Figure 1

Scenario of IAR-${\mathit{pp}}^{\text{'}}$ for the ${}^{208}\mathrm{Pb}$($p,p^{\text{'}}$) reaction (scale of proton energy $E_p,E_{p^{\text{'}}}$ at left). The energies of the particle and hole configurations $\mathit{LJ},\nu$ and $\mathit{lj},\nu$ are taken from Ref. [6]. A single IAR state with spin $\mathit{LJ}=i_{11/2}$ as the second member of the isobaric analog multiplet [${}^{209}\mathrm{Pb}$, ${}^{209}\mathrm{Bi}$, $\dots$] with isospin $T=45/2$ is exemplified by one configuration $i_{11/2}{f}_{5/2}$, but all 45 excess neutrons $\mathit{lj},\nu$, including the $i_{11/2}$ particle, participate equally [see Eq. (3)]. Excitation functions of a typical particle-hole configuration are shown at upper left for each IAR [12]. For the two weakest IAR $i_{11/2},j_{15/2}$ they are barely visible; therefore they are scaled up by a factor of 10 (thick curves). The spin splitting of the neutron particle-hole multiplets is shown in a schematic way at lower left. The relative strength is calculated using Eq. (7) with s.p. widths from Table . The penetrability of the Coulomb barrier for the populating particle $\mathit{LJ}$ can be estimated by comparing the maxima of the excitation functions for the $g_{9/2},g_{7/2}$ (solid) and the $d_{5/2},d_{3/2}$ (dotted) IAR. Similarly the penetrability of the outgoing particles $\mathit{lj}$ can be seen by comparing the mean cross section for the particle-hole configurations $|i_{11/2},\nu \rangle \otimes |\mathit{lj},\nu \rangle$ with spins $I=J-j,\dots ,J+j$ (lower left).Reuse & Permissions

Figure 2

${}^{208}\mathrm{Pb}$($p,p^{\text{'}}$) spectra fitted by GASPAN taken on the $i_{11/2}$ IAR for the energy range $E_x=5.185\text{−}5.290$ MeV. The 5239 level is stronger excited at lower scattering angles Θ.Reuse & Permissions

Figure 3

Spectra of ${}^{208}\mathrm{Pb}$($p,p^{\text{'}}$) for $E_x=4.67\text{−}5.00$ MeV taken at $\Theta ={58}^{°},{72}^{°},{54}^{°}$ on the $g_{9/2},i_{11/2},d_{5/2}$ IAR, with targets T3, T2, T2 (see caption of Table ), respectively. Six levels resonate at $E_p=15.72$ MeV on top of the $i_{11/2}$ IAR (black fill out); the 4709+4711 doublet is resolved by the computer code GASPAN. The energies and the spins of the $i_{11/2}{f}_{5/2}$ multiplet are given in the center panel; the energies are also shown by bars in the higher and lower panel. The counting interval is proportional to $\sqrt{E_x}$ (the momentum of the scattered proton) where one channel corresponds to about 0.3 keV.Reuse & Permissions

Figure 4

Spectra of ${}^{208}\mathrm{Pb}$($p,p^{\text{'}}$) for the $i_{11/2}{p}_{3/2}$ multiplet in the range of $E_x=5.02\text{−}5.29$ MeV. For other details refer to Fig. 3 and the text.Reuse & Permissions

Figure 5

Angle averaged (mean) cross section ${\sigma }_{\mathit{LJ}}^{\alpha I}$ for states containing most of the $i_{11/2}{f}_{5/2}$ strength and $i_{11/2}{p}_{3/2}$ strength. For each state (energy label at left and spin at right), the value ${\sigma }_{\mathit{LJ}}^{\alpha I}$ is shown relative to the mean cross section on all IAR. To obtain partial widths [Eq. (7)], the mean cross section must be reduced by the penetrability ratio $R_{\mathit{LJ}}$ [Eq. (9)] for each IAR $\mathit{LJ}$ given at bottom.Reuse & Permissions

Figure 6

Angular distributions for the 4.71-MeV doublet partner with spin $4^{-}$ and the states with spin $6^{-},7^{-},8^{-}$. The calculated angular distribution for a pure configuration $i_{11/2}{f}_{5/2}$ [Eq. (8)] is shown by the drawn curve, the mean cross section by a dashed line. The calculated curves are corrected for the energy dependent penetrability [Eq. (10)].Reuse & Permissions

Figure 7

Angular distributions states for the $i_{11/2}{p}_{3/2}$ states with spins $4^{-},5^{-},6^{-},7^{-}$. For other details see Fig. 6.Reuse & Permissions

Figure 8

(Upper panel) The centroid excitation energy [Eq. (15)] and the total configuration strength $\sum _I\left|c_{\mathit{LJ},\mathit{lj}}^{\alpha I}\right|{}^2$ are shown. The centroid energies agree with the SSM energies $E_x^{\mathrm{SSM}}=4.210,4.780,5.108$ MeV for the three configurations $i_{11/2}{p}_{1/2},i_{11/2}{f}_{5/2},i_{11/2}{p}_{3/2}$. The total configuration strengths are close to unity using the s.p. widths from Table . (Lower panel) The excitation energies $E_x$ and the partial strength $\left|c_{\mathit{LJ},\mathit{lj}}^{\alpha I}\right|{}^2$ for the states bearing the main strength of the $i_{11/2}{p}_{1/2},i_{11/2}{f}_{5/2},i_{11/2}{p}_{3/2}$ configurations are shown. The cross sections ${\sigma }_{\mathit{LJ}}^{\alpha I}$ from Table are converted to partial strengths by Eq. (7) with s.p. widths from Table and corrected for the energy dependence of the penetrability [Eq. (11)]. For the $i_{11/2}{p}_{1/2}$ multiplet the sum of the partial strengths of the three $5^{-}$ and the three $6^{-}$ states at 4.0 MeV $<E_x<4$.5 MeV is shown [23, 24]. The extremely large cross section of the 4698 $3^{-}$ state on the $i_{11/2}$ IAR is because of the interfering contributions from the $g_{9/2}{p}_{3/2}$ and $d_{5/2}{p}_{1/2}$ configurations [21], therefore it is left out in the determination of the centroid energy and the total configuration strength. The SSM predicts a value $\left|c_{\mathit{LJ},\mathit{lj}}^{\alpha I}\right|{}^2=1$ (dotted line).Reuse & Permissions