Examining empirical evidence of the effect of superfluidity on the fusion barrier

Background: Recent time-dependent Hartree-Fock-Bogoliubov (TDHFB) calculations predict that superfluidity enhances fluctuations of the fusion barrier. This effect is not fully understood and not yet experimentally revealed.

Purpose: The goal of this study is to empirically investigate the effect of superfluidity on the distribution width of the fusion barrier.

Method: Two new methods are proposed in the present study. First, the local regression method is introduced and used to determine the barrier distribution. The second method, which requires only the calculation of an integral of the cross section, is developed to determine accurately the fluctuations of the barrier. This integral method, showing the best performance, is systematically applied to 115 fusion reactions.

Results: Fluctuations of the barrier for open-shell systems are, on average, larger than those for magic or semimagic nuclei. This is due to the deformation and the superfluidity. To disentangle these two effects, a comparison is made between the experimental width and the width estimated from a model that takes into account the tunneling, the deformation, and the vibration effect. This study reveals that superfluidity enhances the fusion barrier width.

Conclusions: This analysis shows that the predicted effect of superfluidity on the width of the barrier is real and is of the order of 1 MeV.

Figure 1

Simulated fusion cross section obtained with the ccfull program, in linear scale (a) and logarithmic (b). The original data are shown with green lines, the blue dots represent the data with noise, and the result of the local regression method $F\left(E\right)$ is shown by the red dashed lines.

Figure 2

Barrier distribution computed from the original ccfull results with the local regression method (green solid line) and with the three-point formula (blacked dotted line). The results obtained from the data with noise are shown using the local regression method (red line with colored band) and the three-point formula (blue points with error bars). The second derivative is computed with the parameter $\mathrm{\Delta }E=2$ MeV. An artificial noise of 5% is applied on (a) and a noise of 2% on (b).

Figure 3

Barrier distribution for the reaction ${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ computed from the experimental cross section [14] with the three-point formula (blue points with error bars) and from the local regression (red curve and shaded area). The value of $L=\mathrm{\Delta }E=1.77$ MeV is used.

Figure 4

(a) Barrier distribution computed by the local regression method for the simulated data with parameter $\mathrm{\Delta }E=L=1$ MeV (solid and dotted lines) and 4 MeV (triangles and crosses) computed from the ccfull calculation with (red dotted line and triangles) and without the collective excitations (blue solid line and crosses). (b) Fluctuations of the barrier distribution from the ccfull calculation with (red triangles) and without the collective excitations (blue crosses) as a function of the three-point derivative parameter $\mathrm{\Delta }E$. A comparison is made with the integration method [Eq. (14)] with (red dotted line) and without collective excitations (blue solid line) as a function of $L$.

Figure 5

${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ ccfull fusion cross section multiplied by the energy (red solid line). The function $g\left(E\right)$ is shown with a dashed black line. The shaded area represents the integral of Eq. (14).

Figure 6

${}^{40}\mathrm{Ca}+{}^{96}\mathrm{Zr}$ ccfull fusion cross section calculation with (red triangles) and without (blue crosses) collective excitation. The function (16) is adjusted to those cross sections and shown, respectively, with a red dotted line and a blue solid line.

Figure 7

Example of application of the integration method for several reactions. The black dashed line represents the $g\left(E\right)$ function and the red bars the experimental data. The experimental data are taken from Refs. [14, 22, 23, 24, 25].

Figure 8

Fluctuations of the barrier ${\sigma }_B$ determined with the integral method as a function of the fit method.

Figure 9

Fluctuations of the barrier ${\sigma }_B$ determined with the integral method as a function of the $z$ parameter.

Figure 10

Fluctuations of the barrier ${\sigma }_B$ determined with the integral method as a function of the estimated barrier from Ref. [11].

Figure 11

Same as Fig. 10 but with ${\sigma }_{exp}$ determined with the fitting procedure.

Figure 12

Experimental fusion radius computed as in Eq. (6). The solid line represent the function $R_B=1.30(A_1^{1/3}+A_2^{1/3})$.

Figure 13

Same as Fig. 10 with a selection of systems for which $R_B^{\exp .}<1.3(A_1^{1/3}+A_2^{1/3})-1$ fm.

Figure 14

Deviation of the experimental centroid of the fusion barrier distribution computed as in (5) from the general trend $0.98z$ as a function of $z$.