Complete identification of states in ${}^{208}\mathrm{Pb}$ below $E_x=6.2$ MeV

The Q3D magnetic spectrograph at the Maier-Leibnitz-Laboratorium of the Ludwig-Maximilians-Universität München and the Technische Universität München (Garching, Germany), was used to study the ${}^{208}\mathrm{Pb}\left(\mathit{p},{\mathit{p}}^{\prime }\right)$, ${}^{206,207,208}\mathrm{Pb}\left(\mathit{d},\mathit{p}\right)$, and ${}^{208}\mathrm{Pb}\left(\mathit{d},{\mathit{d}}^{\prime }\right)$ reactions. One hundred fifty-one states at $E_x<6.20\mathrm{MeV}$ in ${}^{208}\mathrm{Pb}$ are identified and spin and parity assigned. Four states are newly identified and new spins and/or parities are assigned to 25 states. Tentative spin assignments are done to five states at $5.90<E_x<6.10\mathrm{MeV}$. Nearly 50 levels below $E_x=6.20\mathrm{MeV}$ listed by the Nuclear Data Sheets as of 2007 are recognized to be nonexistent or doubly placed. The schematic shell model describing one-particle–one-hole configurations without residual interaction is extended by including two-particle–two-hole configurations. The number of configurations thus predicted at $E_x<6.20\mathrm{MeV}$ nearly agrees with the number of states identified. Several states with dominant two-particle–two-hole configurations are identified. New isobaric analog resonances in ${}^{209}\mathrm{Bi}$ with two-particle–one-hole structure are discovered at $E^{\mathrm{res}}=17.6\mathrm{MeV}$. The excitation energies of 70 states with unnatural parity at $E_x<6.20\mathrm{MeV}$ are found to agree within about 200 keV with one-particle–one-hole configurations predicted by the extended schematic shell model. In contrast, the excitation energies of about 20 natural parity states are more than 0.5 MeV lower than predicted, demonstrating the residual interaction among the configurations to be much larger for natural parity than for unnatural parity.

Figure 1

Level schemes in the four neighbors of the doubly magic nucleus ${}^{208}\mathrm{Pb}$, single-hole levels in ${}^{207}\mathrm{Tl}$ and ${}^{207}\mathrm{Pb}$, and single-particle levels in ${}^{209}\mathrm{Bi}$ and ${}^{209}\mathrm{Pb}$. Dotted lines denote intruder orbits (reverse parity).

Figure 2

Statistics of levels listed by NDS2007 and of identified states in ${}^{208}\mathrm{Pb}$ for $E_x<7.0\mathrm{MeV}$. At $E_x<6.20\mathrm{MeV}$ 151 states are identified (Sec. 4). Above 6.2 MeV spin and parity for only few states are known; consequently, no states are shown. Three large gaps (Sec. 2e) in the mSM are denoted by vertical dashed lines. The deep minimum in panel (b) near the gap at $E_x\approx 6.2\mathrm{MeV}$ corresponds to the difference $N^{\text{predict}}\left(2\right)-N^{\text{ident}}\left(2\right)$ [Eqs. (38) and (52)]. (a) Number of configurations predicted by the mSM (dotted line, Sec. 2a) and by the eSM (solid line; Secs. 2d1, 2d2, 2d3, 2d4, 2d5, 2d6, 2d7); (b) difference between the number of identified states and of configurations predicted by the eSM; (c) the number of NDS2007 levels recognized as “spurious” (Sec. 6) and the appearance of newly identified states.

Figure 3

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $1^{+}$ and $2^{+}$. For details, see Sec. 3c1.

Figure 4

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $3^{+}$ and $4^{+}$. For details, see Sec. 3c1.

Figure 5

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $5^{+}$ and $6^{+}$. For details, see Sec. 3c1.

Figure 6

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $7^{+}$ and $8^{+}$. For details, see Sec. 3c1.

Figure 7

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $9^{+}$ and ${10}^{+}$. For details, see Sec. 3c1.

Figure 8

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $0^{+}$ and ${11}^{+}$. For details, see Sec. 3c1.

Figure 9

Level scheme for states in ${}^{208}\mathrm{Pb}$ with the spin of ${12}^{+}$. The acronym of only five configurations is shown; Table 4 shows all excitation energies and configurations. For more details, see Sec. 3c1. The 6101 state is discussed in Sec. 4b3.

Figure 10

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $0^{-}$ and $1^{-}$. For details, see Sec. 3c1.

Figure 11

Level scheme for states in ${}^{208}\mathrm{Pb}$ with the spin of $2^{-}$. For details, see Sec. 3c1.

Figure 12

Level scheme for states in ${}^{208}\mathrm{Pb}$ with the spin of $3^{-}$. For details, see Sec. 3c1.

Figure 13

Level scheme for states in ${}^{208}\mathrm{Pb}$ with the spin of $4^{-}$. For details, see Sec. 3c1.

Figure 14

Level scheme for states in ${}^{208}\mathrm{Pb}$ with the spin of $5^{-}$. For details, see Sec. 3c1.

Figure 15

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $6^{-}$ and $7^{-}$. For details, see Sec. 3c1.

Figure 16

Level schemes for states in ${}^{208}\mathrm{Pb}$ with the spins of $8^{-}$ and ${14}^{-}$. For details, see Sec. 3c1.

Figure 17

Spectrum for the ${}^{208}\mathrm{Pb}\left(\mathit{d},{\mathit{d}}^{\prime }\right)$ reaction at $3.1<E_x<4.0\mathrm{MeV}$. For details, see Sec. 3c2.

Figure 18

Spectra for the ${}^{208}\mathrm{Pb}\left(\mathit{d},{\mathit{d}}^{\prime }\right)$ reaction at $3.9<E_x<5.0\mathrm{MeV}$. For details, see Sec. 3c4.

Figure 19

Spectra for the ${}^{208}\mathrm{Pb}\left(\mathit{d},{\mathit{d}}^{\prime }\right)$ reaction at $5.0<E_x<5.6\mathrm{MeV}$. For details, see Sec. 3c4.

Figure 20

Spectra for the ${}^{208}\mathrm{Pb}\left(\mathit{d},{\mathit{d}}^{\prime }\right)$ reaction at $5.60<E_x<6.20\mathrm{MeV}$. For details, see Sec. 3c4.

Figure 21

Example for the fit of doublets with gaspan. The lower residuum spectrum belongs to a fit without the seven levels marked magenta; the ovals mark $L$ satellites. For details, see Sec. 3e2.

Figure 22

Example for resolving a doublet. Here, for the 0.5-keV doublet of the 5812 $2^{-}$ and 5813 $3^{-}$ states, the distribution of the values $E_x$ obtained by gaspan is shown without the uncertainties; in Fig. 20 they are shown with the uncertainties. The abscissa shows the excitation energies for $5810<E_x<5815\mathrm{keV}$, the ordinate the run number. The two requested values $E^{\text{req}}=5812.8$ and 5813.2 keV are marked at top and bottom. The 120 runs are divided up into five groups marked by short horizontal lines at left and right. The chosen limits are shown at top with $\mathrm{\Delta }{E}_x^{\mathrm{req}}=0.60$ and 0.30; the asymmetric limits distribute the values $E_x$ over two parts.

Figure 23

Another display of the data shown in Fig. 19. The two figures show the distribution for the deviations from the requested values $E^{\text{req}}=5812.8$ and 5813.2 keV (long arrows at left) for 120 runs. Long-dashed lines ending in open diamonds at right denote the chosen limits $\mathrm{\Delta }{E}_x^{\text{req}}$. For each of the five groups (marked by vertical dotted lines) the mean values $\overline{E_x\left(g\right)-E_x^{\mathrm{req}}}$ [Eq. (39)] are shown by a short dashed line; the value and its uncertainty is shown at top and bottom, respectively. The global mean values ${\overline{E}}_x=5812.75\pm 0.02$ and $5813.24\pm 0.02\mathrm{keV}$ [Eq. (40)] are shown at far right by short arrows; the black arrow at left shows the value $E_x^{\mathrm{req}}$. Near the j${}_{15/2}$ and d${}_{5/2}$ IARs (third and fourth group) the 5813 $3^{-}$ state is more strongly excited than the 5812 $2^{-}$ state at most chosen scattering angles; the scattering angles ${20}^{\circ }\le \mathrm{\Theta }\le {138}^{\circ }$ increase from left to right.

Figure 24

Dependence of the amplitudes $c_{LJ,lj}^{I_M^{\pi }}$ [Eq. (10)] from the anisotropy coefficients $A_K/A_0$ [Eq. (46)] for the (top) 5038 $2_2^{-}$ and (bottom) 5127 $2_3^{-}$ states. The drawn lines show $A_2/A_0$ with the uncertainty $\pm 1\sigma$, the dashed lines show $A_4/A_0$ with the uncertainty [28]. The amplitudes near (top) $c_{\mathit{g}{}_{9/2}\mathit{p}{}_{3/2}}^{2_2^{-}}=-0.4$, $c_{\mathit{g}{}_{9/2}\mathit{f}{}_{5/2}}^{2_2^{-}}=+0.1$ and (bottom) $c_{\mathit{g}{}_{9/2}\mathit{p}{}_{3/2}}^{2_3^{-}}=-0.3$, $c_{\mathit{g}{}_{9/2}\mathit{f}{}_{5/2}}^{2_3^{-}}=+0.5$ fit the angular distributions (Sec. 3g1).

Figure 25

Excitation functions for the ${}^{208}\mathrm{Pb}\left(\mathit{p},{\mathit{p}}^{\prime }\right)$ reaction showing a new IAR at $E^{\mathrm{res}}=17.60\mathrm{MeV}$ with an assumed width of ${\mathrm{\Gamma }}^{\text{tot}}=300\mathrm{keV}$. Excitation functions for the individual states are shown at the bottom in a staggered manner (the base lines are marked at left and right) for the sum at top. The mean cross section is about 5 $\mu \mathrm{b}/sr$ near the new IAR and drops by a factor of five off-resonance. In the top frame, the Lorentzians for the known IARs at $E^{\mathrm{res}}>16.3\mathrm{MeV}$ are shown with arbitrary height by dotted lines together with the values of $LJ$.