1. PROPERTIES OF HIGH-ENERGY ISOSCALAR MONOPOLE EXCITATIONS IN MEDIUM-HEAVY MASS SPHERICAL NUCLEI M. L. Gorelik 1), S. Shlomo 2), B. A. Tulupov 3), M. H. Urin 1) 1) National Research Nuclear University “MEPhI”, Moscow, Russia 2) Cyclotron Institute, Texas A&M University, College Station, Texas, USA 3) Institute for Nu clear Research, RAS, Moscow, Russia

2. The purposes of the presented study: 1. To get within new theoretical approach (the particle-hole dispersive optical model (PHDOM)) the information concerning the properties of the isoscalar monopole excitations in medium-heavy mass spherical nuclei;

3. 2.To examine from a microscopic point of view the applicability of semi-classical collective model transition densities of the isoscalar giant monopole resonance (ISGMR) and its overtone (ISGMR2) in the analysis of the corresponding experimental data. 3.To examine the PHDOM unitarity violation.

4. 1. Within the PHDOM, the continuum-RPA is extended to take into account the spreading effect phenomenologically with averaging over the energy in terms of the imaginary part of an effective optical-model potential. The basic quantity of the model is the energy-averaged p-h Green function (or the effective p-h propagator). The ISM radial component of this propagator,, determines in a wide excitation energy region the energy-averaged ISM radial double transition density, the main innovation of the presented approach:

5. The energy-averaged strength function, corresponding to the ISM external field V(r)Y 00 : Description of the ISGMR (V 1 (r)) and ISGMR2 (V 2 (r)) :  is defined by minimazing energy-weighted sum rule for the field V 2 (r) ; integration - over ISGMR region. (  = 77 fm 2 )

6. The Bethe-Goldstone-type equation for : F(r) – strength of the isoscalar part of the Landau-Migdal p-h interaction:

7. The model parameters The used mean field is consistent with isovector part of Landau-Migdal p-h interaction. The mean field parameters are found from the description of the observed neutron and proton single-quasiparticle spectra in the conside-red nuclei. F(r)=C(f ex  (f ex  f in )f WS (r)) (C=300 MeV  fm 3 ) f WS (r)  Woods-Saxon function, f ex  2.876, f in  0.0875 are determined from the condition that 1  spurious state should be at zero energies and centroid energy of ISGMR,  1  13.96 MeV. Parameter   0.07 MeV  1 (intensity of the used in PHDOM effective optical-model parameter).

9. Parameters of ISGMR and ISGMR2 ISGMR experimental data : centroid energy ω 1  13.96  0.2 MeV total width calculated results: centroid energy total width ISGMR2 calculated results: centroid energy ω 2  22.7 MeV total width Γ 2 = 2.35σ =22.8 MeV (σ is the squared energy dispersion)

10. 2. The experimental study of high-energy particle-hole-type isoscalar monopole (ISM) excitations in medium-heavy mass nuclei is usually connected with the study of ( ,  ')-inelastic scattering at small angels. The analysis of this reaction is widely based on the use of Born approximation (DWBA). The DWBA differential cross section for considered process is given by In certain approximation the energy averaged squared transition matrix element |T fi | 2 is given by Here, V  N (q) is the Fourier transform of the  -nucleon interaction, q=k f - k i, and field V 0,q (r) is the following: The field V 0,q (r) reproduces effects connected with V 1 (r) and V 2 (r).

12. The assumption: double transition density may be factorized? The projected transition densities: According to definition:

14. The energy-averaged semi-classical transition densities: L i (  )  Lorentzian forms (i=1,2) n(r)  the ground state matter density

16. The reduced double transition densities double transition density: projected transition density: semi-classical transition density:

17. The comparison of the reduced ISM double transition density R(r=r',  ) with reduced semi-classical transition densities R sc,1 (r=r') and R sc,2 (r=r')

18. 3. The violation of the model unitarity The signatures of violations: (i) a non-zero value of the calculated strength function, corresponding to the “spurious” external field V 1 (r)=1; (ii) negative values of the strength functions at high excitation energies, that leads to reduction of the total strength. The sources of the model unitarity violation: (i) PHDOM method, used to describe the spreading effect in terms of the imaginary part of the effective optical model potential; (ii) the use of the approximate spectral expansion for the optical-model Green function.

19. The restoration of the model unitarity The first step of the restoration of the model unitarity is the introduction of the factor in the expression for the energy-averaged “free” p-h propagator. P(  ) – real (dispersive) part of the effective optical-model potential used in PHDOM.

20. Then it is necessary to modify properly the double transition density by adding to it the terms, involving the ground-state density, normalized to unity : The modified strength function corresponding to the field V(r) :

21. The results of restoration: (i) the modified strength function of the “spurious” external field is equal to the zero identically; (ii) the modified ISM strength functions for external field V 0 have now no negative values in the considered wide energy interval.

22. The calculated values of obtained with and without account of the unitary restoration ISGMR ISGMR2 1.01 0.95 1.00 1.04

25. Conclusion The applicability of the PHDOM in calculations of the ISM double transition densities and strength functions has been demonstrated. In particularly, it has been shown that in the intermediate excitation energy region (between the energies of the ISGMR and ISGMR2) the double transition density is quite different from that obtained from the collective classical transition densities, which are used commonly for the analysis of hadron inelastic scattering cross sections for the ISM excitations. It has also been shown that to this aim the microscopically justified projected transition densities can be used.

26. Many thanks for your attention!