For the scattering of 28Si on 28Si coupled channel calculations of the elastic scattering and inelastic single excitation of the first 2+ state of 28Si are carried out. The real coupling potentials are calculated in the framework of an adiabatic model. The resulting cross sections reveal structures in agreement with the observed ones and support their interpretation as nuclear molecular resonances

Greiner, Walter

Scheid, Werner

Thiel, Artur

English

Hochschulschriftenserver - Universität Frankfurt am Main

PHYSICAL REVIEW C  VOLUME 29, NUMBER 3  MARCH 1984 Nuclear molecular barrier resonances in the scattering of 28Si  On  28Si  studied by coupled channel calculations Artur Thiel and Walter Greiner Institut für  Theoretische Physik der Johann-  Wolfgang-Goethe-Universität,  Frankfurt um Main, Federal Republic of Germany Werner Scheid Institut für  Theoretische Physik der Justus-Liebig-Universität,  Giessen, Federal Republic of Germany (Received 3 October 1983) For the scattering of 28~i  On  "Si  coupled channel calculations of the elastic scattering and inelas- tic single excitation of the first 2+ state of 28~i  are carried out.  The real coupling potentials are cal- culated in the framework of  an adiabatic model.  The resulting cross sections reveal structures in agreement with the observed ones and support their interpretation as nuclear molecular resonances. I.  INTRODUCTION The experimental differential cross sections for the elas- tic  and  inelastic  scattering of  28~i  On  28~i,  measured  by Betts  et al.'  in  the  energy  range  E,,,,  =45-75  MeV, show similar gross and intermediate structures as those al- ready found in lighter systems, for example, in 12c  On  12c (Ref. 2), 160 on 160 (Ref. 31,  and 12c  On  160 (Ref. 4). Betts et aL5 assigned the angular momenta I =  34-42fi  to the maxima of the gross structures observed in the elastic excitation function of 28~i  On  28~i.  These angular momen- ta are among the highest nuclear spins measured so far in nuclear physics. Correlations of the gross and intermediate structures be- tween  the  elastic  and  inelastic  cross  sections  of  lighter scattering systems have been successfully explained by  the double  resonance  mechanism6"  acting  between  nuclear molecular  resonances  or,  more  simplified,  by  the  band crossing  model.'  In this  Paper, we  will  study the gross and  intermediate  structures  in  the  elastic  and  inelastic cross  sections  of  28~i  On  *'Si  in  the  adiabatic  coupling model of Könnecke et als9 We show that the structures in the cross sections can be explained with the excitation of resonances on top of  the potential barriers  which are the relevant  molecular  resonances in  heavier  scattering  sys- tems.  The excitation of  quasibound resonances inside the potential  pockets,  established already  in  the lighter sys- tems, can be  studied when inelastic excitations of  higher lying states of 28~i  are involved. 11.  THE  ADIABATIC COUPLING MODEL This model serves to calculate the potential energy as a function  of  the  relative  distance  of  the  nuclei  and of  multipole  coordinates  describing  the  deformations  of the nuclear surface.  If  we  restrict  the multipoles to the quadrupoles  only, described by  the coordinates  CY:~ and a;:,  the real potential energy surface can be generally ex- panded for equal nuclei up to quadratic terms as follows: (0) Re11 Potential  Borner Resonances FIG. 1.  (a) The real potential for the system 28Si+28Si. The sum of the real and centrifugal potentials is shown for angular momenta between 34 and 42Zi.  The dashed curve represents the calculation of  p(r)  taken from Seiwert (Ref. ll),  who used the folding  model  with  an  M3Y-two-body  interaction.  On  the right-hand side the energies of the corresponding potential reso- nances near the barriers are indicated and compared with those given by  Betts et al.  (Ref. 5).  (b) The radial potential for the coupling term linear in the quadrupole coordinates [see Eq. (I)]. 864  @ 1984 The Arnerican Physical Society 29  -  NUCLEAR MOLECULAR BARRIER RESONANCES IN THE .  . .  865 In the adiabatic coupling model the potentials  V',  V:", V:',  and ~ji"~'  are determined by the Coulomb interaction and an effective nuclear Yukawa two-body interaction act- ing  on the density,  which  is  homogeneously distributed over the volume of  the composite system for all relative distances  (no compression effects as in the usual folding models due to overlapping nuclear densities; for details of the adiabatic coupling model see Ref. 9).  The parameters of  the potentials are the strength  V.  and range p of  the Yukawa interaction and the radius Ro of  the homogene- ous density distributions of the separated *'Si  nuclei.  The range p is set ,uvl =  1.4 fm-'.  The parameters  V.  and Ro are determined  by  the condition that the energies of  the potential resonances, calculated with  P(r)  of Eq. (1) and lying slightly above the corresponding barriers, agree with those for  2 =34-42fi  obtained  by  Betts et aL5 from the maxima of  the experimental elastic gross structures.  We found the parameters  V.  =  -438  MeV fm and Ro  =4.67 fm. Figure  I(a) shows the sum of  the potential  vO(r)  and the  centrifugal  potential  /(I+  1  )fi2/2~r2  for  1  =34-42 and  the  energies  of  the  barrier  resonances  (calculated without  an absorptive  potential)  in comparison  to those found by  Betts  et  ~1.~  The radius  parameter Ro of  the homogeneous density distribution of  the separated nuclei leads to a barrier at a relative distance of  about 8 fm for angular  momenta  in  the  range  of  34-42%  The  large value  Ro=4.67  fm  corresponds  to  a  radius  where  the charge density of 28~i,  represented by Helm's distribution, has already dropped to 5% of  its maximum value,1° and reflects  the softness of  the Si  nuclei if  they  touch each other.  The same position and height of the potential bar- rier have been obtained by Seiwert,"  who calculated vO(r) by  the usual  folding  model using an M3Y-two-body in- teraction.  This calculation is shown by  the dashed curve in Fig. l(a). After  having  fixed the parameters for  V.  and Ro we can determine all the radial coupling potentials in Eq. (1). In this paper we  only use the coupling potential  propor- tional to vil'(r)  [see Fig. l(b)], which linearly couples the ground state configuration of 28~i+  28~i  to the single exci- tation of the first excited 2+ state of 28~i. 111.  THE ABSORPTIVE POTENTIAL The effects  of  the channels  which  are not  explicitly treated  are described by  an absorptive potential.  For its calculation we  apply the statistical model of  Ref.  12.  In this model the imaginary  potential  is set proportional  to the transition  probability  out  of  the elastic  or inelastic channels  into precompound  channels,  multiplied  by  the level  density  which  depends  on  the  excitation  energy E* =E -  P(r)  and quantum number I of  the total angu- lar momentum. We assume that the radial dependence of  the square of the matrix element is proportional  to the number of  nu- cleons  N(r) in  the  overlap  region  of  the  collision partners.'  For the calculation of N(r)  we approximate the density of  the nucleons of  "Si  by  the following function [@(X)  =  step function]: The parameters po, ?L,  and ro are determined by fitting the model  density  to the measured  charge density  of  28~i.'0 Finally we obtain an imaginary potential with two free pa- rameters, a and ß: 21+  exp  2\/YEi-  I (I  ++I2 W(r,E*,II=aN(r)-  2a2 where with 8 as the moment of inertia of the system. The spin cutoff parameter U and the level density con- stant a  are taken the same as in the work of  Fink et al.' Optical model calculations with  this imaginary  potential have shown that the value of ß has to be taken in such a way  that the contour lines of  constant imaginary  poten- tials in the (E,I)  plane  (at a fixed relative distance r=9 fm, i.e., in  the surface interaction  zone) have  the same slope dE  /dI as the band formed by the barrier resonances (for details see Ref. 13). The strength parameter a of the imaginary  potential  is then fixed by  fitting the order of magnitude of  the calculated elastic excitation function to the experimental one. IV.  RESULTS Figure 2 shows the elastic 90" differential cross section calculated with the optical potential consisting of the real part  vO(r)  and the imaginary potential  (4). For compar- ison we have drawn the experimental data of  Betts et al.' (dashed line).  The theoretical  curve in Fig. 2 clearly re- veals  the  gross  structures  observed  in  the experimental cross section.  These structures arise from the barrier reso- nances shown  in  Fig.  l(a), as an analysis  of  the partial wave functions has shown.  In the case of no absorptive potential  the  resonating  partial  waves  oscillate  rapidly with  small amplitudes inside the potential  pockets (6-12 FIG. 2.  The differential cross section for the elastic scattering of "Si  On  28Si  at 0,.,. =90"  as a function of the incident energy E,.,,.  The theoretical curve  (solid line) represeiits  an  optical model  calculation without coupling of  inelastic channels.  The experimental  data  of  Betts  et al.  (Ref.  1)  are  shown  by  the dashed curve. 866  ARTUR THIEL, WALTER GREINER, AND WERNER SCHEID nodes) and are strongly peaked in the barrier region (8-9 fm). The imaginary potential damps the small oscillations out, but  leaves the surface peaks unaffected.  Therefore, the  28~i+28~i  System  forms molecular  configurations  at larger relative distances than one is used for lighter sys- tems like 12c+12c  and  160+160. In the latter  systems the fitted potentials  are flatter,  and  the positions of  the resonance  states  are found  at smaller  relative  distances than the positions of the barriers. In the next step we  coupled the single excitation of the first 2+ state of 28~i  to the elastic channel.  The reduced matrix  element (  2+  1  1 a2  1 10+ ) ,  needed for the evaluation of  the linear  coupling  matrix  elements, has  been  deter- mined by fitting the order of magnitude of the calculated inelastic single 2+ cross section to the experimental one. We  found  a value of  -0.25.  With  the effective radius Ro=4.67 fm used for the calculation of the real potentials this yields a B(E2,0+-+2+) value of  332 e2  fm4 [experi- mental value 331 e2  fm4 (Ref. 14)].  The same imaginary potential is used in all the channels.  Compared with the calculation  shown in Fig. 2, the imaginary  potential  has been slightly changed by taking a smaller value for ß. Figure 3 shows the resulting elastic and inelastic cross sections for O,.,,  =  90" in comparison with the experimen- tal ones of Betts et al.'  We recognize that the coupling of the single 2+ excitation of 28~i  affects the elastic cross sec- tion  in  the  energy  range  at  about  60  MeV.  The finer structures in the theoretical cross sections arise due to the coupling of the barrier resonances in the considered chan- nels.  The large peak-to-valley ratios of the calculated elas- tic  scattering  cross  section  near  50-55  MeV  may  get smoothed out if more reaction channels are coupled to the considered ones.  If we  had taken 28~i  states with higher excitation energies and spins into account in the coupled channel  calculations, then  quasibound  resonances, which lay more inside the potential  pockets and have a smaller number of nodes than the virtual barrier resonances, could also be excited. V.  CONCLUSIONS Our calculations clearly show that the structures in the cross sections of 28~i  On  28~i  can be traced back to the ex- citation of  resonance states in the relative motion  of  the nuclei.  The  corresponding  molecular  configurations  of the considered system are formed at large internuclear dis- tances in the barrier region of the real potential.  That is a novel effect compared with  the molecular configurations found  in  lighter  systems,  where  the  constituent  nuclei FIG. 3.  The 90" differential cross sections (a)  for the elastic scattering of  28Si+28Si  and  (b) for the inelastic single excitation of  the first 2+ state in 28~i:  28Si[28~i,  "Si(2+ )Iz8Si.  The theoreti- cal  curves  (solid lines) are  calculated  by  coupling  the  elastic channel with  the channels for the single excitation of  the first 2'  state.  The experimental  data  of  Betts  et al. (Ref. 1) are shown by  dashed curves. have a larger overlap of their surfaces. We suppose that also in systems heavier than 28~i  +  28~i molecular  configurations  can exist with  internuclear  dis- tances lying in the barner region.  As shown in Ref.  15, a barrier and a strong absorption behind this barrier are the only  prerequisites  for  the  appearance  of  barrier  reso- nances.  These conditions can be  fulfilled also in heavier systems, like, e.g., U +  U or U +  Cu, as the line structure, observed  in the measurements of  the positrons  emerging from these overcritical systems, reveal~.'~,'~ Thus,  it  is likely that  nuclear  molecules  are a  rather general  phenomenon  appearing  in  many  binary  nuclear systems.  Therefore, the 28~i-28~i  results furnish an impor- tant step in proving this conjecture. ACKNOWLEDGMENTS We are grateful to Dr. Rainer Könnecke for his help and  discussions during the early Stage of  this investiga- tion.  This  work  was  supported  by  BMFT  and  GSI (Darmstadt). IR. R. Betts, S. B. DiCenzo, and J. F. Petersen, Phys. Rev. Lett. 43, 253 (1979). 2D. A.  Bromley, J. A. Kuehner, and  E. Almqvist, Phys. Rev. Lett.  4, 365 (1960). 3~.  V. Maher, M.  W. Sachs, R. H. Siemssen, A. Weidinger, and D. A. Bromley, Phys. Rev.  188,  1665 (1969). 4R. Stokstad et al., Phys.  Rev.  Lett.  28, 1523  (1972); R.  E. Malmin et al., ibid.  28,  1590 (1972). 5R.  R.  Betts,  S. B.  DiCenzo, and  J.  F.  Petersen, Phys.  Lett. m,  117 (1981). 6W. Scheid, W. Greiner, and R. Lemmer, Phys. Rev. Lett.  25, 176 (1970). 'H. J. Fink, W.  Scheid, and W.  Greiner, Nucl. Phys. m,  259 (1972). 8Y. 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Nuclear molecular barrier resonances in the scattering of 28Si on 28Si studied by coupled channel calculations

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