Nuclear shape evolution based on microscopic level densities

By combining microscopically calculated level densities with the Metropolis walk method, we develop a consistent framework for treating the energy and angular-momentum dependence of the nuclear shape evolution in the fission process. For each nucleus under consideration, the level density is calculated microscopically for each of more than five million shapes with a recently developed combinatorial method. The method employs the same single-particle levels as those used for the extraction of the pairing and shell contributions to the macroscopic-microscopic potential-energy surface. Containing no new parameters, the treatment is suitable for elucidating the energy dependence of the dynamics of warm nuclei on pairing and shell effects. It is illustrated for the fission fragment mass distribution for several uranium and plutonium isotopes of particular interest.

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Figure 1

The microscopic level density, $\rho \left(E^{*}\right)$, calculated for ${}^{236}\mathrm{U}$ at angular momentum $I=4\hslash$ and plotted versus the backshifted energy, $E_{\text{shift}}^{*}=E^{*}+E_{\text{sh}}+E_{\text{pc}}$, shown for three selected shapes: (i) the second minimum (red/left) ($E_{\text{sh}}=-5.4$ MeV, $E_{\text{pc}}=-1.5$ MeV), (ii) the outer symmetric fission barrier (blue/right) ($E_{\text{sh}}=+8.7$ MeV, $E_{\text{pc}}=-6.1$ MeV), and (iii) a shape in between (green/middle) ($E_{\text{sh}}=-0.3$ MeV, $E_{\text{pc}}=-2.5$ MeV), either with (solid) or without (dashed) pairing.

Figure 2

Microscopic level densities (solid lines) compared to extrapolated values (dashed lines). Three different deformations for ${}^{236}\mathrm{U}$ are considered: curve $A$ corresponds to the second minimum, $B$ to the asymmetric second saddle, and $C$ to an elongated symmetric shape close to the outer barrier. Extrapolated values are shown for local excitation energies $E^{*}\left(\mathbf{\chi }\right)\ge 6\text{MeV}$. The inset shows the angular momentum distribution of the level density at the asymmetric saddle ($B$) for different excitation energies.

Figure 3

${\text{log}}_{10}\left(\text{number}\text{of}\text{visits}\right)$ to grid points with a given combination of asymmetry ${\alpha }_g$ and elongation $q_2$ for ${}^{236}\mathrm{U}$ with $E_{\text{exc}}=6.55\text{MeV}$. (a) All tracks in the simulation. (b) The tracks ending in symmetric fission. Energy minima are shown as solid circles and saddle points as crosses. The solid black curve shows the mean “scission” elongation. The ridge separating symmetric from asymmetric fission paths is shown as a solid red (dark gray)/dashed green curve, where green/red indicates that the ridge is below/above the total energy, $E_{\mathrm{exc}}=6.55$ MeV, and thus accessible/inaccessible.

Figure 4

Potential-energy curves for ${}^{236}\mathrm{U}$ versus elongation, $q_2$. The energy path leading to asymmetric fission, starting close to the (symmetric) second minimum, is shown by a solid line. The valley leading to symmetric fission (dashed line) is separated from the asymmetric path by the ridge shown by the red (dark gray) line. The thin horizontal line indicates the excitation energy $E_{\text{exc}}=6.55\mathrm{MeV}$.

Figure 5

Contour plot (on a logarithmic scale) of the location of the end points projected onto the $({\alpha }_{\mathrm{g}},q_2)$ plane for fission of ${}^{236}\mathrm{U}$ at $E_{\mathrm{exc}}=6.55\text{MeV}$. Fragment proton numbers (shown on the upper axis) are obtained as $Z_{\text{f}}=46(1+{\alpha }_{\text{g}})$.

Figure 6

Logarithm of the ergodicity ratio, ${\text{log}}_{10}\left(r\right)$, where $r$ is given by Eq. (13), for ${}^{236}\mathrm{U}$ with $E_{\text{exc}}=6.55\text{MeV}$.

Figure 7

Angular-momentum dependence of the yield for ${}^{234}\mathrm{U}$ at two different excitation energies, (a) $E_{\text{exc}}=6.84$ MeV and (b) 11 MeV. The values $I=0,1,\cdots ,9$ for the compound nucleus are considered. The yield curves for the different angular momenta are on top of each other, except for $I=0$ at the thermal-neutron induced fission energy $E_{\text{exc}}=6.84\text{MeV}$.

Figure 8

Fission-fragment charge yields for ${}^{234}\mathrm{U}$ at three different excitation energies: (a) $E_{\mathrm{exc}}=6.84$ MeV, (b) 11 MeV, and (c) 16 MeV. The blue solid curves have been obtained with the microscopic level densities, while the dashed red curves were calculated with the effective level density ${\rho }_{\mathrm{eff}}$ introduced in Ref. [2]. The results for $E_{\text{exc}}=6.84\text{MeV}$ are compared to $(n_{\text{th}},f)$ data [21], while those for $E_{\text{exc}}=11\text{MeV}$ are compared to $(\gamma ,f)$ data [22].

Figure 9

${}^{226}\mathrm{Th}$ charge yields obtained using either the microscopic level density (blue solid line) or effective level-density parametrization [2] (red dashed line), compared to various experimental data: (a) ${}^{226}\mathrm{Th}(\gamma ,f)$ [22], (b) ${}^{235}\mathrm{U}(n_{\mathrm{th}},f)$ [21], and (c) ${}^{239}\mathrm{Pu}(n_{\mathrm{th}},f)$ [21]. It is not within the present scope of the models to treat the odd-even staggering seen in the data.

Figure 10

Calculated charge yields for fission of the nucleus ${}^{236}\mathrm{U}$ at four values of the excitation energy: from $E_{\text{exc}}=6.55$ MeV ($=S_{\text{n}}$, the neutron separation energy) up to 21.55 MeV in steps of 5 MeV.

Figure 11

Symmetric mass yields versus excitation energy for (a) ${}^{226}\mathrm{Th}$, (b) ${}^{234}\mathrm{U}$, and (c) ${}^{240}\mathrm{Pu}$. The calculated yields are obtained with ${\rho }_{\mathrm{mic}}$ (blue circles) or ${\rho }_{\text{eff}}$ as in Ref. [2] (dashed red curve). The squares show various experimental data from Refs. [22] (A), [21] (B), [21] (C), and [23] (D); the charge yields from Ref. [22] have been multiplied by $Z/A$.

Figure 12

Mass yields (logarithmic scale) at symmetry for ${}^{236}\mathrm{U}$ shown versus excitation energy. The calculated results (blue circles) are compared with experimental data A (open squares) from Ref. [24] and B (filled squares) from Ref. [21]. Results of two additional calculations are also shown, one where the level densities are obtained without pairing (green crosses) and one where the pairing strength is increased by 25% (triangles). The potential-energy surface used in the three calculations is the same, consistent with standard pairing.

Figure 13

Level densities at the first three accessible points along the ridge separating the asymmetric and symmetric valleys.