Adiabatic fusion barriers from self-consistent calculations

Adiabatic fusion barriers are studied within the static Hartree-Fock method with the effective Skyrme interactions SkM${}^{*}$ and SLy6. The problem of kinetic energy of the relative motion becoming spurious for separate fragments, relevant for both fusion and fission barriers, is discussed in some detail. Also discussed are the specific assumptions necessary to compensate for the nonuniqueness of the static method. Barriers obtained with two forces agree to within 2 MeV and seem nearly decoupled from errors in binding energies, specific to each force. For a number of reactions, comparisons are made with experimental estimates of barriers and barriers calculated with the frozen densities. The adiabatic barriers are generally lower than the experimental estimates. The offset amounts to less than 3 MeV in lighter systems and varies between 0 and $~10$ MeV in heavy ones. Compared to the data, the frozen-density barriers in heavy systems do not seem more realistic than ours, perhaps, except for tip collisions with deformed heavy targets. We also calculate HF energy surfaces for three heavy systems, looking for a relation between adiabatic potential and the fusion hindrance at large $Z_T{Z}_P$. One can see a link between quasifission and the force opposing fusion, acting inside the Coulomb barrier. One surface illustrates the identity of the adiabatic fusion barrier with the fission saddle of a compound nucleus.

Figure 1

Upper panel: Densities at the barrier for ${}^{244}\mathrm{Pu}$+${}^{48}\mathrm{Ca}$ tip (left, $Q=225$ b, $R=15.47$ fm) and side (right, $Q=133$ b, $R=13.60$ fm, nonaxial shape with the horizontal symmetry axis of ${}^{244}\mathrm{Pu}$) collision. Lower panel: Densities for configurations with a wider neck, developed from the upper ones by decreasing $Q_{40}$ at constant $Q$. Energy gains are 25.7 (left) and 10.5 MeV (right). Contour lines are 0.01 fm${}^{-3}$ apart.

Figure 2

Densities at the barrier for ${}^{48}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$ (left) and ${}^{208}\mathrm{Pb}$+${}^{48}\mathrm{Ca}$ (right); contour lines are 0.01 fm${}^{-3}$ apart.

Figure 3

Tip (squares) and side (circles) fusion barrier for the ${}^{238}\mathrm{U}$+${}^{16}\mathrm{O}$ reaction with the SkM${}^{*}$ force and no pairing. Results for the constrained partition are shown with solid symbols.

Figure 4

Fusion potentials with the SkM${}^{*}$ force, nearly adiabatic (left) and nonadiabatic (right) in their inner part; squares and circles are for no pairing; triangles and diamonds include paring. Results for the constrained partition are shown with solid symbols. The mass-asymmetric fission valley of ${}^{292}114$ is indicated by lines: dashed with $E_{c.m.}(\mathrm{rel})$ subtracted; solid without subtraction.

Figure 5

Scaled fusion potentials $V/B$ obtained with the SkM${}^{*}$ force shown as functions of the effective radius $R_{\mathrm{eff}}=R/(A_T^{1/3}+A_P^{1/3})$ for ${}^{38}\mathrm{Ar}$+${}^{32}\mathrm{S}$ (triangles), ${}^{48}\mathrm{Ca}$+${}^{48}\mathrm{Ca}$ (solid circles), ${}^{90}\mathrm{Zr}$+${}^{40}\mathrm{Ca}$ (solid squares), ${}^{90}\mathrm{Zr}$+${}^{90}\mathrm{Zr}$ (diamonds), ${}^{208}\mathrm{Pb}$+${}^{48}\mathrm{Ca}$ (squares), and ${}^{208}\mathrm{Pb}$+${}^{70}\mathrm{Zn}$ (circles).

Figure 6

Energy surface for ${}^{256}\mathrm{No}$ over the $(Q,Q_{30})$ plane. Contour lines on the base are 1 MeV apart.

Figure 7

Energy surface for ${}^{278}112$ over the $(Q,Q_{30})$ plane. Contour lines on the base are 1 MeV apart.

Figure 8

Energy surface for ${}^{292}114$ over the $(Q,Q_{30})$ plane. Contour lines on the base are 1 MeV apart.