Continuum corrections to the level density and its dependence on excitation energy, n-p asymmetry, and deformation

In the independent-particle model, the nuclear level density is determined from the neutron and proton single-particle level densities. The single-particle level density for the positive-energy continuum levels is important at high excitation energies for stable nuclei and at all excitation energies for nuclei near the drip lines. This single-particle level density is subdivided into compound-nucleus and gas components. Two methods are considered for this subdivision: In the subtraction method, the single-particle level density is determined from the scattering phase shifts. In the Gamov method, only the narrow Gamov states or resonances are included. The level densities calculated with these two methods are similar; both can be approximated by the backshifted Fermi-gas expression with level-density parameters that are dependent on A, but with very little dependence on the neutron or proton richness of the nucleus. However, a small decrease in the level-density parameter is predicted for some nuclei very close to the drip lines. The largest difference between the calculations using the two methods is the deformation dependence of the level density. The Gamov method predicts a very strong peaking of the level density at sphericity for high excitation energies. This leads to a suppression of deformed configurations and, consequently, the fission rate predicted by the statistical model is reduced in the Gamov method.

Figure 1

(Color online) The evolution with ($\varepsilon -\mu )/T$ of the functions f and s [Eq. (8)] used to determine the total energy and entropy. The vertical dashed line indicates the location of $\varepsilon =0$, when $T=E_{\mathrm{sep}}/2$.

Figure 2

(Color online) The dependence of the compound-nucleus single-particle level density $g_{\mathrm{CN}}^{\mathrm{sub}}$ on the nucleon energy $\varepsilon$. The displayed results have been convoluted by a Gaussian resolution of $\mathrm{FHWM}=150\mathrm{keV}$. Results are shown for (a) neutrons in a spherically symmetric potential ($Q=0$), (b) neutrons in a deformed potential ($Q=4$), and (c) protons in a spherically symmetric potential.

Figure 3

(Color online) Evolution of the real energy ${\varepsilon }^R$ for bound (${\varepsilon }^R<0$, ${\Gamma }^R=0$) and narrow Gamov $({\varepsilon }^R>\text{0,}{\Gamma }^R<1\text{MeV})$ states as a function of deformation. The width of the curves reflects the value of ${\Gamma }^R$. Results are shown only for (a) neutron and (b) proton $m^{\pi }=1/2^{-}$ states in ${}^{190}\mathrm{Yb}$.

Figure 4

(Color online) Variation with nucleon energy $\varepsilon$ of the single-particle level density determined by the subtraction method for $m^{\pi }=1/2^{+}$ neutrons in ${}^{190}\mathrm{Yb}$. Results are shown for two neighboring deformations $Q=1.75$ and 1.625.

Figure 5

(Color online) Smoothed single-particle level densities calculated for (a) neutrons and (b) protons in ${}^{160}\mathrm{Yb}$. The curves labeled “sub” were obtained from the subtraction method. The other curves were obtained from the Gamov method with the indicated values of ${\Gamma }_0$ in MeV. The Coulomb barrier for protons is indicated by the arrow in (b).

Figure 6

(Color online) Smoothed single-particle level densities calculated for neutrons in ${}^{160}\mathrm{Yb}$ at three defomations: $Q=-0.5,0,$ and 0.5. (a) Results obtained with the subtraction method; (b) results obtained with the Gamov method for ${\Gamma }_0=1\mathrm{MeV}$.

Figure 7

(Color online) Curves show the variation with deformation Q of the (a) width ${\Gamma }^R$ and (b) energy ${\varepsilon }^R$ of the Gamov states associated with the $j=19/2^{-}$ neutrons in ${}^{160}\mathrm{Yb}$. The solid data points indicate the average energy and width for these states.

Figure 8

(Color online) Variation of the neutron shell correction to the deformation energy determined for ${}^{170}\mathrm{Yb}$. (a) The absolute correction obtained with the indicated smoothing ranges $\gamma$. (b) The data from (a) with the mean correction over all calculated deformations $\langle \delta \left(Q\right)\rangle$ subtracted out.

Figure 9

(Color online) Variation of four calculated energies with deformation for ${}^{170}\mathrm{Yb}$ and ${}^{150}\mathrm{Yb}$. The curve $E_{\mathrm{def}}$ is the deformation energy with shell corrections. For comparison, the liquid-drop deformation energy $V_{\mathrm{ld}}$ is also shown. The higher two curves give the energies corresponding to the critical temperatures $T^{\mathrm{crit}}$ for protons and neutrons. The thick dashed lines indicate the deformation which maximized the level density (subtraction method) for each excitation energy.

Figure 10

(Color online) Variation of the nuclear level density with excitation energy for (a) the seven $A~170$ nuclei, (b) the six $A=60$ nuclei, and (c) the five $A=40$ nuclei. All are calculated up to the excitation energy where $T=E_{\mathrm{cost}}^{min}$. If these maximum excitation energies are within the displayed range, they are indicated by the solid circular symbols and are labeled with the name of the nucleus.

Figure 11

(Color online) Same as for Fig. 10(a), but now the level density for $A~170$ is calculated with the Gamov method using the cutoff decay widths of${\Gamma }_0=1$ and 0.01 MeV. Only curves that are distinguishable from the others are labeled.

Figure 12

(Color online) Same as for Fig. 10(c), but now the level density for $A=40$ is calculated with the Gamov method using the cutoff decay widths of ${\Gamma }_0=1$ and 0.01 MeV. Only curves that are distinguishable from the others are labeled.

Figure 13

(Color online) Level-density parameters $\tilde{a}$ deduced from the calculated entropy and Eq. (62) as a function of the shifted excitation energy per nucleon. Results were obtained using the subtraction method to treat the continuum. The data points, indicated on each curve, are the points at which $E_{\mathrm{cost}}^{min}=T$.

Figure 14

(Color online) Same as for Fig. 13(a), but here the results for $A~170$ were obtained from the Gamov method with (a) ${\Gamma }_0=1\mathrm{MeV}$ and (b) ${\Gamma }_0=0.01\mathrm{MeV}$. Only curves that are distinguishable from the others are labeled.

Figure 15

(Color online) Same as for Fig. 13(c), but here the results for $A=40$ were obtained from the Gamov method with (a) ${\Gamma }_0=1\mathrm{MeV}$ and (b) ${\Gamma }_0=0.01\mathrm{MeV}$.

Figure 16

(Color online) Calculated level-density parameters at $(E^{*}-\delta P+\delta W)/A=0.5\mathrm{MeV}$ are plotted versus $N-N_{\beta }(A)$, the neutron number separation from the $\beta$ valley of stability. The data points were obtained with the subtraction and Gamov methods. For the latter, the three indicated values of the cutoff width ${\Gamma }_0$ were used. For $A=40$ and 60, the fitted variation of the level-density parameter (cases B and C) from Ref. 5 are shown by the dot-dashed and dashed curves, respectively.

Figure 17

(Color online) Asymptotic level-density parameter $\tilde{a}$ as a function of nucleon number A. The data points are the values determined from the calculations of this work. (a) The dashed curve shows a fit to these data with volume and surface components (see text). Curves are also shown for the prescription of Tõke and Światecki[49] and Ignatyuk et al. [50]. (b) The curves show linear dependences of the level-density parameter on A, i.e., $\tilde{a}=A/k$. The value of k for each of the curves is indicated in units of MeV.

Figure 18

(Color online) Asymptotic level-density parameters determined for the $A~170$ systems with Eq. (65). (a) The parameter $\gamma$ was set to infinity; these results are the same as those shown in Fig. 13(a). (b) The value $\gamma =0.035$ was obtained from minimizing the spread in the curves at low excitation energy.

Figure 19

(Color online) Deformation dependence of the level density calculated for ${}^{170}\mathrm{Yb}$ at the indicated excitation energies. The level densities are normalized to the maximum ${\rho }_{max}$ for that excitation energy. For clarity the results for each successive excitation energy are shifted up along the y axis. The thin lines in each case correspond to the shifted x axis. Results obtained with the subtraction method are shown as the thick solid curves, while the dashed curves represent results obtained with the Gamov method for ${\Gamma }_0=1\mathrm{MeV}$.

Figure 20

(Color online) Deformation dependence of the normalized level density as in Fig. 19. Results are shown for ${}^{160}\mathrm{Yb}$ ($E^{*}=300\mathrm{MeV}$) and ${}^{60}\mathrm{Ni}$ ($E^{*}=112\mathrm{MeV}$) with the subtraction method (solid curves) and with the Gamov method for ${\Gamma }_0=1\mathrm{MeV}$ (dashed curves) and 0.01 MeV (dot-dashed curves).

Figure 21

(Color online) Deformation dependence of the normalized level density for three Yb isotopes at the indicated excitation energies. Solid and dashed curves were obtained with the subtraction and Gamov (${\Gamma }_0=1\mathrm{MeV}$) methods, respectively.

Figure 22

(Color online) Deformation dependence of the normalized level density for ${}^{238}\mathrm{U}$ as in Fig. 19.

Figure 23

(Color online) Asymptotic level-density parameters deduced for deformed ${}^{150}\mathrm{Yb}$ nuclei (oblate and prolate shapes) as a function of the relative surface area $B_s$. Results are shown for the indicated thermal excitation energies. For clarity, results at some excitation energies have been shifted up along the y axis by the indicated amounts.

Figure 24

(Color online) Surface coefficient of the level-density parameter obtained for $A~170$ systems as a function of excitation energy. The dotted line indicates the value obtained from the fit in Fig. 17. Other values of the surface coefficient from the prescriptions of Tõke and Światecki [49] and Ignatyuk et al. [50] are also indicated.