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--> --> --> -->A.Fourier nuclear shape parametrization
The axial symmetric shape-profile function of a fissioning nucleus written in cylindrical coordinates $\begin{aligned}[b] \frac{\rho^2_s(u)}{R_0^2} = & a_2\cos(u)+a_3\sin(2u)+a_4\cos(3u)\\& +a_5\sin(4u) + a_6\cos(5u)+\dots, \end{aligned} $ | (1) |
$ \left\{ \begin{array}{ll} q_2 = a_2^{(0)}/a_2 - a_2/a_2^{(0)}\; ,& q_3 = a_3\; , \\[1ex] q_4 = a_4+\sqrt{(q_2/9)^2+(a_4^{(0)})^2}\; ,&\\[1ex] q_5 = a_5-(q_2-2)a_3/10\; ,& \\[1ex] q_6 = a_6-\sqrt{(q_2/100)^2+(a_6^{(0)})^2} &\;\;. \end{array} \right. $ | (2) |
Non-axial shapes can easily be obtained assuming that, for a given value of the
$ \varrho^2(z,\varphi) = \rho^2_s(z) \frac{1-\eta^2}{1+\eta^2+2\eta\cos(2\varphi)} \quad {\rm with} \quad \eta = \frac{b-a}{a+b}\; , $ | (3) |
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B.Macroscopic-microscopic model
In the mac-mic method, first proposed by Myers and ?wi?tecki [28], the total energy of the deformed nucleus is equal to the sum of the macroscopic (liquid-drop type) energy and the quantum energy correction for protons and neutrons generated by shell and pairing effects $ E_{\rm tot} = E^{}_{\rm LSD}+ { E_{\rm shell}} + { E_{\rm pair}} \; . $ | (4) |
$ { E_{\rm shell}} = \sum\limits_k e_k - \widetilde E \; . $ | (5) |
${E_{{\rm{pair}}}} = {E_{{\rm{BCS}}}} - \sum\limits_k {{e_k}} - {\tilde E_{{\rm{pair}}}}.$ | (6) |
$ E_{\rm BCS} = \sum\limits_{k>0} 2e_k v_k^2 - G\left(\sum\limits_{k>0}u_kv_k \right)^2 - G\sum\limits_{k>0} v_k^4 -{\cal E}_0^\varphi\;, $ | (7) |
$ {\cal E}_0^\varphi = \frac{\displaystyle\sum\limits_{k>0}[ (e_k-\lambda)(u_k^2-v_k^2) +2\Delta u_k v_k +Gv_k^4] / E_k^2}{\displaystyle\sum\limits_{k>0} E_k^{-2}}\;. $ | (8) |
$ \begin{aligned}[b] \tilde E_{\rm pair} =& {-\frac{1}{2}\,\tilde{g}\, \tilde{\Delta}^2+\frac{1}{2}\tilde{g}\,G\tilde{\Delta}\, {\rm arctan}\left(\frac{\Omega}{\tilde\Delta}\right) -\log\left(\frac{\Omega}{\tilde\Delta}\right)\tilde{\Delta}}\\& { +\frac{3}{4}G\frac{\Omega/\tilde{\Delta}}{1+(\Omega/\tilde{\Delta})^2}/ {\rm arctan}\left(\frac{\Omega}{\tilde{\Delta}}\right)-\frac{1}{4}G }\; , \end{aligned} $ | (9) |
$ \tilde\Delta = {2\Omega\exp\left(-\frac{1}{G\tilde{g}}\right)}\; . $ | (10) |
$ G = {\frac{g_0}{{\cal N}^{2/3} \, A^{1/3}}}\; . $ | (11) |
In our calculation, the single-particle spectra are obtained by diagonalization of the s.p. Hamiltonian with the Yukawa-folded potential [25,36] using the same parameters as in Ref. [37].
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C.Multidimensional Langevin equation
To study the fission dynamics of atomic nuclei, we use the Langevin equation formalism, which determines the motion of the nucleus in the multidimensional space of deformation parameters $ \left\{ \begin{aligned} {\dfrac{{\rm d}q_i}{{\rm d}t} = }& {\displaystyle\sum\limits_{j} \left[\mathcal{M}^{-1}\right]_{ij}p_j,}\\[3ex] {\dfrac{{\rm d}p_i}{{\rm d}t} = }& {-\dfrac{\partial V}{\partial q_i}-\dfrac{1}{2}\displaystyle\sum\limits_{jk} {\left[\dfrac{\mathcal{M}^{-1}}{\partial q_i}\right]_{jk}}p_jp_k} \\[3ex] & {+ \displaystyle\sum\limits_{jk}\gamma_{ij} \left[\mathcal{M}^{- 1} \right]_{jk}p_k + \sum_{j} g_{ij} \Gamma_j \; ,} \end{aligned} \right. \; $ | (12) |
The inertia tensor is calculated within the incompressible and irrotational liquid drop model using the Werner-Wheeler approximation [39]. For the nuclear surface described by the function
$ \mathcal{M}_{ij}(q) = \pi \rho_m \int\limits_{z_{\min}}^{z_{\max}} \rho^2_s(z,q) \left[ A_i A_j + \frac{1}{8}\rho^2_s(z, q )A_i' A_j' \right] {\rm d} z \; . $ | (13) |
$ A_i = \frac{1}{\rho^2_s(z, q)}\frac{\partial}{\partial q_i}\int_{z}^{z_{\max}} \rho^2_s (z', q) {\rm d}z' \; , $ | (14) |
During the fission process, the temperature of the nucleus changes due to the existence of friction forces. To take this effect into account, another type of the potential must be used, known as the temperature dependent Free Helmholtz energy
$ F(q) = V(q) - a(q)T^2\; ,$ | (15) |
$ T = \sqrt{E^*/a(q)}\; . $ | (16) |
The collective potential in Eq. (4) in the mac-mic approximation is given by the sum of the macroscopic and the microscopic parts
$ V_{\rm mic}(q, T) = {\frac{V_{\rm mic}(q, T = 0)}{1 + e^{(1.5 - T)/0.3}}}\; . $ | (17) |
$ \gamma^{\rm mic}_{ij} = {\frac{0.7\cdot\gamma^{\rm wall}_{ij}} {1+e^{(0.7-T)/0.25}}}\; , $ | (18) |
$ \gamma^{\rm wall}_{ij} = \frac{\pi}{2} \rho_m \bar{v} \int\limits_{z_{\min}}^{z_{\max}} \frac{\partial \rho^2_s}{\partial q_i} \frac {\partial \rho^2_s}{\partial q_j} \left[ \rho^2_s + \frac{1}{4} \left (\frac {\partial \rho^2_s}{\partial z} \right)^2 \right]^{-1/2} {\rm d}z,$ | (19) |
The final term of the second equation in Eq. (12) represents the random Langevin force. Its amplitude
$ \bar{\xi} = 0, \: \bar{\xi}^2 = 2\; , $ | (20) |
The diffusion tensor is obtained using the Einstein relation
$ D_{ij} = \sum_k g_{ik} g_{jk} = \gamma_{ij} \cdot T\; , $ | (21) |
$ T^* = \frac{E_0}{2} \coth \frac{E_0}{2 T}\; , $ | (22) |
The irrotational flow inertia tensor (Eq. (13)) and the wall friction tensor (Eq. (19)) are evaluated using the Fortran codes published in Ref. [45].
All deformation dependent transport coefficients in Eq. (12) were stored for each nucleus at equidistant (
The Langevin calculation of each
$ \begin{aligned}[b] E_{\rm coll} =& {V(q_3,q_4;q^{\rm start}_2)-V(q^{\rm start}_3,q^{\rm start}_3; q^{\rm start}_2)}\\&+ {\frac{1}{2}\sum\limits_{i = 3,4;j = 3,4}{\cal M}_{ij}p_ip_j = E_0} \; , \end{aligned} $ | (23) |
The Langevin trajectory proceeds randomly towards fission within the following rectangular 3D box with the collective variables:
$ \begin{aligned}[b] q^{\rm start}_2 \leqslant & q_2 \\ -0.27\leqslant & q_3 \leqslant 0.27\\ -0.21 \leqslant & q_4 \leqslant 0.21 \end{aligned} $ | (24) |
Figure6. (color online) Fission fragment mass yields of six Ds isotopes. The l.h.s. column corresponds to the case in which the Langevin trajectories begin in the vicinity of the saddle point (low energy fission), while the r.h.s. columns present the estimates made for the spontaneous fission when the trajectories begin around the exit point from the barrier.
Figure7. (color?online)?The same as in Fig. 6 but for Cn isotopes.
Figure8. (color?online)?The same as in Fig. 6 but for Fl isotopes.
Our Langevin estimates of the fission fragment mass yields of
Figure5. (color online) Fission fragment mass yield (red solid line) estimate for the
In Fig. 6, the fission fragment mass yields of the
FigureA4. (color?online)?The same as in Fig. A1 but for
Figures 7 and 8 show similar estimates of the FMY for the