Correlation between nuclear charge radii of Ti and reaction cross sections for $p$-Ti

The nuclear root-mean-square charge radii and nuclear density distributions have been calculated in the framework of relativistic self-consistent mean-field models based on density-dependent meson-exchange relativistic energy density functional for even isotopes of Ti $\left(A=44-56\right)$. The calculations agree well with available laser spectroscopic measurements for ${}^{44,46,48,50}\mathrm{Ti}$, and results for neutron-rich isotopes are reported. The semimicroscopic proton optical potentials for ${}^{44-56}\mathrm{Ti}$ have been derived by folding the target matter density with the extended Jeukenne-Lejeune-Mahaux energy- and density-dependent internucleon interaction and are used to calculate the differential elastic scattering and reaction cross sections at 65 MeV. The calculated differential cross sections reproduce the corresponding experimental values for stable even isotopes, and predictions are presented for ${}^{44,52,54,56}\mathrm{Ti}$. The calculations reveal a prominent kink at the $N=28$ shell closure observed in the variation of reaction cross sections, consistent with mean-square charge radii systematics in the Ti isotopic chain.

Figure 1

Calculated DIRHB rms charge radii relative to $N=28 \left({}^{50}\mathrm{Ti}\right)$ for even Ti isotopes. The corresponding experimental values [5] (where available) are shown for comparison. The dashed line indicates $\delta {\langle r_c^2\rangle }^{N,28}$ values obtained using droplet model [26].

Figure 2

Calculated DIRHB ($L=0$ projected and renormalized) (a) proton densities and (b) neutron densities for even Ti isotopes.

Figure 3

Renormalized real central, imaginary central and real spin-orbit parts of the folded potential using best fit values of ${\lambda }_V$ and ${\lambda }_W$ for $\left(p,p\right)$ scattering from stable even Ti isotopes. The corresponding phenomenological [11] Woods-Saxon (WS) potentials are also shown.

Figure 4

Renormalization constants for (a) real and (b) imaginary central parts of the potential obtained from the present mom calculation for $\left(p,p\right)$ scattering from even Ti isotopes. The ${\lambda }_V$ and ${\lambda }_W$ values from least-squares fit and average values are shown by solid and dashed lines, respectively for ${}^{44-56}\mathrm{Ti}$. The corresponding best fit values for stable isotopes ${}^{46,48,50}\mathrm{Ti}$ are shown by solid circles.

Figure 5

Calculated (set I) and experimental [11] (where available) differential elastic scattering cross section for 65 MeV proton scattering from selected even Ti isotopes.

Figure 6

Calculated total reaction cross section ${\sigma }_R$ for 65 MeV proton scattering from even Ti isotopes. The ${\sigma }_R$ are calculated using the best fit, least-squares fit (set I) and average (set II) values of ${\lambda }_V$ and ${\lambda }_W$. Geometric cross sections and ${\sigma }_R$ from Kox model are indicated by dotted and dashed lines, respectively.