Folding model study of the isobaric analog excitation: Isovector density dependence, Lane potential, and nuclear symmetry energy

A consistent folding model analysis of the ($\Delta S=0,\Delta T=1$) charge exchange $(p,n)$ reaction measured with ${}^{48}\mathrm{Ca}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets at the proton energies of 35 and 45 MeV is done within a two-channel coupling formalism. The nuclear ground state densities given by the Hartree-Fock-Bogoliubov formalism and the density-dependent CDM3Y6 interaction were used as inputs for the folding calculation of the nucleon optical potential and $(p,n)$ form factor. To have an accurate isospin dependence of the interaction, a complex isovector density dependence of the CDM3Y6 interaction has been carefully calibrated against the microscopic Brueckner-Hartree-Fock calculation by Jeukenne, Lejeune, and Mahaux before being used as folding input. Since the isovector coupling was used to explicitly link the isovector part of the nucleon optical potential to the cross section of the $(p,n)$ reaction exciting the 0${}^{+}$ isobaric analog states in ${}^{48}\mathrm{Sc}$, ${}^{90}\mathrm{Nb}$, ${}^{120}\mathrm{Sb}$, and ${}^{208}\mathrm{Bi}$, the newly parametrized isovector density dependence could be well tested in the folding model analysis of the $(p,n)$ reaction. The isospin- and density-dependent CDM3Y6 interaction was further used in the Hartree-Fock calculation of asymmetric nuclear matter, and a realistic estimation of the nuclear symmetry energy was made.

Figure 1

OM description of the elastic proton scattering from ${}^{48}\mathrm{Ca}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets at 40 MeV obtained with the global OP's by Becheetti and Greenlees (BG) [6], Varner et al. (CH89) [7], and Koning and Delaroche (KD) [8]. The data were taken from Refs. [34, 35].

Figure 2

CC description of the charge exchange $A_{g.s.}(p,n){\tilde{A}}_{\mathrm{IAS}}$ reaction measured with ${}^{48}\mathrm{Ca}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets at 35 MeV obtained with $U_1$ deduced from the global OP by Becheetti and Greenlees (BG) [6], Varner et al. (CH89) [7], and Koning and Delaroche (KD) [8]. The data were taken from Ref. [11].

Figure 3

Same as Fig. 2, but for $(p,n)$ data measured at the proton energy of 45 MeV.

Figure 4

Real part $V_{\mathrm{IV}}(E,\rho )$ of the isovector nucleon OP given by the isovector density dependence (20) of the CDM3Y6 interaction in comparison with the JLM results [28] at $E=35$ and 45 MeV.

Figure 5

Same as Fig. 4, but for the imaginary part $W_{\mathrm{IV}}(E,\rho )$ of the isovector nucleon OP.

Figure 6

Imaginary part $W_{\mathrm{IS}}(E,\rho )$ of the isoscalar nucleon OP given by the isoscalar density-dependent interaction (25) compared with the JLM results [28] at $E=35$ and 45 MeV.

Figure 7

OM description of the elastic proton scattering from ${}^{48}\mathrm{Ca}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets at 40 MeV obtained with the complex folded OP [Eq. (28)], and hybrid OP [Eq. (29)]. The data were taken from Refs. [34, 35].

Figure 8

OM description of the elastic neutron scattering from ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ targets at energies of 17–26 MeV obtained with the complex folded OP [Eq. (28)] and hybrid OP [Eq. (29)]. The data were taken from Refs. [47, 48, 49].

Figure 9

Upper part: CC description of the ${}^{48}{\mathrm{Ca}}_{g.s.}(p,n){}^{48}{\mathrm{Sc}}_{\mathrm{IAS}}$ reaction at 35 MeV [11] given by the (unrenormalized) folded $(p,n)$ form factor (30) and that deduced from Eq. (4) using the CH89 parameters [7]. Lower part: OM description of the elastic $p{+}^{48}$Ca scattering at 35 MeV [34] given by the complex folded OP (28), hybrid OP (29), and CH89 global OP [7].

Figure 10

CC description of the ${}^{48}{\mathrm{Ca}}_{g.s.}(p,n){}^{48}{\mathrm{Sc}}_{\mathrm{IAS}}$ reaction [11] at 35 MeV (lower part) and 45 MeV (upper part) given by the renormalized folded $(p,n)$ form factor (30) and that deduced from Eq. (4) using CH89 global OP [7], using two choices [Eqs. (28) and (29)] of the OP.

Figure 11

Real and imaginary parts of the complex folded OP (28) for $p{+}^{48}$Ca system at 45 MeV, scaled by factors $N_{V(W)}$ given by the OM fit of the elastic data at 40 MeV, compared with the CH89 global OP taken from Ref. [7]. The adjusted imaginary CH89 potential has been used in the hybrid choice [Eq. (29)] of the OP.

Figure 12

Same as Fig. 10, but for the ${}^{90}{\mathrm{Zr}}_{g.s.}(p,n){}^{90}{\mathrm{Nb}}_{\mathrm{IAS}}$ reaction [11].

Figure 13

Same as Fig. 10, but for the ${}^{120}{\mathrm{Sn}}_{g.s.}(p,n){}^{120}{\mathrm{Sb}}_{\mathrm{IAS}}$ reaction [11].

Figure 14

Same as Fig. 10, but for the ${}^{208}{\mathrm{Pb}}_{g.s.}(p,n){}^{208}{\mathrm{Bi}}_{\mathrm{IAS}}$ reaction [11].

Figure 15

Density dependence of the nuclear symmetry energy $S(\rho )$ given by the HF calculation of asymmetric NM [12] using different isovector density dependences of the CDM3Y6 interaction and the empirical values deduced from the CC analysis of the $p({}^6\mathrm{He},{}^6{\mathrm{Li}}^{*})n$ reaction [26] as well as the neutron-skin [21] and HI fragmentation [23, 24] studies. See more discussion in the text.