A continuum shell-model calculation based on the collective correlation model has been made for the giant resonance of 12C using the eigenchannel reaction theory. The low-lying negative-parity states of 11C and 11B have been taken into account by corehole coupling. Partial, total, and integrated photoabsorption cross sections are calculated for the region of the giant dipole resonance

Barrett, Ross

Greiner, Walter

Mshelia, Eligäl D.

English

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VOLUME  28, NUMBER  13  PHYSICAL REVIEW LETTERS  27 MARCH  1972 Collective Correlations in the Giant Dipole Continuum of I2C E. D. Mshelia, R.  F. Barrett, and W.  Greiner Institut für  Theovetische Physik der  Universität, Frankfuvi, lih/ankfuyt am  Main,  Germany (Received 10 February 1972) A continuum shell-model calculation based on the collective correlation model has been made for the giant resonance of  "C  using the eigenchannel reaction theory.  The low-lying  negative-parity  states of  "C  and "B  have been taken into account by  core- hole coupling.  Partial, total, and integrated photoabsorption cross sections are calcu- lated for the region of  the giant dipole resonance. The gross structure in the giant dipole reso- nance (GDR) of  light and heavy magic nuclei has been successfully explained in terms of  the lp-lh (one-particle, one-hole) shell model.  Higher- resolution experimental investigations,  for ex- ample, in "C  and "0  have,  however,  revealed more structure than is explicable by the simple lp-lh model.  One approach to the understanding of  the extra intermediate structure in ''C  and several other light nuclei was suggested by Drech- sel, Seaborn,  and Greiner.'  In their theory of collective correlations, they coupled the 1- lp-lh doorway states to the quadrupole vibrations of  the nuclear surface, thereby explaining much of  the observed structure. In this work we investigate the GDR of  12C fol- lowing closely the treatment of  Drechsel, Sea- born,  and Greiner, with the difference that the effect of  the one-particle continuum and the low- lying negative-parity  states of  ''C  and "B  are properly taken into account.  Contrary to the un- realistic bound-state  calculation, the continuum calculation can make explicit predictions of  the various observable quantities such as partial and total cross sections, and the angular distribu- tions of the reaction products. In treating the one-particle continuum we use the eigenchannel method developed by Danos and co-~orkers.~'~  This method has been discussed fully in a recent review arti~le.~  The reliability and accuracy of  the method have been investi- gated by Delsanto,  Roetter, and Wahsweiler5; I and it has previously been applied to the calcula- tion of  the GDR of  all doubly magic nuclei in a lp- lh  continuum shell-model approximati~n.~~~~ In the collective correlation model, the full semimicroscopic Hamiltonian of  the nucleus can be written as where Hph  stands for the p-h Hamiltonian in the lp-lh subspace: Hph=  C elaitai  + 4 C (kll  ~lw~n)a,fa,ta,a, i  klmn ait and ai  are creation and annihilation operators, respectively,  for nucleons in the single-particle state  i.  V is the residual two-body  interaction, and ei is the Hartree-Fock single particle or hole energy.  The residual interaction used by us is the short-range force with the exchange mixture used by Elliot and Flowers8: V,  is the strength of  the residual interaction and has been taken to be -  750 MeV fm3 in order to obtain the position of  the GDR in agreement with experiment. Hquad  is the quadrupole Hamiltonian describing the surface vibrations,  In a harmonic approximation this is where a['j is a tensor of  rank 2 and positive parity for the surface quadrupole collective variables,  and nLz1  the momentum conjugate to a12j. The interaction representing the collective correlations between the GDR and the surface vibrations is given by where D~~[~~  is the P-h dipole operator.  For "C  the numerical values of  the renormalized coupling VOLUME  28, NUMBER  13  PHYSICAL REVIEW LETTERS  27 MARCH  1972 constants K~,  K,„  and K,,  given in Ref. 1 are  K, = -  10.86 MeV fm-',  K„  = -  18.67 MeV fm-',  and K„ = -  23.69 MeV fmm2. In ordinary lp-lh shell-model calculations for 12C,9'10  one cannot describe transitions to excited states of  "C  or "B.  These calculations take the ''C  ground state to be a perfect jj-coupling  shell-mod- el state with a filled P„,  shell, 2nd describe the GDR essentially in terms of  a Single dominant [P,~,-' xd,/,]  state.  Hence, they predict that the residual nucleus will be left in ap„,-'  hole state.  In this work we have taken the transitions to the low-lying excited states of  "C  and ''B  into account.  In a uni- fied-model description Clegg"  has shown that the wave functions for the +',  F,  +-, and F quartet of the low-lying excited states can, to a good approximation,  be described by ap„, nucleon hole coupled to the 2'  first vibrational (i.e.,  one-phonon) state of ''C  at 4.43 MeV,  whereas the 2'  ground state was well described by ap„, nucleon hole coupled to the 0'  ground state.  Thus, following Clegg, our basis wave function can be written in the form where h and p represent holes and particles, respectively,  and { } denotes a 6j Symbol.  The radial quantum numbers are  given by n, and the index U on the ket is to indicate that the particle radial wave functions in the continuum depend on the eigenchannel boundary conditions.  The angular momentum X of  the phonon can take the values 0 and 2.  1 represents the angular momentum of  the low-lying states of  "B  or "C,  and this has been coupled with that of  the particle to  the total angular momentum J'=  1-. The solutions of  (1) were obtained by diagonalization in the basis (6), where the single-particle and hole radial wave functions have been calculated in a Woods-Saxon potential with spin-orbit and Cou- lomb terms parametrized to describe the single-particle level scheme of  "C.  The same single-parti- cle potential parameters were used as  in the lp-lh calculation of  Marangoni and Saruis.12  The ener- gies of HO+H„  have been phenomenologically taken to be those of  the low-lying  states of  "B  and "C. In this way, the effect of  anharmonicity of  the quadrupole surface vibrations is partialiy included.  The parameters Eu,  = A(C,/B,)~~~  = 4.43 MeV and the vibrational amplitude 8, = (5fi~,/2C,)'/~  = 0.4 were taken from the low-lying  energy spectrum of  "C.13  The calculation of  the photo cross sections is performed in exactly the same way as for a normal lp-lh eigenchannel calculation.  The states consisting of  a phonon coupled to the lp-lh states are not directly excited by the incoming photons since the collective dipole operator is approximated by the p-h dipole operator.  Their influence is felt only through their coupling by nieans of Hdip,quad  to the lp-lh states.  On the whole there were 22 coupled channels.  The configurations considered for both protonc and neutrons are [[p3/2-1~  0+]312-~~112]1-,  [[p3/,-'x O+]~/~-X~~/~]~',  [[P~/~-~X  0+]9/2-~d512]1-,  [[P~/~-~X  2+]1/2'  xs  1/211-  7 The results of  the calculation are  shown in Figs. 1-3.  Figure 1 shows the total photoabsorption cross section.  The peaks at 17.7,  19.5,  and 20.9 MeV of  Fig. l(b)  are  in good agreement with the experimental photoabsorption cross section of  , Fig. 1  (a), and the splitting of  the strength in the giant resonance region is also reproduced by the calculation.  None of  the vibrational satellites is as high in energy as the experimental peak at 25.5 MeV,  however,  a fact which is also true of the bound-state calculation of  Drechsel, Seaborn, and Greiner.  The theoretical integrated photo- absorption cross section to 27.15 MeV is about twice the experimental value in both the lp-lh model [~i~.  l(c)]  and the collective correlation -- I  model [3?ig.  l(b)l. This is to be expected.  The extra states in the collective correlation model do not contain any dipole strength in themselves and produce their effect by modulating the strength already contained in the lp-lh doorway states. In Fig. 2, the (Y,  n)  partial cross section is com- pared with the experimental result of  Fultz et a1.16 and a recent result by van de Vijver et aZ.16 The splitting of  the giant resonance observed ex- perimentally is again reproduced.  Peaks at 22, 23.4,  and 24.3 MeV in the calculated cross sec- tion correspond to those observed by  Fultz et al. at 22,  23.2,  and 25.5 MeV.  As mentioned earlier, the last one is too low by about 1 MeV. VOLUME  28, NUMBER  13  PHYSICAL REVIEW LETTERS  27 MARCH  1972 ENERGY [Medl FIG. 1.  Total photoabsorption cross section of  "C, (a)  Experimental result of  Shevchenko and Yudin  (Ref. 14).  (b) Theoretical result with collective correlations including excited states of  "C  and  "B.  Solid curve, total cross section; dot-dashed curve, ground-st,ate cross section; dotted curve, contribution of  excited states.  (C) Usual lp-lh eigenchannel continuum calcu- lation (without collective correlations) with residual force of  V, =- 980 MeV  fm3. Finally in Figs. 3(b) and 3(c)  we present the re- sults of  the (y,p,)  and (Y,#)  partial cross sections, respectively.  The experimental result of  the in- verse reaction llB(P,  y,)12C  of  Allas et aE.17 is shown in Fig. 3(a) for coinparison.  Besides the peaks in the giant resonance region already dis- cussed earlier, the peaks at 17.7,  19.5,  and 20.9 MeV are observed experimentally.  The latter two are not present in a simple lp-lh calculation and have been identified as one-phonon satellites of  the 17.7-MeV [P„,-"xs„,]  peak. The integrated differential cross sections at 90" associated with transitions to the ground and excited states have been calculated and compared with the experiments of  Medicus et ~1.'~  and of Fultz et aZ.l5  The results are summarized as follows:  About 91% of  the integrated photoneutron cross section up to 27.15  MeV is associated with the ground-state transition, while 7% is associ- ated with the first excited state and the remaiii- ing 2% with higher excited states,  The experi- mental results of  Medicus et al., which included, apart from negative parity also positiva-parity states, up to a 28-MeV  limit of  integration are 88%,  5%,  and 7%, while those of  Fultz et al. are 22  24  26  28  30 ENERGY  [MeV] FIG. 2.  Total photoneutron cross section of  "C. (a),(b)  Experimental data of  Fultz et  al. (Ref. 15) and van de Vijver et  al. (Ref. 16), respectively.  (C) Result with collective correlations. 83%, 6%,  and 11%.  For the (y,p)  process, the corresponding theoretical values are  80%, 8%, and 12% while Medicus et al. give 89%, 5%,  and 6%.  The overali agreement is thus semiquantita- ENERGY [MeV] FIG.  3.  Photo proton cross section of  ''C.  (a) Ex- perimental result of  inverse photoreaction '1~(~,yo)12~ according to Allas et al. (Ref. 17).  (b),  (C) Theoretical ground state (y,Po)  and total (Y,$) cross sections, re- spectively. VOLUME  28, NUMBER  13  PHYSICAE REVIEW LETTERS  27 MARCH  1972 tive.  The inclusion of  higher collective states (two and more phonons), which have been neglect- ed here, and their coupling to the particle-hole configurations might further improve the agree- ment between theory and experiment. In conclusion,  the results of  the eigenchannel calculation for the GDR of  I2C including collective carrelations and tbe low-lying  states of  the mass- 11 nuclei exhibit a marked improvement in the agreement with experiment over those based on a simple lp-lh model.  The intermediate struc- ture of  the GDR is better explained, and the cal- culations of  tne partial (Y,$)  and (Y,%)  Cross sec- tions to excited states of  the residual nuclei show a semiquantitative agreement with experiment. Application of  the model to other nuclei thus seems warranted. This work has been supported by the Bundes- ministerium für Bildung und Wissenschaft, and by the Deutsche Forschungsgemeinschaft.  One of  us (R.F.B.  ) acknowledges financial support from the Alexander von Humboldt-Stiftung in the form of  a postdoctoral fellowship.  Computing facilities were provided by the Zentrales Rechen- institut,  Frankfurt am Main. We gratefully acknowledge fruitful discussions with Dr. P. P. Delsanto. 'D.  Drechsel, J. B. Seaborn, and W.  Greiner,  Phys. Rev. 162, 983 (1967). 2~.%inos  and W.  Greiner, Phys. Rev, 146,  708 (1966). 3~.  G.  Wahsmeiler,  W.  Greiner, and M. Danos,  Phys. Rev.  170, 893 (1963). 4~.F.~arrett,  L.  C. Biedenharn, M. Danos,  P. P. Delsanto,  W.  Greiner, and H. C.  Wahsweiler, to be published. 5~.  P.  Delsanto,  M.  F. Roetter, and H.  G.  Wahswejl- er, 2. Phys. 222,  67 (1969). 'X.  E.'.  Barrett, A.  Rabie, P. P. Delsanto, and A.  Pi- azza, Phys. Lett. 35B,  551 (1971). 'R.  F. Barrett anä~,  P. Delsanto,  Nucl. Phys. M, 641 (1971). 8~.  P. Elliot and B. W.  Flowers,  Proc. Roy.  Soc., Ser. A 242,  57 (1957). 9~.  Vinh-Mau  and G.  E. Brown,  Nucl. Phys. E,  89 (1962). ''V.  Gillet and N.  Virih-Mau,  Nucl. Phys, 54, 321 (1964). "A.  B.  Clegg, Nucl.  Phys. E, 353 (1962). I2M. Marangoni and A.  M.  Saruis, Nucl. Phys. &%, 649 (1969). 13~he  reduced transition probabilty B(E2;0+-2+)  has recently been nieasured by electron scattering and has been found to be 39.41 12.20 fm4, from  which ß0  is cal- culated to be about 0.4.  H.  Theissen, Lnstitut fiir Tech- nische Kernphysik,  Darmstadt (private  communication). I4v.  Shevchenko and N.  Yudin,  At. Energy Rev. 3, 3 (1965). 15s. C.  Fultz, J. T. Caldwell, B. I,. Berman, R.  L. Bramblett, and R.  R.  Harvey, Phys. Rev. 143,  790 (1966). '%. van de Vijver, H. Ferdinande,  G. Knuyt,  R. Car- chon and J. Devos, J. Phys. (Paris), Colloq. E,  225 (1971). 'IR.  G. Allas, S. S. Hanna,  L.  Meyer-Schutzmeister, and R.  E. Segel, Nucl.  Phys. 58,  122 (1964). 18~.  A.  Medicus,  E. M.  Bowey,  D.  B. Gayther, B.  H. Patrick, and E. J. Winhold,  Nucl.  Phys. e,  257 (1970). Two-Pomeranchukon Cuts and Vanishing of tlie Sriple-Pomerailchukon Coupling* I. J.  Muzinich,  Frank E. Paige,  T. L. Trueman, and Ling-Lie Wang Brookhaven National  Laboratory,  Upton, New  York 11973 (Received 15 February 1972) It is shown that the Gribov-Migdal  lower bound on the magnitude of  the two-Pomeran- chukon cut is not valid rinless both the triple-Pomeranchukon coupling gppp(t  ,qi2,qZ2) and its derivative dg(t  ,q2,q2)/dq2  vanish when t =qI2  =qZ2  = 0.  A  numerical estimate of the cut is given.  We give the connection between the behavior of gpp, and the validity of the Bronzan and Jones unjtarity condition on the discontinuity of the cut at t = 0. The importance of  the triple-Pomeranchukon coupling gppp  has recently become apparerit both in phe- nomen~logy'"~  and theoretical. models  .42  It is generally believed that gppp  vanishes when the masses of all three Reggeons coupled at the vertex vanish,  although the mechanism for this is not yet ~lear.~-' We will assume that gp„  vanishes linearly with the Pomeranchukon masses; and, sine qua non,  the Pomeranchukon is a factorizable simple Regge pole with a(0)  = 1. We will then obtain a formula con- necting gppp  to the discontinuity of  the two-Pomeranchukon cut (Fig. 1). It follows that the Gribov-Mig- da1 lower bound on the cut magnitude4 is not generally valid;  and, as a result, the absorption model 

Collective correlations in the giant dipole continuum of 12C

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