Nuclear Jacobi and Poincaré transitions at high spins and temperatures: Account of dynamic effects and large-amplitude motion

We present a theoretical analysis of the competition between the so-called nuclear Jacobi and Poincaré shape transitions as a function of spin at high temperatures. The latter condition implies the method of choice, a realistic version of the nuclear liquid drop model, here the Lublin-Strasbourg drop model. We address specifically the fact that the Jacobi and Poincaré shape transitions are accompanied by the flattening of the total nuclear energy landscape as a function of the relevant deformation parameters, which enforces large-amplitude oscillation modes that need to be taken into account. For that purpose we introduce an approximate form of the collective Schrödinger equation whose solutions are used to calculate the most probable deformations associated with the nuclear Jacobi and Poincaré transitions. We discuss selected aspects of the new description focusing on the critical-spin values for both types of these transitions.

Figure 1

Axial-basis cutoff stability test for three nuclei representative for the mass ranges illustrated in this article. The curves correspond to minimizing the nuclear energy at each ${\alpha }_{20}$ over other axially symmetric deformations with increasing $\lambda \le {\lambda }_{\max .}$ Here we select, as a measure of stability, the energy differences in Eq. (15). By definition, vertical lines indicate the elongation at which the biggest contribution equals 20 keV. Observe that, characteristically, the convergence is not monotonic (behavior of the form “the higher the $\lambda$ value, the smaller the discrepancy” does not apply here). (For comments, see Figs. 2 and 3 and discussion in the text.)

Figure 2

(Left) Neck radius as a function of the nuclear elongation for increasing multipolarity in the nuclear surface expansion, in units of the corresponding spherical radius $R_0$. Positions of the vertical lines are the same as those in Fig. 1; for more details, see the text. (Observe a perfect stabilization of the nuclear neck-radius curves in terms of the multipole expansion.) (Right) Illustration of the decomposition of the nuclear neck radii in terms of the increasing multipolarity. Observe a nonmonotonic behavior of various contributions with increasing ${\lambda }_{\max .}$.

Figure 3

Illustration of the shape fluctuations in the neck region in terms of the increasing order of the axially symmetric multipole contributions for ${}^{70}\mathrm{Se}$ as an example at ${\alpha }_{20}=2.5$. Given the fact that the model cannot claim the adequacy of the description of the neck area at the zone illustrated, the fluctuations shown which remain at the level of 1% of the original radius can be considered “harmless” in the context.

Figure 4

Total energy projections for the nuclei indicated at spins in the vicinity of the Jacobi shape transitions. We use the “traditional” quadrupole deformation parameters $(\beta ,\gamma )$. The vertical axis corresponds to $\gamma ={60}^{\circ }$ (oblate), with the nuclear spin directed along the symmetry axis. The path to fission ($\gamma =0^{\circ }$ axis) has ${30}^{\circ }$ inclination with respect to ${\mathcal{O}}_x$ axis. Downsloping lines correspond to $\gamma =-{60}^{\circ }$, oblate shapes with the nucleus turning about an axis perpendicular to the symmetry axis. Finally, the $\gamma =-{120}^{\circ }$ axis corresponds to the prolate shapes with the nuclear spin aligned with the symmetry axis. At each $\{\beta ,\gamma \}$ point the energy was minimized over the even-$\lambda$ deformations ${\alpha }_{\lambda 0}$ for $\lambda \le 16$.

Figure 5

Energy differences for ${}^{46}\mathrm{Ti}$ at spins, representing the energies from the preceding figure minus the projections minimized in addition with respect to ${\alpha }_{42}$, according to the notation introduced in the text: $E_{22}-E_{22,42}$ (left column). The three maps on the right-hand side represent the difference $E_{22,42}-E_{22,42,62}$. Other minimization conditions are same here as in Fig. 4.

Figure 6

Analogous to the preceding figure, here for ${}^{142}\mathrm{Ba}$, representative for heavier nuclei studied in this paper.

Figure 7

Neck radii of the nuclei indicated as a function of the quadrupole deformation. Plotted values are normalized to the radii of the corresponding spherical nuclei, the former defined by $R_0=r_0{A}^{1/3}$, with $r_0=1.2$ fm. The horizontal line shows the geometrical scission reference defined by $R_{\mathrm{scission}}=0.3R_0$. Let us emphasize that the latter quantity can be seen as a somewhat arbitrary reference value.

Figure 8

Illustration similar to the one in Fig. 7, but with the neck radius expressed in fm. At the level of the neck size of the order of 1.5 fm, there is hardly any space left for any single nucleon to move orthogonally with respect to the elongation axis.

Figure 9

LSD-C model energies as functions of ${\alpha }_{20}$, minimized over $\left\{{\alpha }_{\lambda 0}\right\}$ for $\lambda \in [3,12]$ for the $a_{\mathrm{neck}}$ values indicated, compared with the energies from of Myers and Świa̧tecki [58] and labeled (Cong. MS). Experimental values, squares, from the references given in Table 6 are placed at either the scission elongations defined by the condition $R_{\mathrm{neck}}=0.3\times R_0$ (all but one), or the saddle-point deformation in the case of ${}^{173}\mathrm{Lu}$; cf. the text for more details).

Figure 10

Dependence of the nuclear energy calculated using the LSD-C expression with $a_{\mathrm{neck}}=0.5$ at the large elongation regime and at spin $L=0$ $\hslash$, as a function of the basis cutoff. The energies have been minimized over deformations ${\alpha }_{\lambda 0}$ for $\lambda \le {\lambda }_{\max }$ indicated. To increase the legibility of the present illustration, a linear reference has been subtracted as indicated in the description of the vertical axis. By convention, the curves stop at the deformation at which $R_{\mathrm{neck}}<0.3\times R_0$ for all but the ${}^{173}\mathrm{Lu}$ case, in which the saddle-point deformation comes first.

Figure 11

Dependence of the nuclear minimum energy on the basis cutoff parameter ${\lambda }_{\max }$ as indicated as a function of increasing spin. Minimization performed over ${\alpha }_{\lambda 0}$ for $\lambda \le {\lambda }_{\max }$. The macroscopic energies include the shape-dependent congruence energy with $a_{\mathrm{neck}}=0.5$. Curves end if the scission condition is met for spins lower than the spins at which the barrier vanishes, otherwise at the spins of the vanishing barriers. Notice that the stability is obtained already at ${\lambda }_{\max }=8$.

Figure 12

Illustration of the typical behavior of the classical moment of inertia at the deformations corresponding to the minimum of the potential energy for increasing spin, at three typical values of the nuclear radius parameters $r_0$; for details, see the text and Fig. 11. Results for other nuclei have very similar form.

Figure 13

Illustration of the typical impact of the uncertainties in the classical nuclear moments of inertia on the total energy minimization result—here in terms of the total energies at the nuclear energy minima for the three radius-parameter values indicated—plotted relative to a parabolic reference to increase the legibility of the figure.

Figure 14

Fission barrier heights obtained with (dashed line) and without (solid line) congruence-energy contributions by using, in the former case, ${\text{a}}_{\mathrm{neck}}=0.5$. The barrier heights are defined either as the nuclear energy at the scission point—if the saddle point corresponds to the stronger elongation (compared to the scission point)—or else to the saddle-point energy.

Figure 15

Total energy surfaces for increasing spin in ${}^{120}\mathrm{Cd}$, using a “traditional” shape-coordinate representation with the quadrupole deformation parameters $(\beta ,\gamma )$ of Bohr. The vertical, upsloping straight lines correspond to $\gamma ={60}^{\circ }$ (representing oblate shapes, with the spin of “noncollective” origin aligned with the symmetry axis), whereas the “path to fission” ($\gamma =0^{\circ }$ axis) has ${30}^{\circ }$ inclination with respect to the ${\mathcal{O}}_x$ axis. Downsloping straight lines correspond to $\gamma =-{60}^{\circ }$ deformation and represent oblate shape configurations in which the nucleus is rotating collectively about an axis perpendicular to the symmetry axis. Finally, the vertical, downsloping axes correspond to prolate axially symmetric configurations with spins aligned with the symmetry axis. (The minimization over axial-deformation coordinates ${\alpha }_{\lambda 0}$ with $\lambda \le 12$; the insets contain the actual spin and the energy minimum in MeV using the normalization to zero at $I=0\hslash$.)

Figure 16

Illustration equivalent to the one in Fig. 15 but using an alternative set of coordinates, ${\alpha }_{20}$ and ${\alpha }_{22}$ rather than $\beta$ and $\gamma$, the former better suited for the formulation of the equations of motion of the collective model (discussed here and in the next section). [These coordinates make it possible to write the kinetic energy operator in the Hamiltonian of Eq. (11) in a convenient uniform fashion.] Surfaces of the potential energy, for spins around the critical-spin value for the Jacobi shape transition in ${}^{120}\mathrm{Cd}$, minimized over the axial-deformation parameters ${\alpha }_{\lambda 0}$ with $\lambda \le 12$. Compared to Fig. 15, horizontal lines correspond to $\gamma =0^{\circ }$ (axial) deformation, the straight lines with the positive inclination correspond to the $\gamma ={60}^{\circ }$ axis, whereas the line with the negative inclination corresponds to the $\gamma =-{60}^{\circ }$ axis. Prolongation of each of the axes on the opposite side of the center corresponds to changing the geometry: oblate $\to$ prolate and vice versa.

Figure 17

Contour-plot representation of the absolute values of the wave-function solution to the 2D collective Schrödinger equation, Eq. (22), with the constant mass-parameter approximation as discussed in the text. (Top) The energy eigenvalue corresponds approximately to 0.5 MeV above the potential minimum; (middle) approximately 1 MeV above the minimum; (bottom) approximately 1.5 MeV above the minimum. The color scale corresponds to the unit 0.02.

Figure 18

Three lowest-collective-energies solutions of the Schrödinger equation (22) in terms of the variables ${\alpha }_{20}$ and ${\alpha }_{22}$, here presented as functions of spin for ${}^{120}\mathrm{Cd}$. The mass parameter is adapted in such a way that the lowest collective energy lies 0.5 MeV above the potential minimum at the critical spin. The symbols $n=1,2$, and 3 enumerate the energies of the first three collective solutions, $E_1$, $E_2$, and $E_3$, of Eq. (22).

Figure 19

Static values of the axial-symmetry quadrupole deformation ${\alpha }_{20}$, circles, taken at the minimum of the total energy landscape and the dynamical most probable quadrupole deformation, $\sqrt{\langle {\alpha }_{20}^2\rangle }$, squares. The vertical bars give the $\pm \sigma$ deviation intervals around the positions of the centers. The actual values of $\sigma$ are defined as in Eq. (26). In the present case of the Gaussian-form probability distributions, the deformations contained within the intervals shown represent, approximately, the 68% probability level. The results, from the top to the bottom, correspond to the three choices of the mass parameter as in Fig. 17.

Figure 20

Illustration similar to the one in Fig. 19 but for the nonaxial quadrupole deformation parameter ${\alpha }_{22}$. Again the $\pm \sigma$ intervals define the portion of the deformation axis in which the system is to be found with the probability of approximately 68%. (When comparing with the preceding figure, notice the differences in the vertical scale units.)

Figure 21

Example of the Poincaré-type shape evolution with spin using the 2D projections $({\alpha }_{20},{\alpha }_{30})$, analogous to the one in Fig. 16 for the Jacobi-type transition. Two-dimensional projections corresponding to these two figures were obtained using the same full space of collective coordinates over which minimization has been performed. Observe that, strictly speaking, the static Poincaré transition takes place at the very highest spins only, and this close to the fission critical spin (disappearance of the fission barrier). However, the evolution of the dynamic effects with lowering of the octuple valley extending to the “north” is clearly visible.

Figure 22

Example of the pear-shape octupole ${\alpha }_{30}$ evolution with spin. The wave functions (solid lines) correspond to the left-hand side scale, whereas the potentials (dot-dashed lines) correspond to the right-hand scale.

Figure 23

Example of the pear-shape octupole ${\alpha }_{30}$ evolution with spin, in terms of the static and the dynamic deformations. As before, the static deformations are taken at the equilibrium (minima), whereas the dynamic ones, defined as ${\overline{\alpha }}_{30}$, are given by relation analogous to the one in Eq. (23).

Figure 24

Squares, differences between the dynamical (most likely) quadrupole deformations [in Eq. (24) denoted with the symbol $\overline{\alpha }]$ obtained from the solutions of the 2D Schrödinger equation within the space of ${\alpha }_{20}$ and ${\alpha }_{30}$ variables minus the same quantity calculated from the 2D solutions within the space of ${\alpha }_{20}$ and ${\alpha }_{22}$. Let us emphasize that in the former case, the minimization has been performed, among others, over the ${\alpha }_{22}$ triaxiality parameter. Positive values of the curve express a “stronger flatness” in the direction of ${\alpha }_{20}$ with ${\alpha }_{30}\ne 0$ compared to ${\alpha }_{22}\ne 0$. Solid dots, differences between the dynamical (most likely) octupole deformations obtained using 2D solutions of the Schrödinger equation in the space of ${\alpha }_{20}$ and ${\alpha }_{30}$ deformations and the analogous 1D projection estimates as those in Fig. 23. Observe that the differences are relatively small and nearly constant, suggesting that the 1D approximation offers a good representation of the most probable octupole deformation in the studied case.

Figure 25

Analogous to the illustration in Fig. (24), but for the differences between the dispersion parameters (cf. the preceding figure).

Figure 26

Collective wave-functions solutions of the collective Schrödinger equation [cf. Eqs. (11, 12, 13)]. (Top) The mass parameter $B_{3030}$ has been chosen in such a way that the solution of the 1D projection on the ${\alpha }_{30}$ axis corresponds to $E_{\mathrm{vib}.}=0.5$ MeV. (Bottom) Similar to the previous case, but with the $B_{3030}$ implying $E_{\mathrm{vib}.}=1.5$ MeV. For further details, see text.

Figure 27

Illustration of the differences between the two wave functions shown in Fig. (26). The one corresponding to the larger value of the mass parameter $B_{3030}$ has lower energy minus the one corresponding to the smaller value of the discussed mass parameter. Observe that the discussed quantity is very small and, to be able to present the difference, the corresponding difference has been multiplied by the factor of 10.