It is suggested that the experimentally observed intermediate structure in the cross section of elastic O16 + O16 scattering is due to quasibound molecular states of the ion-ion system while the gross structure originates from virtually bound molecular states

Greiner, Walter

Lemmer, Richard

Scheid, Werner

English

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VOLUME  25, NUMBER  3  PHYSICAL REVIEW LETTERS  20 JULY 1970 QUASIMOLECULAR STRUCTURE IN  ELASTIC 016  + 016 SCATTERING  * Werner Scheid,  Walter Greiner,  and Richard Lemmerf Institut für Theoretisclie Physik der Universität Frankfurt arn Main,  Frankfurt an1 Main, Germang (Received 6 April 1970) It is suggested that the experimeiitally observed intermediate structure in the cross section of  elastic 0j6  + 0''  scattering is due to quasibound molecular states of  the ion- ion system while the gross structure originates from virtually bound molecular states The excitation function for the elastic scatter- ing of  01%01'  as  measured by the Yale groupl exhibits three distinguishable types of  structure in the energy range 17 to 35 MeV (c.m,):  the gross structure, the intermediate,  and the fine structure ("grass").  The gross structure with a large width of  about 4 MeV has been success- fully analyzed by an optical m0de1,l"~  The inter- mediate and fine structures are due to the com- bined effects of  direct nuclear excitations,  com- pound elastic scattering, and transfer reactions, Comparing the elastic Cross sections of  the Cl2 + Cl2  and the 016  +Ola reactions5 it is suggested that the stronger intermediate structure in the Cl2  +Cl2  case arises from the lower stability of the Cl2  shells against deformation.  To form the compound nucleus the nuclear deformation has to change during the collision,  so that collective states are mainly excited,  Hence,  inelastic ex- citations of  collective 016 states should be strong- ly coupled both to the elastic entrance and to the compound-nucleus  channels.  In the following we propose to describe the intermediate structure by inelastic excitations of  low-energy states in the individual nuclei,  A similar coupling has been mentioned first by Block and Malik.'  For a survey of  statistical interpretations of  the inter- mediate structure we refer to Ref. 1. The starting point of  our considerations is the molecular-type potential shown in Fig, 1  which FIG. 1. Real and imaginary parts of  the ion-ion potential.  The left-hand side shows an elastic channel at energy Ei  that de-excites by energy E*  into a quasibound state.  The positions of  the virtual and quasibound states are found numerically by a phase-shift  analysis.  They are shown on the right.  The de-excitation E* is compensated for by possible excitations of  the oi6  core as shown in the left part of  the figure. PHYSICAL REVIEW LETTERS is derived on the basis of  a Thomas-Fermi mod- el of the nucleus and used in the elastic 016 +016 scatteringe3  As can be Seen this potential exhib- its quasibound states up to a certain angular mo- mentum and energy (1  = 10, E = 16 MeV;  all ener- gies in the c.m.  system).  The positions of  these states in Fig. 1 are computed quantum mechani- cally by a phase-shift analysis.  In these states the nuclei can form a quasimolecule7  which then can tiecay through the Coulomb and centrifugal barriers as  well as into the compound nucleus, Because of  the potential barrier the quasimolecu- lar states are not directly reached in the elastic scattering.  However,  indirect excitation of these quasibound states is possible according to the following mechanism:  After the ions have crossed the potential barrier they lose kinetic energy by inelastic excitations of  low energy lev- els of  one or both of  the ions.  Thereby they drop into the potential and are able to form a quasi- molecule,  if their relative energy Ei-E*  coin- cides with the energy of  a quasibound state (see Fig. 1).  This idea is supported by the calcula- tions of  Imanishi8 who found that the resonances Seen in the Cl2  +ClZ  reaction below the Coulomb barrier7  can be interpreted as molecule-like compourid states of  two Cl2  nuclei,  one being in the ground state and the other one in the first ex- cited state. The potential of  Fig. 1  has virtual states for angular momenta exceeding 1  = 10 that produce the wide gross structure of  the elastic cross sec- tion.  Each peak of  the gross structure is gener- ated by essentially one partial wave in resonance in a virtual state.  Since experimentally the in- 1 EXCiTED  QUASI-BOUND  * OR  VlRTUAL  STATE  f ANG. MOM.,ENERGY  > = B INTRlNSlC  EXClTATlON  P OF  THE  0'*0'  SYSTEM  2 STATE ;  ENERGY "l  m termediate structure is closely associated with the main structure ure  conclude that the quasi- bound states are fed through the virtual states as doorway states.  Compared with all other ingoing partial waves only that partial wave resonating in a virtual state has a large amplitude to excite quasibound states.  Therefore the intermediate structure is generated by a double-resonance event:  First an elastic partial wave resonates with its corresponding virtual state.  Then a qua- sibound state is excited by this wave.  For this to happen it is necessary that the coupled low- energy excitation accounts for the difference in energy and angular momentum of  the virtual and quasibound state.  So entrance- channel enhance- ment of  the intermediate structure occurs. As an example we consider the peak around 20,5 MeV  in the 90" differential 0''  +016  cross section,  Its main structure originates from the 1  = 12 partial wave according to a phase analysis using the potential of  Ref.  3.  In Fig. 2 211 those low-energy excitations are summarized which connect quasibound states of  different angular momenta with the 1 = 12 partial wave within its resonance width (from 19 to 23 MeV).  There are obviously only a small number of  allowed inelas- tic excitations with the correct angular momen- tum and energy,  in good agreement with the num- ber of  peaks observed as intermediate structure in this particular energy range.  The description of  the gross structure by a particular resonating partial wave is very sensitive to the form of  the real part of  the elastic-scattering potential.  So Rickertsen et al.4 using somewhat different and deeper potentials attribute an 1  = 14 and 1  = 16  par- POSITION  OF THE  RESONANCE 19  20  21  22  23  24 c.m. ENERGY  (MeV) FIG. 2.  Composition of the quasibound states that can be excited by the  1 =I2  virtuai state within its widths (V denotes virtual,  qb denotes quasibound,  and gs denotes ground state of  016). VOLUME  25, NUMBER  3  PHYSICAL REVIEW LETTERS  20 JULY 1970 tial wave to the peak at 20.5  MeV.  This problem is relevant for a spectroscopic analysis of  the cross section,  but does not affect the interpreta- tion of  the intermediate structure. The Hamiltonian of  the system is composed of the kinetic energy of  the relative motion,  the in- trinsic Hamiltonians of  two noninteracting 016 nuclei,  and the interaction between the relative and the internal motion: states defined by H016~,=~,~„  a=(I,n).  (2) The wave function $ of  the composite 016  + 016 system is symmetrized with respect to the simul- tanous exchange of  all nucleons,  thus neglecting exchange of  individual nucleons.  This symmetri- zation is suggested because in the entrance chan- nel the two unexcited nuclei act as  bosons and the symmetry is conserved by the Hamiltonian H =T(Y)  +~~16(1)  +H016(2)  + wint(F,  1, 2).  (1)  (1).  Thusg The sets of  internal coordinates of  both 0''  nu- clei are abbreviated by 1 and 2.  We expand the  S =  2  R~JXI(~)[~'Y~@VJX(~~  2)11M 2,J,X,I (3) solutions of  H with respect to a complete set of  1  ,  with If  both 016 nuclei are in the Same state (a,  = a,) the sum of  the angular momenta I  +J  has to be an even number. We divide the interaction energy into the optical potential for the elastic channel describing the gross structure of  the elastic scattering of  two 016 nuclei and the residual interaction which couples excited 016 states to the entrance channel: Using (3) and (5) we find the coupled system of  differential equations for the radial function R(r) of the relative motion, The reduced matrix elements of  the residual interaction are  functions of  the relative distance between the 016 nuclei.  In general nonvanishing diagonal elements occur.  They produce a state-dependent  con- tribution to the potential U(r) in the excited states.  If  for simplicity the 016 excited states are assumed as  vibrational states described by  collective surface coordinates the transition potential QLM(r,  1, 2) of Eq.  (5) can be obtained in a similar fashion as  used in the calculation of  the real part of  U(T).~$'  Intro- ducing the S matrix the wave function V, Eq. (3), has the asymptotic form # = ~A~(J~~~~~~-s~~~o~I)(~~'Y,I~~~~~~~),~, CC' where J„ Oc = [l/(~~~~~r)]e*'~~(~,r  iF  ,)  are Coulomb functions.  For the transition where an 016 nucleus in the state pl  =(I„  MI,  n,) hits an 016 target in the state p2 and one of  the scattered 016 nuclei is mea- sured in the direction 8, cp  (c,m.) in the state ylf, the other in the opposite direction in the state P,', the symmetrized differential cross section is given by n  (1.m ljrm  ,>IM)  [(1  + 6a14)(1 +  6u1~„~)1'2(~2in16~~~~~~~~-~11~1~~,i~~r)~21m1~z~  (8) Here the Coulomb scattering amplitude is denoted by fc.  Since usually the magnetic quantum num- bers are  not measured, one has to average over the initial states and to sum over the final states. VOLUME  25, NUMBER  3  PHYSICAL REVIEW LETTERS  20 JULY 1970 Permanent address:  Physics Departnient,  Rand Afrikaans University, Johannesburg,  South Africa. 'R.  H. Siemssen,  J. V.  Maher,  A. Weidinger,  and D.  A.  Bromley,  Phys. Rev.  Lett. 2,  369  (1967), and 20,  175 (1967); J. V.  Maher,  M.  W. Sachs,  R. H. Siemssen,  A. Weidinger,  and  D.  A.  Bromley,  Phys. Rev. 188,  1665 (1969). 'R.  J. Munn,  B.  Block,  and  F. B.  Malik,  Phys. Rev. Lett. 3,  159 (1968);  K. A.  Brueckner,  J. R. Buchler, and  M.  M. Kelly,  Phys. Rev. 173,  944  (1968). 3~,  Scheid,  R. Ligensa,  and W. Greiner, Phys. Rev. Lett. G, 1479 (1968); W. Scheid and  W. Greiner, Z. Phys. 226, 364  (1969). 4~.  Rickertsen,  B.  Block, J. W. Clark,  and  F. B. Malilc,  Phys. Rev.  Lett, 3,  951 (1969); R.  A.  Chatwin, J. S. Eck,  A. Richter, and  D.  Robson,  Phys. Rev, 180, 1049 (1969), and  Phys. Rev. C I,  795  (1970). 5~.  A.  Brornley,  in Proceedings of  the International Conference on Nuclear Reactions Induced by  Heavy Ions,  Heidelberg,  Germany,  15-18 July 1969 (tobe published) . %. Block and  F. B. Malik,  Phys. Rev. Lett, 2, 239  (1967). 7~.  Almqvist,  D. A. Bromley,  and J. A. Kuehner, Phys. Rev. Lett. 4,  515  (1960)- 8~.  Imanishi,  Phys. Lett. G,  267 (1968), and Nucl. Phys. M,  33  (1969). 9~.  Scheid and  W. Greiner, to be published. "U.  C. Voos,  W. v. Oertzen,  and R. Bock,  Nucl. Phys. s,  207  (1969). ANISOTROPY AND POLARIZATION IN THE GRAVITATIONAL-RADIATION  EXPERIMENTS* J. Webert Institute for Advanced Study, Princeton, New  Jersey 08540 (Received 8 September 1969; revised manuscript received 24 June 1970) Large anisotropy is observed for gravitational-radiation-detector  intensity as a func- tion of  sidereal time, with peaks in the direction of  the galactic center and in the oppo- site direction consistent with 12-h antenna symmetry.  The 12-h sidereal-time histo- grams exhibit anisotropy exceeding 6 standard deviations from the mean. Earlier paperslt2  have discussed the concep- tion,  analysis,  and development of  the gravita- tional-radiation antenna.  A  coincidence experi- ment involving detectors at  Argonne National Laboratory and the University of  Maryland was carried  in 1969,  and gave positive results, Evidence has been presented to support conclu- sions that the coincidences are  not accidental and are  not caused by seismic or electromagnet- ic effects.  Results of  a study of  cosmic-ray in- teractions,  at the Maryland site, have been pub- li~hed.~ The antennas are  mass quadrupoles and have some directivity.  Their axes are horizontal and point east-west at each site.  The earth rotates the "antenna beam" among the stars.  The time of  each coincidence is therefore some measure of  the direction of  the source.  Observation of coincidences over a period of  months would be expected to show anisotropy and give information concerning possible sources. excited by a delta function at t = 0.  The equation of  motion is The final velocity 2,  just after t = 0 is given in terms of  the initial velocity 2,  just before t = 0, by integration of  (I), as 2, will vary over a wide range in consequence of the thermal fluctuations.  For small signal-to- noise ratio the final velocity 4,  and the energy added in consequence of  external excitation will therefore also vary over a considerable range. A procedure will now be described for measur- ing,  to a good approximation, the square root of the source intensity by  observing the detector output,  which is mainly noise.  Each gravitation- al-radiation detector employs a chain of  linear amplifiers followed by a synchronous square-law detector.  All voltages referred to are outputs Intensity measurements.  -1t  will be proved that  of  the linear amplifiers just before the square- for the detectors (which are harmonic oscilla-  law detector.  Let V„  be the voltage caused by tors) the effect of  sudden gravitational-radiation  the Jth  coincidence if  no noise were present, and excitation is to add the gravitational induced  let VJA  be the noise voltage just before excita- velocity to the initial velocity associated with the  tion,  for channel A. Let C„  and V„ be the cor- thermal fluctuations.  The sudden velocity in-  responding quantities for channel B.  Let (VJA2) crease decays with the relaxation time of  the de-  and (VJB2)  be mean-squared values with aver- tector cylinder.  Consider a harmonic oscillator  ages taken only over a 3-sec interval just pre- 

Quasimolecular structure in elastic O16 + O16 scattering

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