It is shown that nuclear matter is compressed during the encounter of heavy ions. If the relative velocity of the nuclei is larger than the velocity of first sound in nuclear matter (compression sound for isospin T=0), nuclear shock waves occur. They lead to densities which are 3-5 times higher than the nuclear equilibrium density &#961;0, depending on the energy of the nuclei. The implications of this phenomenon are discussed

Greiner, Walter

Müller, Hans

Scheid, Werner

English

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VOLUME  32, NCMBER  13  PHYSICAL REVIEW LETTERS  1 APRIL  1974 or due to a characteristic of  the mechanism in- dependent of  this set of  values. After extrapolating (d~/di2)~„  at larger angles, as indicated by the dashed line in Fig. 3,  the in- tegrated cross section for the production of quasi-Kr  is about 500 mb.  It is a large part of the reaction cross section (calculated value 870 mb),  and this seems to confirm the idea that the quasifission reactions occur instead of  the com- plete-fusion  reactions.  One should remember, however,  that the method used for the angular distribution is based on a hypothesis which is reasonable but not entirely proved for angles far away from 553. More work-with more intense beams if pos- sible-has  to be done in order to get more de- tailed information on the features of  the quasi- fission reactions induced by Kr ions.  Never- theless we hope the indications we have obtained will stimulate theoretical work which is now un- derway for understanding the reaction mecha- nism.  In particular,  the angular distribution should help in deciding among the various pic- tures for the potential energy between the two nuclei which can be considered in order to ex- plain the quasifission reactions." We thank N.  Rowley for discussions and for reading the manuscript. Note added.*e  have recently made angular- distribution measurements at large angles.  The maximum at 65" has been confirmed but the de- scent above 80" is not as steep as the dashed curve on Fig.  3:  At  120" the cross section has decreased down to 4 mb,/sr. *Chercheur, Institut Interuniversitaire des Sciences Nucleaire,  Universite Libre de Bruxelles,  Brussels, Belgium. tD6tach6 de l'Universit6 de Clermont-Ferrand,  Cler- mont-Ferraqi,  France. 'J. Galin,  D. Guerreau,  hl. Lefort,  J. Peter,  X. Tar- rago,  and  R. Basile,  Nucl. Phys. M,  461 (1970). 'A.  G. Artukh,  G.  F. Gridnev,  V. L. hfikheev, V. V. Volkov,  and J. Wilczynski,  Joint Institute for Nuclear Research,  Dubna,  Report No.  E7-6970,  1973 (to be published) . 3~.  G.  Moretto,  D. Heunemann, R. C. Jared,  R. C. Gatti,  and  S. G. Thompson,  International Atomic En- ergy Agency Symposium on  Fission,  Rochester, New York,  1973 (to be published),  Paper No.  ~~174/75. 4~.  Peter,  F. Hanappe,  C. Ng6,  and  B.  Tamain, Institut de Physique Nucleaire,  Orsay,  Report No. IPNO-RC-73-07,  1973 (to be published) . 5~.  Tamain,  M.  Lefort,  C. ~g6,  and J. Peter, J. Phys. (Paris) 33,  Colloq. G,  Vol. 2,  50  (1972). 'A.  J. Sierk and J.  R. Nix,  International Atomic En- ergy Agency Symposium on Fission,  Rochester,  New York,  1973 (to  be published),  Paper No.  SM174/74. 'M.  Lefort,  C. Ng6,  J. Peter,  and B. Tamain,  Nucl. Phys. M,  166 (1973). 'K.  Dietrich,  private communication. Nuclear Shock Waves in Heavy-Ion Collisions Werner Scheid,  Hans Müller,  and Walter Greiner kstitut für  Theoretische Physik der Universität Frankfurt,  Frankfurt arn Main,  Germany (Received 19 November 1973) It is shown that nuclear matter is compressed during the encounter of  heavy ions.  If  the relative velocity of  the nuclei is larger than the velocity of  first sound in nuclear matter (compression sound for isospin T =O),  nuclear shock waves occur.  They lead to densities which are 3-5  times higher than the nuclear equilibrium density P,,,  depending on the ener- gy  of  the nuclei.  The implications of  this phenomenon are discussed. The possibility of  compression of  nuclear mat- ter in nucleus-nucleus collisions in one of  the most interesting aspects of  heavy-ion  physics. It has been discussed earlier in connection with the sudden nucleus-nucleus potentialsl and their energy dependen~e,~  which seems to confirm the experimentally deduced heavy-ion potentials of the Yale gro~p.~  These earlier considerations are valid as long as  the relative heavy-ion velo- city U, does not exceed the velocity of  first sound in nuclear matter C,;  i.e.,  for V,  < C,.  Here the first sound is an isospin T = 0 compression wave while the second sound describes an isospin T = 1 wave where a proton-neutron separation travels in constant nuclear matter density p, = p,  +P,.~ In order to study local compression effects for VOLUME  32, NUMBER  13  PHYSICAL REVIEW LETTERS  1 APRIL  1974 the cases V,  > C,  in head-on nucleus-nucleus col- lisions,  we propose the following very simplified model: We restrict ourselves to two identical nu- clei whose volume is divided into three parts [see Fig. 1  (a)], namely an ellipsoid with axes a and b sandwiched between two cut-off  spheres with ra- dius R and relative distance r.  The system will clearly be rotationally symmetric around the z axis,  connecting the centers of  the two spheres and of  the ellipsoid.  The nuclear matter is as- sumed to be homogeneously distributed over the different volumes.  Compression (p >P,) should only occur in the ellipsoidal region.  The four coordinates a, b,  Y, and R define the geometry of  the system.  We choose,  however,  R = const in the collision so that we are dealing with three degrees of  freedom only.  This approximation is rather inessential for the results, but simplifies the theoretical considerations considerably. If  the density in the cut-off  spheres is  the equi- librium density po, the density in the ellipsoidal region follows from matter conservation as where a = (s -  r)/2R and ß = s/2a,  with s the dis- tance between the intersection points of  the two spheres with the ellipsoid [see Fig. l(a)].  The various types of  shapes shown in Fig. l(b)  are possible, depending on the relative size of  the ellipsoid and the spheres; they can be charac- terized by the quantity 6 = b2 -  R2  + ($Y)~.  (2 We will later exclude the case 6 < 0,  because it corresponds to an ellipsoid with compressed matter embedded completely within the two spheres,  which is unphysical.  The distance s is given by for 6 > 0,  where E = b2/a2 -  1, and s  0  for 6 < 0. Because the solution of  the full hydrodynamical problem is too complicated,  we  make the follow- ing approximations for the calculation of  the ki- netic energy E = $~/~~v~d~:  (I)  The velocity field of the matter in the cut-off  spheres is homoge- neous and equal to the velocity of  the respective centers,  i.e.,  -.  I  .-. v=*zY~~.  (3) FIG. 1.  (a)  Geometrie parameters of  the model. (b)  Two  cases 6 >O  and  6 < 0.  The unphysicai situation 6 <O is excluded by  forces of  constraints. (11) The velocity field in the ellipsoid is irrota- tional and can thus be derived from a potential through the equations The boundary condition V = b and  ez  = 0 along the circle z = 0,  2  + 9  = b2 gives for the solution of  (4) which allows with (3) the calculation of  the kinet- ic energy in the form where qi  ={Y,  a,  b)  . If  the Coulomb energy is disregarded, the po- tential energy is given by the compression ener- gy4 and the heat energy of  the Fermi gas5 in the ellipsodial regi~n,~  i.e., =SwPd7=  ~(~~)p,~~ah~ = (C  /2P0)(~,  -   PO)^^ i: aab2,  (7 ) with 2  -2j3  W)  = (~/2pg)(  P -  pJ2  + +PT  P 9 T being the temperature in the compressed re- gion.  To avoid unphysical shapes with  0 < 0,  a VOLUME  32, NCMBER  13  PHYSICAL REVIEW LETTERS  1 APRIL  1974 potential of  constraints SvO6' is added for 6  < 0 to the potential energy.  It forces the system to shapes with 6 >0.  The strength V,  has negligible influence on the results. Since we have introduced regions of  different densities p,  and p,,  the following boundary condi- tions along the surface between p,  and p,  have to hold7: The brackets [lj  denote the jump of  the quantity J  across the surface, i.e.,  [f]a  f,  -  j,;  ii is the unit vector on the surface, and U =?  -5  -G, where G  is the speed of  advance of  the surface.  The pressurep is related to the density of  the poten- tial energy according to (7): From Eqs. (8)  it follows that Restricting ourselves, for simplicity, to the case that p does not deviate too much from po, we dis- regard the temperature dependence.  Further- more,  because the velocity field is already given as a function of  the coordinates a,  b,  and r, the above Eqs. (8) connect the velocities d,  b, and +. For simplicity we only fulfill these equations at the points z = i  a,  x = y = 0  and obtain where C,  = (c/M)"'  is the velocity of  first sound, with M the nucleon mass.  This indicates that the coordinate a is not an independent degree of  free- dom in this model. The total energy of the system, has to be constant in time,  i.e.,  dE/dt=O.  The solutions of  (11)  have to fulfill the additional con- dition (10),  which serves as  an equation of  mo- tion.  Introducing the momenta  fik =C,  mkiqii  the three equations of  motion to be solved are with The calculations are carried out for '60-'60 scattering.  The two parameters of  the model are chosen as  C =Y  MeV,  which corresponds to a compressibility K = 100 MeV,  and p,  = 0.138  fmm3 (see Ref. 2).  The initial conditions for 1 =0  are 3- = 2(R -AR) with AR = 2, R,  p, = 2p0,  11  = (zRAR)'",  .  I 11  = -  z(R  /1,)?1.  The initial compression p, = 2p,  in the surface Zone does not influence the results, as can be Seen from pl(t) in  Fig.  2.  The initial conditions for all other coordinates rc,  P, and ci are determined by Eqs.  (I), (lo), and (11). Figure 2 shows the results for the time depen- dence of T.  (I,  1, T ,.,.,  and P,.  Here T,., means the distance between centers of  mass of the two halves of the nuclear system [see Fig, l(a)].  Clearly,  the model can only be used as long as  SY+R  >U is fulfilled.  Obviously the radi- FIG. 2.  Parameters 7,  a,  6,  T,,,;.,  and  p $/P, as a function of time for energies E,„.  = 1, 10, 100, and 1000 MeV. VOLUME  32, NUMBER  13  PHYSICAL REVIEW LETTERS  1 APRIL  1974 FIG. 3.  Potential energy as a function of  the center- of-mass distance.  It is strongly energy-dependent and shows a strong soft core, particularly for high ener- gies. us a(t) increases with nearly constant velocity, in contrast to b(t) which increases faster for high- er  energies.  This means that matter is pushed outwards perpendicular to the relative motion of the two nuclei.  Note that for small energies the density p,  is first decreasing rapidly and then again increasing slowly at larger times.  The fast decrease in the beginning is the adjustment of  the compressed nuclear density from the ini- tial condition p,  = 2p0 to the final density.  There- fore the results are independent of  the initial con- ditions. Already for a bombarding energy of  E,.,.  = 100 MeV,  which corresponds to a laboratory energy of Elah  /A  = 12.5 MeV/nucleon  of  the ''0  projectile, a density compression of  p, /C,  = 2 results.  This indicates a nuclear matter shock wave,  which be- Comes even more pronounced (p, = fjp,)  for very high nucleus-nucleus energies (3  100 MeV/nu- cleon).  In this latter case it is necessary to in- vestigate the relativistic hydrodynamics.  Figure 3 shows the potential energy, which is stored in the compressed matter of  the ellipsoid,  as  a func- tion of  Y,.,.  .  It obviously leads to a strong ener- gy-dependent  soft-core potential.  The soft core reaches the magnitude of  a few MeV for de  ions of  10 MeV to several hundred MeV for  ions of  1000 MeV. The strongly compressed nuclear system shows up experimentally in the strong energy-dependent soft-core potential for nucleus-nucleus scatter- ing.  It should be observable in the backward di- rection of  elastic scattering.  This elastic Cross section is expected to be very small because of the large number of inelastic channels available as a result of the highly compressed matter.  The systematic investigation of  the elastic and inelas- tic excitation functions in nucleus-nucleus colli- sion should,  however,  give information on the nuclear compressibility.  The time dependence of  a(t) and b(t) (Fig. 2) indicates that the shock wave is expanding faster along the beam direc- tion [a(t) grows faster than b(t)].  Therefore the fragments are expected to peak forward and back- ward in the c.m.  system,  i.e.,  forward in the lab system. It should be realized that the results are  valid for all nuclei and not only for '80-'60  collisions, i.e.,  in particular also for Pb-Pb or U-U  colli- sions at high energies.  In these latter cases one can also expect strong nuclear shock waves with p,= (3-5)p,.  If  matter is compressed to such an extent,  these systems can be overcritical for the pionic field in analogy to similar phenomena for the electron-positron fields in superheavy inter- mediate molecules8:  Because of  the pion-nucleus interaction the binding energy of  bound  pion-atom- ic states can become larger than 2m,c2  ("diving of  pionic bound  states"),  and n+-n  -  pairs or n0 pairs may be produced without cost of  energy. Since pions are bosons,  such pairs can be cre- ated until either a repulsive pion-pion  interaction (V'  term) or  a "pionic  Pauli principle"  can pre- vent further production.  A "pionic  Pauli princi- ple"  can occur, for exampfe,  if  the pions are para-particles;  i.e.,  they obey,  for example, para-Fermi statistics, which would  stabilize the pion vacuum.  Depending on the degree of  such a para-statistics,  the energy in the nuclear matter of  the highly compressed overcritical system will condense into pionic matter.  It is not yet clear whether such mixed pionic-nuclear  matter can be stable as, e.g.,  a pionic drop (new vacuum state for hadronic matter) or whether it will dis- sipate into many pions (multiple pion production, pionic showers).  In any case, these new "de- grees of  freedom" for overcritical nuclear mat- ter seem to give not only a clear indication for the intermediate highly compressed nuclear mat- ter, but also new,  exciting prospects for high- energy heavy-ion  physics,  in that fundamental PHYSICAL REVIE\Y7 LETTERS properties of  hadronic matter can eventually be tested. *Work supported by the Bundeministerium für For- schung und  Technologie and by the Gesellschaft für Schwerionenforschung (GSQ  . 'K.  Pruess and W.  Greiner, Phys. Lett. s,  197 (1970); W. Greiner and W. Scheid,  Suppl. J.  Phys. (Paris) 2,  C6-91 (1971). 2~.  Müller,  W.  Scheid,  and W. Greiner, to be pub- lished. 3~.  A. Bromley,  in Pvoceedings of  the International Conference on Nuclear Reactions  Induced by  Heauy Ions, Heidelberg,  Gemany, 1969, edited by R. Bock and W.  R. Herring (North-Holland,  Amsterdam,  1970), P. 27. "uclear  hydrodynamics has been discussed and justi- fied by hl. Danos,  University of Maryland Technical Re- port No. 221,  1961 (unpublished);  SV.  Wild,  Sitzungsber. Math.-Naturwiss.  K1.  Bayer. Akad. Wiss. München 1955, 371;  A.  E. Glassgold,  W. Heckroth,  and K. M.  Watson, Ann. Phys. (Paris) 5, 1 (1959). 'L.  D. Landau and  E. M.  Lifshitz,  Statistical Physics (Pergamon,  London,  1958). 'J.  D.  Walecka, Phys. Rev. 126,  653 (1962); K. Woeste, Z. Phys. 133,  370 (1952); H. A. Bethe,  Annu.  Rev. Nucl. Sci. 2,  93 (1971). 'J. Serrin, in Handbuch der Physik,  edited by S. Flügge (Springer,  Berlin,  1959), Vol. 8 Part I, P. 125. 'B.  Müller,  H.  Peitz, J. Rafelskl,  and W.  Greiner, Phys. Rev. Lett. E, 1235 (1972);  B. Mhiler,  J. Refel- ski,  and W.  Greiner,  Z.  Phys. 257,  62,  183 (1972). ERRATUM MEASUREMENTS OF  ELECTRON DENSITY EVOLUTION AND BEAM SELF- FOCUSING IN A LASER-PRODUCED  PLASMA.  L.  C.  Johnson and T. K.  Chu [phys. Rev. Lett. -  32,  517 (1974)l. Reference 1 is incorrect.  It should read as follows: 'J. M.  Dawson,  R.  E. Kidder,  and A.  Hertzberg, Princeton University Plasma Physics Laboratory, MATT  Report No.  782,  1971 (unpublished); J. M. Daw- son,  A. Hertzberg,  R. Kidder,  G. Vlases,  H. Ahlstron, and L. Steinhauer,  in Proceedings  of the Faurth Con- ference on  Plasma Physics und Controlled Nuclear Fusion, Madison,  TRsconsin,  I971 (International Atomic Energy Agency,  Vienna,  Austria,  1972), Vol. I, P. 673. 

Nuclear shock waves in heavy-ion collisions

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