Scission configurations and their implication in fission-fragment angular momenta

The generation of sizable angular momenta in fragments formed in low-energy nuclear fission is described microscopically within the general quantum-mechanical framework of orientation pumping due to the Heisenberg uncertainty principle. Within this framework, we make use of the results of Skyrme-Hartree-Fock plus BCS-pairing calculations of fragment deformabilities to deduce a distribution of fission-fragment spins as a function of the fragment total excitation energy. We consider a fragmentation corresponding to a pair of deformed fragments and for which fission data are available. The properties of the scission configurations determine to a large extent the fission-fragment spins. This is why we pay particular attention to quantitatively defining the scission configurations and to studying the various implications of such a specific choice. A fair qualitative agreement with data is demonstrated and discussed within the limits of the simple scission-configuration model used here.

Figure 1

Potential energy of deformation $E_{\mathrm{def}}^{(i)}$ of ${}^{106}\mathrm{Mo}$ and ${}^{146}\mathrm{Ba}$ with respect to their respective ground-state values as a function of the axial quadrupole moment $Q_{20}^{(i)}$.

Figure 2

Expectation value of ${\hat{\mathbf{J}}}^2$ (in $\hslash {}^2$) as a function of the quadrupole moment for ${}^{106}\mathrm{Mo}$ and ${}^{146}\mathrm{Ba}$.

Figure 3

Contour map of $J_f$ (in ℏ) as a function of fragment elongation for the fragmentation ${}^{106}\mathrm{Mo}+{}^{146}\mathrm{Ba}$. Contours range from 1 (innermost contours labeled “1”) to 14 (outermost contours labeled “14”), with a $1\hslash$ step.

Figure 4

Contour maps of $D_{\mathrm{c}.\mathrm{m}.}$ at scission. The first contour in the bottom left corner corresponds to $D_{\mathrm{c}.\mathrm{m}.}=13$ fm for the upper panel ($\eta =0.1$%) and to $D_{\mathrm{c}.\mathrm{m}.}=12$ fm for the lower one ($\eta =0.5$%). As one increases the fragment deformations the $D_{\mathrm{c}.\mathrm{m}.}$-values increases by 1 fm.

Figure 5

Contour maps of TXE at scission from 0 to 35 MeV for ${}^{106}\mathrm{Mo}+{}^{146}\mathrm{Ba}$ scission configurations obtained with the YPE nuclear potential for two different η values and three different values of $f$. Dots indicate the location on each level contour of the scission configuration having the lowest value of the potential energy $E_{\mathrm{sc}}$ defined by Eq. (20).

Figure 6

Variation of the scission-configuration energy $E_{\mathrm{sc}}$ along the various TXE contours (whose values are indicated in MeV in the legend) for the six combinations of η and $f$ values.

Figure 7

Same as Fig. 6, but for the quadrupole moment of the light fragment $Q_{20}^{(1)}$.

Figure 8

Same as Fig. 6, but for the quadrupole moment of the heavy fragment $Q_{20}^{(2)}$.

Figure 9

Same as Fig. 6, but for the available energy at scission $E_{\mathrm{free}}$.

Figure 10

Same as Fig. 6, but for the expectation value of ${\hat{\mathbf{J}}}^2$ of the light fragment $\langle J_1^2\rangle {}_{\mathrm{intr}}$.

Figure 11

Same as Fig. 10, but for the heavy fragment.

Figure 12

Same as Fig. 6, but for the scission configuration probability $w$ with $\Theta =1$ MeV.

Figure 13

Same as Fig. 12, but with $\Theta =15$ MeV.

Figure 14

Available energy at scission $E_{\mathrm{free}}$ as a function of the fragment total excitation energy TXE for $\eta =0.1$% and $\eta =0.5$%. Results are shown for $\Theta =15$ MeV only.

Figure 15

Same as Fig. 14, but for the total energy of deformation of the fragments $E_{\mathrm{def}}$.

Figure 16

Same as Fig. 14, but for the center-of-mass distance $D_{\mathrm{c}.\mathrm{m}.}$ and the tip distance $d$.

Figure 17

Variation of total kinetic energy of the fragments TKE as a function of the average center-of-mass distance ${\overline{D}}_{\mathrm{c}.\mathrm{m}.}$ for the six combinations of η and $f$ values.

Figure 18

Variation of average angular momentum in units of ℏ at scission with the total excitation energy for the ${}^{106}\mathrm{Mo}+{}^{146}\mathrm{Ba}$ fragmentation. Dots mark the actual calculated results, curves serve to guide the eye.

Figure 19

Same as Fig. 18, but for different values of Θ with $\eta =0.1$% and $f=0.5$.