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The EDF in the point-coupling version of CDFT can be written as
$ \begin{aligned}[b] {{E}_{\rm CDF}}=&\int{{\rm d}}\mathbf{r}{{\varepsilon }_{\rm CDF}}(\mathbf{r}) \\ =&\sum\limits_{k}{\int{\rm d}}\mathbf{r}\upsilon _{k}^{2}{{{\bar{\psi }}}_{k}}(\mathbf{r})(-{\rm i}\mathbf{\gamma }\mathbf{\nabla} +m){{\psi }_{k}}(\mathbf{r}) \\ &+\int{{\rm d}\mathbf{r}\left( \frac{{{\alpha }_{S}}}{2}\rho _{S}^{2}+\frac{{{\beta }_{S}}}{3}{{\rho}_S^{3}}+\frac{{{\gamma }_{S}}}{4}\rho _{S}^{4}+\frac{{{\delta }_{S}}}{2}{{\rho }_{S}}\Delta {{\rho }_{S}} \right.} \\ &+\frac{{{\alpha }_{V}}}{2}{{j}_{\mu }}{{j}^{\mu }}+\frac{{{\gamma }_{V}}}{4}{{({{j}_{\mu }}{{j}^{\mu }})}^{2}}+\frac{\delta_V}{2}{{j}_{\mu }}\Delta {{j}^{\mu }}+\frac{{e}}{2}{{\rho }_{p}}{{A}^{0}} \\ &\left. +\frac{{{\alpha }_{TV}}}{2}j_{TV}^{\mu }\cdot {{(j_{TV})}_{\mu }}+\frac{{{\delta }_{TV}}}{2}j_{TV}^{\mu }\cdot \Delta (j_{TV})_{\mu } \right) \end{aligned}$
(1) with the local densities and currents
$\begin{aligned}[b] {{\rho }_{S}}(\mathbf{r})&=\sum\limits_{k}{v_{k}^{2}}{{\bar{\psi }}_{k}}(\mathbf{r}){{\psi }_{k}}(\mathbf{r})\\ {{j}^{\mu }}(\mathbf{r})&=\sum\limits_{k}{v_{k}^{2}}{{\bar{\psi }}_{k}}(\mathbf{r}){{\gamma }^{\mu }}{{\psi }_{k}}(\mathbf{r})\\ j_{TV}^{\mu }(\mathbf{r})&=\sum\limits_{k}{v_{k}^{2}}{{\bar{\psi }}_{k}}(\mathbf{r}){{\gamma }^{\mu }}\tau_3{{\psi }_{k}}(\mathbf{r}), \end{aligned} $
(2) where ψ is the Dirac spinor of the nucleon, and
$ \rho_p $ and$ A^0 $ are the proton density and Coulomb field, respectively. The coupling constants$ (\alpha, \beta, \gamma, \delta) $ are determined via PC-PK1 parameterization [50] in this study. The subscripts indicate the symmetry of the couplings, where S stands for scalar, V for vector, and T for isovector.From the variation in the EDF with respect to the densities and currents, we can then obtain the relativistic Kohn-Sham equation, which has the form of a single-particle Dirac equation,
$\begin{array}{*{20}{l}} \{-{\rm i}\mathbf{\alpha}\cdot\mathbf{\nabla} +V(\mathbf{r})+\beta [M+S(\mathbf{r})]\}\psi_k(\mathbf{r})=\varepsilon_k\psi_k(\mathbf{r}). \end{array}$
(3) The single-particle effective Hamiltonian contains local scalar
$ S(\mathbf{r}) $ and vector$ V(\mathbf{r}) $ potentials, which are functions of densities and currents,$ \begin{aligned}[b] S(\mathbf{r})=&\alpha_S \rho_S+\beta_S\rho_S^2+\gamma_S\rho_S^3+\delta_S\Delta\rho_S\\ V^\mu (\mathbf{r})=&\alpha_V j^\mu+\gamma_V(j_\nu j^\nu)j^\mu+\delta_V\Delta j^\mu+eA^\mu\frac{1-\tau_3}{2}\\ & +\tau_3(\alpha_{TV}j^\mu_{TV}+\delta_{TV}\Delta j_{TV}^\mu). \end{aligned}$
(4) Pairing correlations between nucleons are treated using the Bardeen-Cooper-Schrieffer (BCS) approach with a δ pairing force [51]. Owing to the broken translational symmetry, we must consider the center-of-mass (c.m.) correction energy for the motion of the c.m., and the phenomenological formula
$ E_{\rm c.m.}=-\dfrac{3}{4}\cdot 41A^{-1/3} $ is adopted. Finally, the total energy reads as$ \begin{array}{*{20}{l}} E_{\rm tot}=E_{\rm CDF}+E_{\rm pair}+E_{\rm c.m.}. \end{array} $
(5) To calculate the multi-dimensional PES in a large deformation space, we must solve the Dirac equation (Eq. (3)) with high precision and efficiency. One way to achieve this is by expanding the Dirac spinor in a two-center harmonic oscillator (TCHO) basis, which contains eigenfunctions in an axially symmetric TCHO potential,
$ \begin{array}{*{20}{l}} V\left(r_{\perp}, z\right)=\frac{1}{2} M \omega_{\perp}^{2} r_{\perp}^{2}+ \begin{cases}\dfrac{1}{2} M \omega_{1}^{2}\left(z+z_{1}\right)^{2}, & z<0 \\ \dfrac{1}{2} M \omega_{2}^{2}\left(z-z_{2}\right)^{2}, & z \geq 0\end{cases} \end{array}$
(6) in the cylindrical coordinate system. A TCHO can be regarded as two off-center harmonic oscillators connected at
$ z=0 $ , while$ z_1 (z_2) $ and$ \omega_1 (\omega_2) $ denote the distance from$ z=0 $ to the center of the left (right) harmonic oscillator and its frequency, respectively. In practice, we set$ z_1=z_2= 2.24\sqrt{4.32{{\beta }_{2}}-2.38}-3.21 $ fm and$ \omega_1 = \omega_2= 3.81 $ MeV/$ \hbar $ . Details can be found in Refs. [52, 53].The entire map of the energy surface in 3D collective space for fission is obtained by imposing constraints on the three collective coordinates: quadrupole deformation
$ \beta_2 $ , octupole deformation$ \beta_3 $ , and the number of nucleons in the neck$ q_N $ $ \begin{equation} \langle E_{\rm tot}\rangle+\sum\limits_{k=2,3}C_k(\langle\hat Q_k\rangle-q_k)^2+C_N(\langle\hat Q_N\rangle-q_N)^2, \end{equation} $
(7) where
$ \langle E_{\rm tot}\rangle $ is the total energy of CDFT,$ \hat Q_2 $ ,$ \hat Q_3 $ , and$ \hat Q_N $ denote the mass quadrupole and octupole operators, and the Gaussian neck operator, respectively, and$ q_k $ and$ q_N $ are the constraint values of these operators. The Gaussian neck operator is generally chosen as$\hat Q_N= \exp[-(z- z_N)^2/a^2_N]$ , where$ a_N $ = 1 fm, and$ z_N $ is the position of the neck determined by minimizing$ \langle\hat Q_N\rangle $ [33]. The left and right fragments are defined as parts of the whole nucleus with$ z\leq z_N $ and$ z\geq z_N $ , respectively.Once the constraint on
$ q_N $ is adopted, the variation in the configurations around the scission becomes smooth and continuous. Therefore, the Coulomb energy between the left and right fragments is calculated using$ \begin{equation} E_{\rm C} (\beta_2, \beta_3, q_N)={\rm e}^2\int {\rm d}\mathbf{r} {\rm d}\mathbf{r^\prime} \frac{\rho^L_p(\mathbf{r})\rho^R_p(\mathbf{r^\prime})}{|\mathbf{r}-\mathbf{r^\prime}|}, \end{equation} $
(8) where
$ \rho_p^L $ ($ \rho_p^R $ ) is the proton density of the left (right) fragment at the configuration$ (\beta_2, \beta_3, q_N) $ , which is also smooth around the scission and can be used to estimate the total kinetic energy (TKE).
Three-dimensional potential energy surface for fission of 236U within covariant density functional theory
- Received Date: 2023-02-06
- Available Online: 2023-06-15
Abstract: We calculate the three-dimensional potential energy surface (PES) for the fission of the compound nucleus