The strong coupling of the giant resonance to the surface vibrations in C12 results in the splitting of the single one-particle, one-hole, 1- collective state into several components, thus improving the agreement between theory and experiment to a very large extent

Drechsel, Dieter

Greiner, Walter

Seaborn, James B.

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLUME  17, NUMBER  9  PHYSICAL REVIEW LETTERS  29 AUGUST  1966 COLLECTNE CORRELATIONS IN Cl2? D. Drechsel, J. B.  Seaborn,  and W.  Greiner Institut für Theoretische Physik der Universität Frankfurt am iviain,  Frankfurt ain Main, Germany (Received 25 July 1966) Tlie strong coupling of  the giant resonance to the surface vibrations in ci2  results in the splitting of  the single one-particle,  one-hole,  1-  collective state into several com- ponents, thus improving the agreement bet~veen  theory and experiment to a very large extent. It is well known that pure single-particle transitions do not account for collective nucle- ar  states.  Elliot and Flowersl and other~~-~ showed that the inclusion of  residual nuclear forces is necessary for the understanding of the excitation energy and dipole strength of the giant resonance.  A number of  detailed one- particle,  one -hole  (lp-  lh) investigations fol- lo~ed.~-~  For example,  in the case of  Cl2  one collective 1-  state has been obtained.  However, the experimental photoabsorption Cross section shows evidence for three to four major peaks in the region from 18 to 30 MeV which cannot be explained by  simple lp-lh calculations. On the other hand,  the collective model for giant resonances explains such structures in a very natural way by  the coupling of  giant-di- pole and surface-quadrupole degrees of  free- dom,  as  has been shown recently.*-l0 It is our aim to demonstrate in this Letter the importance of  such collective correlations in light nuclei.  We  therefore,  firstly,  formu- late the idea of  collective correlations in the particle-hole framework and,  secondly,  give results obtained for Cl2. The Hamiltonian of  the dynamic collective lp-lh calculations is the interaction with the collective surface degrees of  freedom (surface phonons) contained in HQ + HDQ . The surface phonons might be thought of  as complicated,  coherent superpositions of  lp-lh, 2p-2h,  3p-3h,  etc., configurations coupled to 0'  and 2+  in the particle-hole framework.  Of Course,  if the complete particle-hole  Hamilto- nian,  Hph,  is diagonalized in the full Hilbert space consisting of  lp-lh,  2p-2h,  3p-3h,  etc., configurations,  one obtains in principle an ex- act solution of  the nuclear problem.  However, such a procedure is  both nontransparent and impossible to carry through because of  the tre- mendous number of  many-particle,  many-hole configurations.  Therefore,  a more physical approach to the problem is necessary. Guided by  the collective Hamiltonian (I), we assume that from all many-particle,  many- hole configurations only a few,  namely,  the collective ones describing surface modes,  are essential for the structure of  the giant reso- nances.  We  therefore propose the following Hamiltonian: theory is of  the formg H=H +H  +H D  Q  DQ' where H  (l' shall describe the particle-hole qh (1)  Hamiltonian in the lp-lh, 1-  subspace,  and where the first term, Hg,  describes the giant resonances; HQ describes the surface oscilla- tions;  and  is the interaction between the giant resonances and the surface oscillations. If  HD~  is neglected,  the solution of  (1)  yields only one dipole state carrying all strength, and many states of  the form one dipole phonon-many quadrupole phonons carrying no  strength.  E, however,  HDQ is taken into account,  all these states mix and,  as  a result,  several states with appreciable dipole strength occur. The situation is similar in the particle-hole model,  where the diagonalization of  the resid- ual force with lp-lh states usually gives one collective dipole state.  What is  missing in such H  = f&  [&~21~~~21]tOl+~J~~2[a~21xa  [2l]COl, Q  2 H  =K1[a~~lX[(Y[~lXN[~'][~l][~l DQ +K  [[a[21Xa[21][~'x[a[21xa[21  [O]  L01 2 0  I  I +K22[[a  ~21Xar21]~21x[(yr"xa  L21  [21 r01  11, Kl=  -1.588C1,  Km=  -0.708C„  Kz2=  -0.936Cl.  (3) C, is calculated in the collective model.1°  ar2] and a[ll  are the usual collective coordinates for surface quadrupoles and giant dipoles,  re- spectively.  The latter ones are expressed in terms of  particle-hole  operators by the require- VOLUME  17, NUMBER  9  PHYSICAL REVIEW LETTERS  29 AUGUST  1966 FIG. 1.  Photoabsorption cross section of  ci2.  The experimental points give the total photoabsorption cross section [V. Shevchenko and N. Yudin,  At.  En- ergy Rev. 3, 3 (1965)l; the dashed line gives the (y,n,) cross section at 90" in arbitrary units [M. Danos and E. G.  Fuller, Ann. Rev. Nucl.  Sci. 3,  29 (1965); V. V. Verbinski and J. C. Courtney,  Nucl.  Phys. 73,  398 (1965)l.  (a) The dipole strengths calculated with col- lective correlations, (b)  with usual particle-hole  cal- culations. ment that the dipole operator has  to be the Same in the collective and particle-hole  language: where and A/l,  and M, are constant~.~  Substituting W[''  in HDQ  (3),  we finally obtainl' - where the coupling parameters are K is the symmetry energy of  the Bethe-Weiz- säcker mass  formula, A  is the atomic number, and R, is the equivalence radius.  The last three terms in (6)  describe the collective correla- tions between the lp-lh states and the surface vibrations.  They act like an  additional inter- action between the various lp-lh, 1'  states. The solutions of  (6)  were obtained by diago- nalization in the basis consisting of  products of  lp-lh states and phonons.  The parameters EW,=  R(c,/B,)'~ and B,= (5f~iw,/2~,)~'~  are tak- en  from  the low-energy  spectrum of  C12.12  The residual force of  Gillet4 has been used with a strength V,= -35  MeV.  The results are shown in the upper part of  Fig. 1.13  For comparison we also have plotted the results of  usual lp-lh calculations.  The nearly quantitative agreement of  the three major peaks occurring at 22.1, 22.9,  and 24.5 MeV with the experimental pho- toabsorption cross section demonstrates the importance of  collective correlations in Cl2. Similar results are expected for other nuclei. We acknowledge constructive discussions with M. Danos. TThis work has been supported by the Deutsche For- schungsgemeinschaft with a contract for studies in nuclear structure. 'J. P. Elliot and B. W. Flowers, Proc. Roy. Soc. (London) M,  57 (1957). 2~.  E. Brown and M. Bolsterli,  Phys. Rev. Letters 3, 472  (1959).  - 3~.  Fallieros, R. A. Ferrell, and M. K. Pal, Nucl. Phys. i5, 363 (1960). %, Gillet,  dissertation,  Paris, 1962 (unpublished). 5~.  Vinh-Mau  and G. E. Brown,  Nucl.  Phys. E,  89 (1962). 6~.  E. Brown, L. Castillejo,  and J. A. Evans, Kucl. Phys. G, 1  (1961). 'L.  N.  Bolen and J. M. Eisenberg,  Phys. Letters 9, 52  (1964). '5. Le Tourneux, Phys. Letters l3, 323 (1964). 'M.  G. Huber,  H.  J. Weber,  M. Danos,  and W. Grein- er, Phys. Letters i5,  529 (1965); H. J.  Weber,  M. G. Huber,  and W. Greiner,  Z. Physik 192,  182,  223 (1966). ''T.  Urbas and W. Greiner, "On  the Interaction of Collective Degrees of  Freedom"  (to be published). "Note  that the collective correlations are complete- ly different from the interaction of  particle-hole  states VOLUME  17, NUMBER  9  PHYSICAL REVIEW LETTERS  29 AUGUST  1966 and surface vibrations proposed by  E. Boelier, W. M.  =44  fm4  and a transition radius of  about 3.3 fm; thus De  Muijnck,  C. C. Jonker, Comptes Rendus du Con-  Po is about 0.43.  F. G~idden,  private conimunication. grgs International de Phgsique ~ucleaire,  Paris, 1964,  "~he  large vibrational  amplitudes suggest that an- edited by  P. Gugenberger (Centre National de la Re-  harmonic terms might be of  some importance.  In cherche Scientifique, Paris,  19641, Vol. 11,  p. 405.  fact, it is expected that theg will improve the agree- "~ecent  electron scattering experiments give B@  2)  ment behveen theory and experiment. DOUBLE  POLES AND NONEXPONENTIAL DECAYS IN ATOMIC  PHYSICS* K. E. Lassila Institute for Theoretical Physics,  Department of  Physics,  Stanford University,  Stanford, California, and Physics Department, Iowa State University,  Ames, Iowa and Vesa Ruuskanen Department of  Nuclear  Physics, University of  Helsinki,  Helsinki, Finland (Received 25 July  1966) Atomic resonance fluorescence is examined as a scattering problem,  and Goldberger- Watson double poles are found to occur in situations producible in the laboratory.  Sys- tematic study of  this effect and associated nonexponential decay thus  appears possible. It was pointed out by Goldberger and Watsonl that the evidence supporting purely exponential decay for every unstable particle is "far from convincing."  These authors showed,  in partic- ular, that when the S  matrix has multiple poles, the decay amplitude for the associated state becomes a polynomial in time multiplied by the usual exponential factor.  Since then,  sev- eral model situations have been discussed where double poles can occ~r.~  In the present note physical situations with double poles are pre- sented that can be produced and studied in the laboratory. The transition matrix,  or T matrix, under discussion is that for resonance scattering of light through more than one excited state.  The calculational method and some results have been discussed el~ewhere~,~;  we now  extend these to the case where the T matrix has a dou- ble pole.  This approach3 is similar to that of the Lee mode15 in that the calculation begins from a second-quantized Hamiltonian in which the unperturbed part gives the atomic energy levels and photon energies,  and the interaction part describes transitions between the levels with photon emission and absorption.  That such an approach is appropriate had been noted ear- lier also by Källen.6 Resonance flourescence through two excited levels is probably the case of  most immediate experimental interest.  This process is described by a T matrix of  the following form (subscripts 1 and 2  refer to the two excited states; a sin- gle ground state is assumed,  and the incident 1  photon energy is W): where wj is the energy (E=  1) of  the excited state lj), rj is the corresponding width,  and Vij is the  breviations for the absorption and emission matrix element of  an external (or possibly in-  matrix elements,  respectively,  connecting the ternal) perturbation coupling the excited states.  excited state  I i) with the ground state. The fi  and gi,  in Franken's7 notation,  are  ab-  A double pole in T requires W+= W_.  This 

Collective correlations in C12

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