Angular and energy distributions of fragments emitted from fast nucleus-nucleus collisions (Ne--> U at 250, 400, and 800 MeV/N) are calculated with use of nuclear fluid dynamics. A characteristic dependence of the energy spectra and angular distributions on the impact parameter is predicted. The preferential sideward emission of reaction fragments observed in the calculation for nearly central collisions seems to be supported by recent experimental data

Greiner, Walter

Maruhn, Joachim

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLUME  44, NUMBER  11  PHYSICAL REVIEW LETTERS  17 MARCH  1980 semi-infinite  system.  However,  the wavelength of  such oscillations is of  the order of  6,5  fm, which is much larger than wavelength of  the Thrope-Thouless-type  oscillations.  The two kinds of  oscillations are thus quite distinguisha- ble and the two types of  calculations can be con- sidered ~omplementary.~ Of  particular interest is the appearance of  the first bump on density distribution long before the regime of  nuclear matter is reached.  This might be surprising but it is not caused by a mistake in calculation.  It corresponds to the first Zero of the highest-lying momentum state in the Fermi ~ea.~  It agrees with considerations based on a Thomas-Fermi calculation in a linear p~tential.~ It is also in agreement with considerations due to diffuseness of  the one-body p~tential.~  Further- more, it can also occur in a slab cal~ulation,~  if the range of  the two-body  interaction is chosen long enough.  However, in this last case the Skyrme-type interaction in Ref.  3 was found not good enough to produce accurate density oscilla- tions, as has already been pointed out.'' The author wants to thank Professor M.  Baran- ger for supervision in studying semi-infinite nu- clear matter.  He also wants to thank Professor J, W, Negele for many discussions~ 'H.  D.  Keller, Nucl.  Phys. M,  141 (1971), and ref- erences cited therein.  For more recent work, See D.  G.  Ravenhall,  C. D.  Bennett, and C. J. Pethick, Phys. Rev.  Lett. 3,  978 (1972). 'B.  D.  Day,  Phys. Rev. 136,  B1594 (1964). 3~.  Bonche,  S. Koonin,  and J. W.  Negele,  Phys. Rev. C 13,  1226 (1976). 4~.  Khakpour,  Ph.D.  thesis, Massachusetts Institute of  Technology, 1975 (unpublished). 5~.  W.  L. Sprung and P. K. Banerjee, Nucl.  Phys. A168,  273 (1971). 6~.  W.  Negele,  Phys. Rev. C I,  1260 (1970).  See also X.  Campi and D.  W, L. Sprung, Nucl. Phys. g,  401 (1972). 7~.  A.  Thorpe and D. J. Thouless, Nucl.  Phys. M, 225 (1970). 8~.  A.  Broglia,  A. Molinari,  and T. Regge, Ann.  Phys. (NY) E, 289 (1976).  See also T. Regge,  Low Temp. Phys. 9,  123 (1972). 9~.  A. Bethe,  Phys. Rev. 167,  879 (1968). 'OJ.  W.  Negele,  in Proceedings of the Saclay Meeting on Electron Scattering at Intermediate Energy, Centre d'~tudes  Nucleaires de Saclay, France, 1975  (unpub- lished), and Massachusetts Institute of  Technology Center for Theoretical Physics Report No.  CTP-505, 1975 (unpublished). Collective Sideward Flow of Nuclear Matter in Violent High-Energy Heavy-Ion Collisions Horst Stöcker Gesellschaft für Schwerionenforschung, D-61 00 Darmstadt, Germany and Jouchim A. Maruhn and Walter Greiner Institut für  Theoretische Physik  der Universität Frankfurt, 0-6000 Frankjkrt um Main,  Germany (Received 11 December 1979) Angular and energy distributions of fragments emitted from fast nucleus-nucleus  col- lisions (Ne-  U at 250,  400,  and 800 MeV/N) are  calculated with use of nuclear fluid dy- namics.  A characteristic dependence of the energy spectra and angular distributions on the impact parameter is predicted.  The preferential sideward emission of reaction fragments observed in the calculation for nearly central collisions seems  to be supported by recent experimental data. Sideward emission of  nuclear matter was one  mentum transfer.  Quite early experiments2# sup- of  the first predictions of  fluid-dynamical  model  ported this hypothesis:  Preferential sideward calculations of  fast nuclear collisions,  implying  emission of  (mainly) a, particles was observed in considerable transformation of  the incident kinet-  the irradiation of  Cr-sr particle-track detectors ic energy into compression and heat energ~."~  with light nuclei at E l,b  = 0.2-4.2  GeV/N  when se- The outflow of  matter from the highly compressed  lecting nearly central collisions (only azimuthally "shock"  region reflects a large transverse-mo-  symmetric many-pronged  stars).  This was inter- C3  1980 The American Physical Society  725 VOLUME  44, NUMBER  11  PHYSICAL RE  VIEW LETTERS  17 MARCH  I980 preted as  being due to a strong,  sideward-travel- ing compression wave,  called the Mach shock wave  .2*  Additional indications for quasihydro- dynamical behavior in fast nuclear collisions seem to be found in various other recent experi- ment~.~- To investigate some of  these phenomena quan- titatively, we performed three-dimensional  non- relativistic fluid-dynamical  calculations of  the collisions of  neon projectiles with uranium nuclei at various bombarding energies, a system exten- sively studied experimentally.  The Euler equa- tions were integrated n~merically.~'  '  They en- Sure the conservation of  particle number,  mo- mentum,  and energy.  The important input to these equations is the nuclear equation of  state, i.e.,  the functional dependence of  the internai en- ergy W@,  T) =E  -E 1, and the pressure p  =p2(a  W/ ap),  (at constant entropy s) on the local density p and temperature T.  W@,  T) is split into the compression energy E,@, T = 0) and the thermal energy E,@, T).  For E,@) a parabolic expansion around the ground-state density with a compres- sion constant K = 200 MeV and for  E .(P  ,T) a Fer- mi-gas Ansatz  is used."  The Coulomb and Yuka~va  potentials are treated explicitly.'  They allow for a realistic treatment of  the binding and surface energies of  the nuclei.  Ho~vever,  it must be emphasized that the reaction dynamics is prac- tically not influenced by the Coulomb and Yukawa f orces at high collision energies (E l,t,-  400 M~V/ N):  We find that they are only important for the stability of  the incident nuclei.  The evolution of the system is  followed in time and the energy spectra and angular distributions are calculated by integrating the matter flux over all space. The computation is  stopped at a late stage when the compressed matter has expanded so that the maximum density is  less  than P,-  0.5~~  where thermal contact between the nucleons becomes questionable because of their large distances. The final spectra and angular distributions were checked to be independent of the exact value of  p,. Figure 1 shows snapshots of  the density and temperature contours and of  the velocity field in the scattering plane in a late stage of  the reac- tion Ne -  U at E l,b=  400 MeV/N  for kvo different impact parameters b =O and 6 fm.  For head-on collisions [~ig.  l(a)]  two zones can be distin- guished:  Near the collision axis, where the pro- jectile directly hits the target, a strongly com- pressed, highly excited head shock is  formed. In contrast to the oversimplified geometrical "fireball"  models, the head shock pushes matter 400 MeVln b=6  frn __--  : ,/  H/--  * ,-T  Y  #/ / .....  Y N ;,---~--  P..: T >?/1(i  Mo\/ FIG. 1.  Density and temperature contours and veloc- ity field (arrows) in the scattering plane at a late stage of the reaction.  (a)  Central collision;  (b) Intermediate impact paramet  er. sidewards, thus initiating a strong,  sidetvard- traveling compression wave (called shock2). The small velocities everywhere are remarkable: No projectilelike fragments are observed; the for- ward momentum is transferred to the combined target-plus-projectile  matter.  The angular dis- tribution of  emitted fragments shown in Fig. 2(a) is sidewards peaked at the Mach-shock-wave  an- gle OMS=  70'.  Most of  the contribution to the peak arises from particles with small kinetic energies E kin  = 201 10 MeV (dashed line) and these also dominate the energy distribution [I?ig.  2(b)]. Practically no fast particles can be Seen.  It should be  stressed, though,  that the locations of the maxima in the angular distributions and ener- gy spectra are  not well determined because of VOLUME  44, NUMBER  11  PHYSICAL REV  IEW LETTERS  17 MARCH  1980 400 MeVln b=O  mox b=8 )  ö ............ i bifml  .,<TI-'---,]  .  .......- b  Ifml  (b) l FIG.  2.  Impact-parameter  dependence of  (a)  angular distribution (inset:  position of maximum), and  (b) en- ergy spectra of  reaction products. 500 the rather poor statistics characteristic of  hydro- dynamic-type calc~lations.~*  Also,  only the sharp peak indo/dO  of  Fig. 2(a) at b =O corre- sponds to a peak in do/dSt,  whereas in the double- differential cross section d20/dSZ dE peaks appear even for b # 0, as  is  the case f or the experimen- tally observed peaks in Ref. 9. In nearly central collisions the maximum of the cross section is  still located at small for- ward and small perpendicular momenta P/M  50.2 close to the target rapidity.  Forward emission is suppressed.  For collisions of  equal nuclei the sideward emission is seen at fonvard angles in the laboratory system though mainly with large kinetic energies (above the center-of-mass  ener- gy).=  In Fig. l(b) the analogous situation is shown for b =6  fm.  Here two nuclear residues are kicked apart by the highly compressed head shock Zone ("bounceoff"  effect6).  The velocity field shows that the target residue slowly moves to QIab  = 90'  and is expected to fission because of  its large def ormation and angular momentum.  The -  400 MeVln J b=O -  : projectile residue is deflected to forward angles = 30") and is azimuthally anticorrelated with the target residue. The two kinds of  events just presented-the Mach shock wave for nearly central collisions and bounceoff  for the intermediate impact param- eterscan  be distinguished by  their characteris- tic dependence of  the calculated fragment angular distributions and energy spectra on the impact parameter [see Figs. 2(a) and 2(b)l.  For increas- ing impact parameter ,  the angular distributions broaden.  The maxima of  da/d8  corresponding to the deflected projectile fragments shift to small- er angles [see the inset to Fig. 2(a)].  The con- tribution of  the high-energy tails of  the spectra (up to the incident energy) increases with b.  This increase is partly due to the deflected projectile residue, and partly to the exploding head-shock matter.  Low-energy particles with E hs  5 M~V/ N  coming mostly from the target residue domi- nate for intermediate b.  For b = 8 fm, two dis- tinct contributions are observed in the matter distribution,  one arising from the bounced-off projectilelike fragments at large transverse and forward momenta,  the second bigger one from the residual target at low parallel and perpendic- ular momentum (i.e.,  near the target9s  initial position). For b =4  and 6 fm, the double-differential cross sections show an important feature (see Fig. 3): There are maxima in the angular dis- tributions that move to  f orward angles with in- creasing fragment kinetic energy [see Fig. s(~)]. At  b S 4 fm, the broad low-energy backward peak caused by the shock wave is much higher than the narrow forward peak in the medium kinetic-en- ergy windows. Experimental evidence for the bounce-off  ef- fect in intermediate-impact-parameter  colli- sions was foundg:  In Ne-  Au  reactions slow heavy-target residues, moving to 90" in the lab- oratory system, were detected in coincidence with 180" azimuthally correlated jets of  fast, light particles, which can be attributed to the bounced-off  projectile matter.  To investigate the nearly central collisions,  Stocket al. select- ed events with high multiplicities in Ne -  U col- lisions.1°  For an estimated mean impact parame- ter of  b=  4 fm, they find maxima in the proton angular distributions,  The low-energy particles show a broad 90"-peaked  distribution, which be- Comes narrower and shifts to forward angles with increasing kinetic energy, in qualitative agreement with the results presented here.  One ............  b-2  - b  =4  -----  - b  = 6 - b=8 P 1 100  2 00  300  400  E kln  i~e~lnl VOLUME  44, NUMBER  11  PHYSICAL REVIEW LETTERS  17 MARCH  1980 FIG. 3.  Double-differential Cross section, plotted as  angular distributions for different fragment lrinetic energies, for 4-  and 6-fm  impact parameters.  The right-hand-side  plot  shows the position of the maximum as a function of the laboratory angle. interesting possibility is the search for super- dense abnormal states of  nuclear matter via ir- regularities in the excitation function of  the Mach-shock-wave  angle5  OMS (E  ab) and of  other measurable quantities (pion production rate, temperature,  et~.).~ In conclusion we have discussed strong-com- pression phenomena in violent nucleus-nucleus collisions in nuclear fluid dynamics.  We find predominant sideward emission of  matter for small impact parameters.  The characteristic features of  the recent experimental data seems to support the theoretical results.  Additional theoretical and experimental efforts are neces- sary, however,  to extract conclusive information on the properties of  dense nuclear matter from these phenomena. 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Collective sideward flow of nuclear matter in violent high-energy heavy-ion collisions

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