The entropy of hot nuclear systems is deduced from the mass distribution of fragments emitted from high energy proton- and nucleus-induced reactions via a quantum statistical model. It is found that the entropy per baryon, S/A, of intermediate rapidity ("participant") fragments is higher than the entropy of target rapidity ("spectator") fragments. The spectator fragments exhibit S/A values of &#8773; 1.8 independent of the projectile energy from 30 MeV/nucleon up to 350 GeV. This value of the entropy coincides with the entropy at which nuclear matter becomes unbound

Jacak, Barbara V.

Stöcker, Horst

Westfall, Gary D.

English

Hochschulschriftenserver - Universität Frankfurt am Main

PHYSICAL REVIEW C  VOLUME 29, NUMBER 5  MAY 1984 Rapidity dependence of entropy production in proton- and nucleus-induced reactions on heavy nuclei B.  V. Jacak, H. Stöcker, and G. D. Westfall National Superconducting  Cyclotron Laboratory, Michigan Stute Uniuersity, East Lansing, Michigan 48824 (Received 13 December 1983) The entropy of  hot nuclear Systems is deduced from the mass distribution of  fragments emitted from high energy proton- and nucleus-induced reactions via a quantum statistical model.  It is found that the entropy per baryon, S/A,  of  intermediate rapidity ("participant") fragments is higher than the entropy of  target rapidity ("spectator") fragments.  The spectator fragments exhibit S/A  values of  s  1.8 independent of  the projectile energy from 30 MeV/nucleon  up to 350 GeV.  This value of the entropy coincides with the entropy at which nuclear matter becomes unbound. The  study  of  the  entropy  produced  in  proton-  and heavy ion-induced reactions on heavy nuclei can provide a method  of  studying  the  collective  behavior  of  nuclear matter.'-4  ~~drod~narnical~~~  and  intranuclear  cascade calculations7  have  demonstrated  that  the  entropy  per baryon, S/A, created in high energy nucleus-nucleus col- lisions remains constant during the expansion of  the sys- tem.  Information  concerning  the  entropy  produced  in these  reactions  can  be  obtained by  comparing the frag- ment production cross sections with a quantum statistical model.  Thus fragments observed experimentally can car- ry information concerning not only the late stages of the reaction  but  also the hot, compressed stage of  the reac- tion. In high energy proton-nucleus interactions, the target is excited and decays by  emitting nucleons and nuclear frag- ments as well as fissioning and emitting y rays.  In central collisions of  high  energy  heavy  ions, a  compressed and highly excited interaction region4,8'9  is formed which sub- sequently expands and ~001s.~  The system expands until the constituents  are no longer interacting.  At  this  time the momentum  distributions and  particle  concentrations are frozen out.  Particles emitted from this system appear predominantly at rapidities intermediate between those of the projectile and target,  while  fragments from proton- nucleus and peripheral nucleus-nucleus reactions are emit- ted at rapidities  close to the projectile and target  rapidi- ties. The  entropy  created  in  high  energy  proton-  and nucleus-induced reactions  can be  extracted  from  the re- sulting  fragment  production  cross  sections  using  the quantum  statistical model  as  proposed  in  Ref.  3.  This method  was used  in Ref. 4 to determine the entropy via the  intermediate  rapidity  fragments  11  A 114 emitted from high  energy  nucleus-nucleus reactions.  The quan- tum statistical model assumes a system of N neiitrons and Z Protons in thermal and chemical equilibrium at a given entropy  per  baryon  S/A  and  nuclear  density, p, which then specifies the temperature T.  Included in the calcula- tion are nucleons, pions, delta resonances, stable and y- unstable nuclei up to A =20,  and the known particle un- stable nuclear states up to A =  10.  The predictions of this rnodel become  less  reliable above  A =  10, thus  compar- isons with data are done only up to A =  14.  The N and Z of  the initial system are chosen to be those of  the target nucleus  for  proton-induced  and  peripheral  nucleus- induced  reactions.  For  nucleus-nucleus  reactions  the overlapping volume of  the two nuclei at the impact pa- rameter  with  the  most  weight is used.  Interactions are taken  into account  via the excluded volume of  the frag- ments.  The S/A  for a given beam, target, and incident energy is extracted using a least Squares fit to the observed fragment distributions.  The fragment distributions calcu- lated  for  a  given  S/A  are  insensitive  to the  assumed breakup density for 0. 3po 5  p 1  0.  7po.  3'4 A large body of data exists concerning high energy pro- ton  induced  reactions  on  heavy  nuclei.  We  concentrate here on fragmentation or spallation of  target nuclei lead- ing  to the emission of  fragments  with  1 5  A 540.  ''-I2 These data are complernented with measurements of  tar- get rapidity fragments from high energy nucleus-induced reacti~ns.~"~-'~  Rad  iochemical measurements will not be discussed here because they generally report data for frag- ments  with  A 2 40.  In Fig.  1 relative production  cross sections are shown for target rapidity  from the reactions 400 MeV/nucleon Ne +  u,~  2.1 GeV/nucleon  Ne +   AU,'^ 480  MeV  p +  Ag,12  and  80-350  GeV  p +  ~e."  The solid lines in Fig.  1 represent the quantum statistical cal- culation  described  above  carried  out  at  a  density  of p =  0. 3po.  Although data were measured up to A =  30 for the p +  Xe case, only the cross sections that can be com- pared  with  the present model are shown.  The fits agree well with the observed fragment yields for all four cases, except for Z =2 fragments from 480 MeV p +  Ag.  The entropies obtained from the fits to these fragment distri- butions  as well  as those extracted  from the reactions of 30  MeV/nucleon  C +   AU,'^  55 -  110  MeV/nucleon C +  Ag,15  250  MeV/nucleon  Ne +   AU,'^  2100 MeV/nucleon  Ne +  U,' and 4.9 GeV p +  Ag and U (Ref. 11) are shown in Fig. 2(a) as a function of  the projectile kinetic energyhucleon.  The depicted errors reflect the er- rors from the fit as well as known systematic errors in the data.  The  fits  generally  encompass  fragments  with 3525  10  and  appear  to  be  independent  of  both projectile-type and energy.  The average value for S/A  in these cases is 1.84k0.16.  Not shown in this figure is S/A 29  1744  -  @ 1984 The American Physical Society 29  -  RAPIDITY DEPENDENCE OF ENTROPY PRODUCTION IN . . .  1745 FRAGMENT  Z 107  80-350 GeV p + Xe  -  -  -I  106 ;? W 7  105- IO-~~~  103A5c3iLL~3ecI! FRAGMENT  Z  FRAGMENT  A FIG. 1.  Target rapidity fragments from the reactions of  (a) 400 MeV/nucleon  Ne +  U (Ref. 9), (b) 480  MeV  p +  Ag  (Ref. 121,  (C)  2.1  GeV/nucleon  Ne +  Au  (Ref. 14), and  (d) 80-350 GeV  p +  Xe  (Ref. 10). The histograms  represent fits using  a quantum statistical calculation as described in the text. extracted  from 80-350  GeV p +  Xe (Ref. 10) which is 1.46I0.67.  This constant value of  1.84 for the extracted S/A  coincides with  the entropy  necessary  to excite the target nuclei to their binding energy. Shown in Fig. 2(b) are the extracted entropies for inter- mediate rapidity fragments (1  <  A < 14) from the reaction of  Ar +  Au  at  42,  92,  and  137  MeV/nucleon.  These values are higher  than those extracted from target frag- ments  and increase somewhat with  bombarding  energy.4 No  data  exist for intermediate rapidity  fragments  with A  6 at energies above 137 MeV/nucleon.  The data of Ref. 9 do not  extend to intermediate  rapidities for frag- ments heavier than 4~e. One may treat these fragments by assuming production in a fireball created by the overlapping volume of the pro- jectile nucleus with the target nucleus.  The excitation of the fireball can be calculated taking into account the slow- ing from Coulomb repulsion between the two nuclei and the resulting  entropy  can be  calculated  using the Fermi gas model.  These calculations are shown in Fig. 2(b) as solid lines for three densities, p= l.Opo, 2.0po, and 3.0~~. The maximum density of the fireball system will increase with beam energy and a more sophisticated model such as hydrodynamics would incorporate this effect. The  extracted  entropies  from  intermediate  rapidity fragments with  1 I  A I  3 and  1 I  A 14  from high energy nucleus-nucleus  reactions  are  also  shown  in  Fig.  2(b). Typical  fits to intermediate  rapidity  light  particles  only are shown  in  Fig.  3.  In Fig.  3(a) light fragments with 1  A 14  from the reaction of  137 MeV/nucleon Ar +  Au (Ref. 4) are fitted,  while  fragments  with  1<  A 14  and 1 <  A j 3 are fitted in Figs. 3(b) and 3(c),  respectively, for the sytsem 393 MeV/nucleon  Ne + u.'~  Additional  sys- tems reported in Fig. 2(b) are 35 MeV/nucleon C +   AU,'^ 42,  92,  and  137 MeV/nucleon  Ar +  AU:  100 and  156 HEAYI FRAGMENTS  1  1 nA=  1-14 *A= 1-4 OA=I-3 MlDRAPlDlTY  FRAGMENTS FIG. 2. Extracted entropy per baryon (S/A) using a quantum statistical model  as  described  in  the text.  (a) Target rapidity fragments from  the reactions 480  MeV  p +  Ag  (Ref. 121,  2.1 GeV/nucleon  Ne + Au  (Ref.  141,  30  MeV/nucleon  C +  Au (Ref. 13), 55-  110  MeV/nucleon  C +  Ag  (Ref. 13, 250  and 2100  MeV/nucleon  Ne+Au  (Ref.  14),  400  and  2100 MeV/nucleon  Ne +  U  (Ref. 9), and  4.9  GeV  p +  Ag  and  U (Ref.  11); (b) midrapidity  fragments  from  the  reactions  35 MeV/nucleon  C +  Au  (Ref. 161, 42, 92, and  137 MeV/nucleon Ar +  Au  (Ref. 4), 100 and  156 MeV/nucleon  Ne + Au  (Ref. 81, 241  and  393  MeV/nucleon  Ne+ U  (Ref.  171,  and  2.1 GeV/nucleon  Ne + Pb  (Ref. 18).  The dashed  and  solid  lines represent the weighted average for fragments with 1  5  A 53  and 11  A 14,  respectively.  The  dot-dash line  represents  a power law fit for fragments with  1 5  A 1  14.  The three grouped solid lines  represent  the  entropy calculated  using  fireball geometry and  a Fermi  gas  model  at three different densities, p=l.Opo, 2. Opo, and 3.0~~. MeV/nucleon  Ne +   AU,^  241  and  393  MeV/nucleon Ne +  u,'~  and 2.1 GeV/nucleon  Ne +  ~b."  The average value of  S/A  for fragments with  1  A  3 and  1 5 A  4 are 4.2410.32  and  3.6010.12,  respectively, independent of  the  incident  energy  and  projectile  nucleus.  These values can be  compared  to the entropies extracted  from Proton and deuteron cross sections using a model in which the system is assumed to be a dilute gas of  nucleons and deuterons' where an S/A of  =4.7  was deduced. The difference between the entropy extracted using the same quantum statistical model compared to light particle cross  sections  and  cross  sections  for  fragments  with 1  A  4 appears to be a paradox because these fragments seem  to have  common  origins.  The apparent  tempera- tures of  these  fragments  are similar to each  other at a 'l 0- I  4  I  '18 I; I;' Frogment A 393 MeV/n  Ne + U C  a: ioO - V) U) 10-~  '  '  '  '  '  02460246 Frogment  A FIG.  3.  Intermediate  rapidity  fragments  from  137 MeV/nucleon  Ar +  Au  (Ref. 4) and 393 MeV/nucleon  Ne +  U (Ref. 9). The solid lines represent fits using a quantum statisti- cal  model  as  described  in  the  text  where  fragments  with 1  5  A 5  4 are fitted in (a)  and (b)  while in (C) only fragments with 1  A 5  3 are fitted. given bombarding energy, while the extracted source ve- locities  vary  from 0.5  times  the projectile  velocity  for A 5 3 to 0.3 times the projectile  velocity for the heavier fragments.  However,  the present  light particle inclusive data are dominated  by  more peripheral  collisions  where the small number of nucleons contained in the interaction volume between the two nuclei exclude the formation of heavier emitted fragments.  The macrocanonical approach inherent in the quantum statistical model requires many particles in the System and is inapplicable for peripheral collisions.  In contrast, the thermodynamic limit  is ap- proached for near-central collisions2 which is where medi- um mass fragments are produced.14 It is clear that to ex- tract the entropy from high energy nucleus-nucleus reac- tions it is essential to treat complex fragments as well as light particles. The distributions of fragment masses from high energy proton- and nucleus-induced reactions have been previous- ly interpreted in terms of a power law dependence on the mass number of  the emitted fragment10 which would be indicative of a liquid-vapor phase transiti~n~~'~~'~~'~~'~-~~. It is clear from the present work and other ~ork~~>~~  that the mass yields can be described without the need to in- voke a liquid-vapor phase transition. In  Summary,  the  entropy  produced  in  high  energy proton- and nucleus-induced  reactions on heavy nuclei is extracted using a quantum statistical model for the emis- sion of complex fragments.  The S/A extracted from tar- getlike fragments is 1.84110.16 independent of beam ener- gy, projectile, and target.  For intermediate rapidity com- plex fragments from nucleus-nucleus reactions, S/A tends to  increase  with  bombarding  energy  from  2.2  at  42 MeV/nucleon  to 2.4 at 137 MeV/nucleon.  However, the lack  of  intermediate rapidity  complex fragment data at higher energies hampers a definite conclusion concerning the energy dependence of  S/A  from these Systems.  En- tropie~  extracted for intermediate rapidity  light particles appear to be independent of beam energy, projectile, and target mass and are in the range of S/A =  3 -  5.  This ef- fect could be due to the fact that inclusive data were used to extract the entropy.  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Rapidity dependence of entropy production in proton- and nucleus-induced reactions on heavy nuclei

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