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
The purposes of the presented study:
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;
1.
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.
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:
The energy-averaged strength function
,
corresponding to the ISM external field V(r)Y00:
Description of the ISGMR (V1 (r)) and ISGMR2
(V2 (r)) :
 is defined by minimazing
energy-weighted sum rule for the
field V2 (r) ; integration - over ISGMR region.
( = 77 fm2 )
The Bethe-Goldstone-type equation for
F(r) – strength of the isoscalar part of the
Landau-Migdal p-h interaction:
:
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(fex  (fex  fin )fWS (r)) (C=300 MeV  fm3)
fWS (r) Woods-Saxon function, fex 2.876, fin  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).
Parameters of ISGMR and ISGMR2
ISGMR
experimental data :
centroid energy
ω1  13.96  0.2 MeV
total width
calculated results:
centroid energy
total width
ISGMR2
calculated results:
centroid energy
ω2  22.7 MeV
total width
Γ2 = 2.35σ =22.8 MeV
(σ is the squared energy dispersion)
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 |Tfi|2 is given by
Here, VN (q) is the Fourier transform of the -nucleon interaction,
q=kf - ki , and field V0,q(r) is the following:
The field V0,q(r) reproduces effects connected with V1(r) and V2(r).
The assumption: double transition
density may be factorized?
The projected transition densities:
According to definition:
The energy-averaged semi-classical
transition densities:
Li () Lorentzian forms (i=1,2)
n(r)  the ground state matter density
The reduced double transition
densities
double transition density:
projected transition density:
semi-classical transition density:
The comparison of the reduced ISM double transition density R(r=r',)
with reduced semi-classical transition densities Rsc,1(r=r') and Rsc,2(r=r')
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 V1(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.
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.
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) :
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 V0 have now no
negative values in the considered wide
energy interval.
The calculated values of
obtained with
and without
account of the unitary restoration
ISGMR
1.01
0.95
ISGMR2
1.00
1.04
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.
Many thanks for your
attention!