  other energies,  where  the statistical error  bars  are 3%,  the  yields  are  systematically  higher  (by  about 10°/o)  with the medium equation of  state than with the stiff equation of state.  For the time being we feel un- able,  though,  to  extract  a  nuclear  equation  of  state from the data.6 We  have  also  investigated  the  observed sidewards peaking  [Fig.  3(b)]  in  multiplicity  selected  Nb(400 MeVInucleon) +Nb  collisions3:  The  flow  tensor13 Fij= zvpipj/2mv  is  determined  on an  event-by-event basis and the direction of the maximum kinetic-energy flow  is  determined.  Cascade  calculations  produce forward-peaked  di~tributions.~.'~  We  would  like  to emphasize again the important role of the binding po- tential:  If  the  binding  potential  is  neglected  as  in Cugnon's original program,8,'"inite  flow angles may be obtained18 as a result of  the sidewards expansion of the  unbound  projectile  and  target  nucleons.  This is demonstrated in Fig. 3(c),  which shows the flow-angle distributions  obtained  with  the original  Cugnon  pro- gram8  (i.e.,  unstable  nuclei)  and  the  modified  ver- sion6<l3  of his code with "frozen"  Fermi momenta. Figure  3  (a)  shows  the  flow-angle distribution  for Nb(400 MeV/nucleon) +Nb at impact Parameter b = 1 fm with  the medium  and the stiff  equation  of  state. Observe the strong sidewards maximum at large angles in both cases and the drastic influence of the equation of  state:  The average flow angle is  43"  with  the stiff equation of state, while  it is 28" with the medium one. As a consequence of rhis  substantial variation  of  the flow with the nuclear potential, it seems feasible to ex- tract the nuclear equation of  state from a comparison of the data with high-statistics calculations, whence the experimental trigger conditions and efficiency cuts are carefully simulated in the calculations. In  conclusion,  a  microccopic  theory  which  incor- porates both a self-consistent mean field and two-body collisions  which  respect  the  Pauli  principle  explains both  pion  multiplicity and collective flow observed in central nucleus-nucleus collisions.  The results exhibit a significant influence of both the nuclear equation of state and the Pauli principle. Discussions with  J. Aichelin, G. Bertsch, J. Harris, and R. Stock are acknowledged.  This work  was sup- ported by the National Science Foundation. (a)Present address:  Telco  Research.  h'ashville,  Tenn. 37203. 0  20  4  0  60  8  0 THETA (DEGREES (b) 0  30  60  90 BF (deg)  ---  7  - L  (C)  I 1.0- n 7  CASCADE I  unbound L/ I  bound 0  2  0  40  60  80 8  (deg) FIG. 3.  Flow-angle distributions dN/(/cosHf  for 93Nb(400 MeV/nucleon) +93Nb:  (a)  The  present  theory  with  the medium and hard equation of state; if  the Pauli principle  is neglected, the maximum in the angular distribution shifts to larger angles (by about 20").  (b) "Plastic-ball"  data (Ref. 3) for various multiplicity  bins.  (C) Results obtained with  the standard cascade  with  use of  unstable  nuclei  (Ref. 8) and the "frozen"  Fermi-momenta approximation (Ref. 13). 'W. Scheid, H. Müller, and W. Greiner, Phys. Rev. Lett. 32. 741 (1974); H. Stöcker, J. A. Maruhn, and W. Greiner. Phys. Rev. Lett. 44, 725 (1980); J. I. Kapusta and D. Strott- man,  Phys.  Lett.  103B,  269  (1981);  G.  Buchwald, G. Graebner,  J.  Theis,  J.  A.  Maruhn.  W.  Greiner,  and H. Stöcker.  Phys.  Rev.  Lett.  52.  1594  (1984). and  Phys. Rev. C 28. 2349 (1983). Vor r \IE  54.  NL  ~~BLR  3  PHYSICAL REVIEW LETTERS  28 J \\L  AU>  1985 *J. J. Molitoris.  J. B.  Hoffer,  I{.  Kruse, and  11.  Stöcker. Phys. Rev. Lett. 53, 899  (1984); R. Malfliet, University  of Groningen,  Kernfysisch  Versneller  Institute  Report,  1984 (to be  published); S. M. Kiselev, Institut of Theoretical and Experimental Physics  Report  Ko. ITEP-123.  1984 (unpub- lished).  Transverse  peaking  of  cross  sections  ~inci  a  mass dependence, which is not obtained froni hydrodynamics. has been  predicted  for  central  collisioni  in  earlier  niany-body simulations:  A. R. Bodmer, C. N. Panos. and A. D.  Mac- Kellar, Phys. Re\. C 22,  1025 (1980): see also E. Halbert. Phys. Rev. C 23, 295 (1981). 3H. 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 H. Weirrian. Phys. Rev. Lett. 52, 1590 (1984). 4R. E. Renfordt, D. Schall, R. Bock. R. Brockmann. J. W. Harris.  A. Sandoval.  R. Stock.  H. Stroebele.  D. Bangert, W. Rauch, G. Odyniec.  H. G. Pugh.  and  L. S. Schroeder. Phys. Rev. Lett. 53,  763 (1984): D.  Keane ei 01..  to be pub- lished. SH.  Stöcker. W. Greiner. W. Scheid. Z. Phys. A  286.  121 (1978);  P.  Danielecwicz,  Nucl.  Ph1.s.  A314.  465  (1979); H. Stöcker. J. Phys. Lett. 10, 11 1 (1984). 6R. Stock. R. Bock, R. Brockmann. J. W. Harris.  .4. San- doval,  H.  Stroebele,  K.  L.  Wolf.  H.  G.  Pugh.  L.  S. Schroeder, M. Maier.  R. E. Renfordt, A. Dacal. and M.  E. Ortiz, Phys. Rev. Lett. 49, 1236 (1982): A. Sandoval  6.1  01.. Phys. Rev. Lett. 45.  874 (1980): J. W. Harris  er  U/.. to be published. 'Y.  Yariv and Z. Fraenkel, Phys. Rev. C 20. 2227 (1979). and  C  24. 488  (1981); V. D. Toneev and  K.  K. Gudinia, Phys. A400. 173 (1983). 8J. Cugnon, D. Kinet, and J. Vandernieulcn.  Nuci. Phys. A379. 553 (1982); J. Cugnon, T. Mituzani. and J. Vander- meulen, Nucl. Phys. A352, 505 (1981). 9G. Bertsch,  H. Kruse, and S. Das Ciupta.  Phys. Rev. C 29. 673 (1984). I0H. Kruse. B. V. Jacak, and H.  Stöcker. unp~iblished. "S.  Nagamiya.  M. C. Lemaire. E. Moeller.  S. Schnetrer. G. Shapiro. H. Steiner, and  I. Tanihata.  Phys. Rev. C  24. 971  (1981); B.  Jacak, G.  D. Westfall. C. K. Gelbke. L. H. Harwood,  W. G. Lynch,  D. K.  Scott.  H. Stöcker,  M. B. Tsang, and  T.  J.  M. Symons.  Ph's.  Rev.  Lett.  51,  1846 (1984). '*D. R. F. Cochran, P. N. Dean.  P.  A. M. Gram. E. .4. Knapp.  E. R. Martin,  D. E.  Nagle,  R. B.  Perkins.  W. J. Shlaer, H. A. Thiessen, and E. D. Theriot, Phys. Rev. D 6. 3085 (1972). '3M. Gyulassy, K. A. Fraenkel, and H. Stöcker. Ph~s.  Lett. llOB, 185 (1982). 141.  Montvay.  and  J.  Zinianyi.  Kucl.  Phys.  A312.  290 (1979). 1SR.  Malfliet  and  B.  Schürmann.  Phys.  Rev. C  28.  1136 (1983). I6M. Cahay, J.  Cugnon, and J. Vandermeulen. Vucl. Phys. A411, 524 (1983). '?Y.  Kitazoe,  M.  Gyulassy.  P.  Daiiielewicz,  H.  Toki. Y. Yamamura, and M. Sano, Phys. Lett. 138B, 341  (1984); Phys. Rev. C 29. 828 (1984). This cascade version assumes a  Zero  resonance  lifetime  and  neglects  pion  absorption, which results in largely different pion yields iRef. 8). 185. Cugnon. and D. L'Hote. unpublished. 

Microscopic theory of pion production and sidewards flow in heavy ion collisions

Abstract

Similar works

Full text

Available Versions

Hochschulschriftenserver - Universität Frankfurt am Main