Capture barrier distributions: Some insights and details

The “experimental barrier distribution” provides a parameter-free representation of experimental heavy-ion capture cross sections that highlights the effects of entrance-channel couplings. Its relation to the $s$-wave transmission is discussed, and in particular it is shown how the full capture cross section can be generated from an $l=0$ coupled-channels calculation. Furthermore, it is shown how this transmission can be simply exploited in calculations of quasifission and evaporation-residue cross sections. The system ${}^{48}\mathrm{Ca}+{}^{154}\mathrm{Sm}$ is studied in detail. A calculation of the compound-nucleus spin distribution reveals a possible energy dependence of barrier weights due to polarization arising from target and projectile quadrupole phonon states; this effect also gives rise to an entrance-channel “extra-push.”

Figure 1

Experimental $T_0^{\mathrm{tot}}$ of Quint et al. [9] for ${}^{100}\mathrm{Mo}+{}^{100}\mathrm{Mo}$ demonstrates the concepts of the dynamical and adiabatic barriers, and of $D_{\inf }$ and the extra-push energy $E_{\mathrm{x}}$. The solid curve shows the fit of Ref. [12] to these data, and the dashed curve shows a single barrier cross section which fits the lowest energy points. Parts (a) and (b) show the same curves on logarithmic and linear scales, respectively.

Figure 2

For ${}^{100}\mathrm{Mo}+{}^{100}\mathrm{Mo}$ and a single barrier with $B=202.4$ MeV, we show $E{\sigma }_0$ and $E\sigma$, where ${\sigma }_0$ and σ are the exact optical-model $s$-wave partial cross section (dashed curve) and total capture cross section (solid curve), respectively. The results of Eq. (11) are practically indistinguishable from the exact results on this scale: (a) corresponds to a surface diffuseness $a=0.6$ fm and (b) to $a=1.2$ fm. Note the significant deviations in the Wong cross sections below the barrier (dotted curves) using a curvature calculated at the barrier top.

Figure 3

Same quantities as in Fig. 2, but on a linear scale and at higher $E$. For $a=0.6$ fm, the integral (11) is almost indistinguishable from the exact results and is not shown. For $a=1.2$ fm, there are small differences because the barrier for high $l$ values shifts to smaller radii. In this respect, the Wong cross section gives a very similar result to the integral.

Figure 4

For ${}^{100}\mathrm{Mo}+{}^{100}\mathrm{Mo}$ with coupling to the first quadrupole-phonon state in each nucleus [1,1], we show (a) $E{\sigma }_0$ (dashed curve) and $E\sigma$ (circles), both calculated with CCFULL, and the results of Eq. (19) (solid line), and (b) the corresponding barrier distribution ${\mathit{dT}}_0^{\mathrm{tot}}/\mathit{dE}$.

Figure 5

Same as in Fig. 4, but for the system ${}^{48}\mathrm{Ca}+{}^{154}\mathrm{Sm}$ with quadrupole couplings to the first six excited states [0,6] of the ${}^{154}\mathrm{Sm}$ ground-state rotational band ($0^{+}$ to ${12}^{+}$).

Figure 6

For the system ${}^{16}\mathrm{O}+{}^{238}\mathrm{U}$, we show the difference between including the barrier dependence of the radius and ignoring it [see Eq. (24)]. Both curves are normalized to unity. The insets show schematically how the high barriers correspond to a more compact configuration than do the low ones (see Sec. 4b).

Figure 7

Expression (11) reproduces very well the logarithmic derivative far below the Coulomb barrier. This is an uncoupled calculation for ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ with $a=1.65$ fm. The Wong approximation does not reproduce the correct behavior in this energy region.

Figure 8

Calculated capture barrier distribution for the [1,7] ${}^{48}\mathrm{Ca}+{}^{154}\mathrm{Sm}$ calculation. The dot-dashed line shows the part of this distribution which contributes to quasifission, and the dashed curve is the QF barrier distribution used, including the factor ${\gamma }_{\mathrm{QF}}=0.2$ (see text and Fig. 9).

Figure 9

The measured and calculated capture, quasifission, and evaporation-residue cross sections for ${}^{48}\mathrm{Ca}+{}^{154}\mathrm{Sm}$ (see text and Figs. 8, 10 for more details).

Figure 10

$P_{\mathrm{sur}}(l,E^{*})$ as a function of $l$ for the ${}^{202}\mathrm{Pb}$ compound nucleus at three different excitation energies $E^{*}({}^{202}$Pb)$=51.25,53.75$, and 56.25 MeV (corresponding to $E=142$, 144.5, and 147 MeV in the ${}^{48}\mathrm{Ca}+{}^{154}\mathrm{Sm}$ reaction). The symbols come from the HIVAP calculation described in the text. The solid curves are from the parametrization (30) with $[{\gamma }_{\mathrm{sur}},L_{\mathrm{sur}},{\Delta }_{\mathrm{L}}]=$ [0.91, 37.4, 21.7]; [0.89, 35.4, 21.7]; [0.87, 33.2, 21.7] in order of increasing energy. The upper and lower curves are used in the integral expression for ${\sigma }_{\mathrm{ER}}$, below 142 MeV and above 147 MeV, respectively, with a linear interpolation between these energies.

Figure 11

Solid lines are exact CCFULL calculations of the spin distribution $(2l+1)T_l$ of the compound nucleus. Dashed lines are the sums of uncoupled optical-model calculations with barrier heights and weights defined for $l=0$ (see Fig. 12 and Sec. 6). The discrepancies are relatively small for critical $l$ values typical of high-spin state experiments, but they show a significant variation of the barrier weights at high $E$ (see Sec. 6).

Figure 12

Exact $T_0^{\mathrm{tot}}$ (solid line) which corresponds to the system of Fig. 4, along with the sum of uncoupled calculations (dashed line) with the barrier weights and heights $(w_{\alpha },B_{\alpha })=$ (0.038, 193.80), (0.295, 202.96), and (0.667, 212.30).

Figure 13

(a) For a quadrupole phonon with $E^{*}=0.53$ MeV, $T_0^{\mathrm{tot}}$ (dashed line) is very different from its adiabatic value (solid line) for $l=0$, in particular the weight of the lowest barrier is greatly reduced. With no Coulomb coupling (dot-dashed line), however, the lowest barrier is actually enhanced relative to the adiabatic value. (b) For higher $l$, the barrier occurs at higher energy where the weights come closer to their adiabatic values. Calculations for $l=170$ and 200 ħ show that the weights vary little over a limited range of angular momenta/energy.