Collective-subspace requantization for sub-barrier fusion reactions: Inertial functions for collective motions

Time-dependent density functional theory (TDDFT) is a standard tool for studies of a variety of heavy-ion reactions Neg82 ; Sim12 ; NMMY16 , such as nuclear fusion above the Coulomb barrier energy GN12-P ; Was15 ; EN15-P , induced fission reaction BMRS16 , quasi-fission reaction UOS16 , and multi-nucleon transfer reactions Sim10 ; SY16 . It also provides an important method in studies of nuclear response, namely the (quasiparticle) random-phase approximation ((Q)RPA) in the linear regime. In addition, the properties of the giant resonances have been extensively investigated with TDDFT real-time simulations NY05 ; Eba10 ; SBMR11 ; SL13-2 . A universal description of nuclei in the entire mass region and a unified microscopic description of nuclear structure and reaction are great features of the TDDFT method.

On the other hand, the current TDDFT models of nuclei have a problem in a proper account of fluctuation effects. The problems are especially serious in transitional regions where the nuclear shape fluctuates and is not well-defined. In nuclear reaction studies, a prominent weak point is evident for the sub-barrier fusion and the spontaneous fission reactions, which never takes place in the straightforward TDDFT calculations. The theory lacks a part of important quantum fluctuation associated with the nuclear shape. In order to recover the missing quantum fluctuation, theories beyond the mean field are required.

The most popular method of improving the mean-field theory is the generator coordinate method (GCM) RS80 ; Sch19 . The GCM is a quantum mechanical approach and is capable of describing the quantum fluctuations along the chosen generator coordinates. Increasing the number of generator coordinates, in principle, the solution converges to the exact eigenstates of the Hamiltonian. However, this is not the case for the energy density functionals. Although many of the nuclear energy density functionals are based on density-dependent effective interactions, the exact ground state of the Hamiltonian is not physical. Therefore, we must carefully choose a small number of generator coordinates to avoid such unphysical states. This also prevents us from utilizing a variational method to determine the generator coordinates SONY06 ; Fuk13 . The GCM has both numerical and theoretical problems NMMY16 , especially when we use the effective interactions which depend on a fractional power of density AER01 ; DSNR07 ; Dug09 .

Our strategy for a solution to these problems is the requantization of collective degrees of freedom which are selected according to the TDDFT dynamics. Similar approaches have been adopted in studies of low-lying excitation spectra and in the fission dynamics, e.g., the five-dimensional collective Hamiltonian (5DCH) method instead of the GCM on the $(\beta,\gamma)$ plane KB67 ; Del10 ; NMMY16 and the collective model of spontaneous fission Sad20 . However, in these models, the shape variables are chosen by hand and the inertia parameters are approximated by the cranking formula Ing54 ; Ker61 . We aim to improve the models in terms of the microscopic point of view.

We determine a collective subspace which is decoupled from the other intrinsic degrees of freedom. For this purpose, we use the adiabatic self-consistent collective coordinate (ASCC) method MNM00 ; HNMM07 ; HNMM09 ; Nak12 ; NMMY16 . The solution of the ASCC equations provides a set of canonical variables $(q,p)$ to specify the one-body density $\hat{\rho}(q,p)$, and the generators of collective variables $(\hat{Q}(q),\hat{P}(q))$. The collective potential is given by the energy values at $p=0$, $V(q)=E[\rho(q,p=0)]$, and the expansion with respect to the $p$ provides the kinetic energy with the inertial masses, $M^{-1}(q)=\partial^{2}E/\partial p^{2}|_{p=0}$. This leads to the collective Hamiltonian

$$H=\frac{p^{2}}{2M(q)}+V(q).$$

(1)

Since the scale of the coordinate $q$ is arbitrary, we can adjust it so as to make the collective inertia constant, $M(q)=\mu$. In the study of the sub-barrier nuclear fusion, we expect that the coordinate $q$ should be proportional to the distance between the projectile and the target nuclei, $r$, when two nuclei are far apart. Thus, it is intuitive to write down the collective Hamiltonian in the relative coordinate $r$ and its conjugate momentum $p_{r}=p(dq/dr)$. It should be noted that the collective inertia $M(r)$, given by $\{M(r)\}^{-1}=(dr/dq)^{2}\mu^{-1}$, is no longer constant, especially at $r\lesssim R_{\rm touch}$ where $R_{\rm touch}$ is a touching distance of two nuclei.

Assuming the axial symmetry with respect to the axis of the relative coordinate $\mathbf{r}$ between the projectile and the target, two angles to fix the direction of $\mathbf{r}$ describe the rotational motion, and their moments of inertia are calculated solving the Thouless-Valatin equation TV62 at each point on the collective subspace $r$. Eventually, the following collective Hamiltonian is obtained.

$$H=\frac{p_{r}^{2}}{2M(r)}+\frac{L^{2}}{2\mathscr{I}(r)}+V(r),$$

(2)

where $L^{2}$ is the square of the angular momentum produced by the rotation of the coordinate $\mathbf{r}$. In the end, the canonical quantization is performed, replacing $p_{r}$ by $-i\partial/\partial q$ ($\hbar=1$) with the Pauli’s prescription and $L^{2}$ by the angular momentum operator $\hat{L}^{2}$. It should be noted that the point transformation from $q$ to $r$ does not change the physics, as far as the inertial functions and the conjugate momentum are properly defined.

In this paper, we focus our discussion on the inertial functions for various collective motions. In Sec. 2, we briefly review the ASCC method and present methods of calculating the inertial functions in Sec. 3. We especially, describe the derivation of the cranking formulae based on the TDDFT equation of motion, which can clarify the missing correlations in the cranking formulae. In Sec. 5, we show a somewhat surprising property of the moment of inertia for the pair rotation. The conclusion and the perspectives are given in Sec. 6.

Using the energy density functional $E[\rho]$ and the local (normal-mode) generator $Q(q)$, the collective subspace for the low-energy collective motion is defined by

$$\biggr{[}h[\rho(q)]-\lambda Q(q),\rho(q)\biggr{]}=0,$$

(3)

where $h_{ij}\equiv\delta E/\delta\rho_{ji}$. The Lagrange multiplier $\lambda$ corresponds to the gradient of the energy, $\lambda=\partial E[\rho(q)]/\partial q$. The generators, $(Q(q),P(q))$, are given by a solution of the moving-frame random-phase approximation (MF-RPA).

$$\left\{A(q)+B(q)\right\}\left\{A(q)-B(q)\right\}Q(q)=\omega^{2}Q(q),\quad\quad P(q)=M(q)\left\{A(q)-B(q)\right\}Q(q),$$

(4)

where the matrices $A(q)$ and $B(q)$ are given by

$$A_{php^{\prime}h^{\prime}}(q)=(\epsilon_{p}-\epsilon_{h})\delta_{pp^{\prime}}\delta_{hh^{\prime}}+\frac{\partial h_{ph}}{\partial\rho_{p^{\prime}h^{\prime}}},\quad\quad B_{php^{\prime}h^{\prime}}(q)=\frac{\partial h_{ph}}{\partial\rho_{h^{\prime}p^{\prime}}}.$$

(5)

The MF-RPA equation determines only the particle-hole and hole-particle matrix elements of the generators, $(Q(q),P(q))$. In this paper, we assume that their particle-particle and hole-hole matrix elements vanish. The single-particle energies $\epsilon_{i}$ in Eq. (5) are the eigenvalues of $h-\lambda Q(q)$. The derivatives in Eq. (5) are calculated according to the finite-amplitude method (FAM) NIY07 ; AN11 ; Sto11 , especially using the m-FAM scheme in Ref. AN13 .

The MF-RPA equation (3) cannot fix the magnitude of $Q(q)$. Since these equations are scale invariant concerning $Q(q)\rightarrow\alpha Q(q)$, $P(q)\rightarrow\alpha^{-1}P(q)$, and $M(q)\rightarrow\alpha^{-2}M(q)$, we can arbitrarily normalize $Q(q)$. In actual calculations, we set $M(q)$ a constant value, $M(q)=\mu$, to determine the normalization of $Q(q)$ according to the weakly canonicity condition, $\mathrm{Tr}\left(\left[Q(q),P(q)\right]\rho(q)\right)=i$.

The ASCC equations consist of Eqs. (3) and (4) which should be solved self-consistently. Among solutions of Eq. (4), we select a normal mode of the lowest energy keeping the axial symmetry. Note that $\omega^{2}$ can be negative. The numerical computation produces a series of density $\{\rho(q_{0}),\rho(q_{1}),\cdots\}$ on the discretized collective coordinate $q_{n}=q_{0}+n\times\Delta q$. This defines the collective subspace, namely the reaction path in the present study. The potential in Eq. (1), $V(q)=E[\rho(q)]$, then, transformed into $V(r)=V(q(r))$ in Eq. (2).

In this paper, we focus our discussion on the properties of collective inertias and their effect on the nuclear dynamics.

In a somewhat different context, we have recently found a property, similar to Eq. (29), that the moments of inertia of the pair rotation decreases as a function of the pairing gap (deformation in the gauge space) $\Delta$.

$$\frac{\partial\mathscr{I}_{\rm pair}}{\partial\Delta}<0,$$

(31)

where the moment of inertia is calculated, using the ground-state energies with different neutron (proton) numbers, as

$$\frac{1}{\mathscr{I}_{\rm pair}}=\frac{E(N+2)-2E(N)+E(N-2)}{4}.$$

(32)

This behavior can be qualitatively understood as follows, using the BCS treatment of the pair model. The particle number $N$ is written as

$$N(\lambda,\Delta)=\int_{\lambda-\Lambda}^{\lambda+\Lambda}\rho(\epsilon)g(\epsilon-\lambda,\Delta)d\epsilon,\quad\quad g(\tilde{\epsilon},\Delta)\equiv\frac{1}{2}\left(1-\frac{\tilde{\epsilon}}{\sqrt{\tilde{\epsilon}^{2}+\Delta^{2}}}\right),$$

(33)

where $\rho(\epsilon)$ is the single-particle level density and $\lambda$ is the chemical potential. We truncate the model space for the pairing correlations to include the single-particle levels in the section of $2\Lambda$ centered around the chemical potential. The moment of inertia is given by

$$\mathscr{I}_{\rm pair}=\frac{\partial N}{\partial\lambda}=\int_{\lambda-\Lambda}^{\lambda+\Lambda}\rho(\epsilon)\frac{\partial g(\epsilon-\lambda,\Delta)}{\partial\lambda}d\epsilon=-\int_{-\Lambda}^{\Lambda}\rho(\tilde{\epsilon}+\lambda)\frac{\partial g(\tilde{\epsilon},\Delta)}{\partial\tilde{\epsilon}}d\tilde{\epsilon}.$$

(34)

Here, we keep the model space invariant when we differentiate the equation with respect to $\lambda$. Then, its derivative with respect to $\Delta$ is

$$\frac{\partial\mathscr{I}_{\rm pair}}{\partial\Delta}=\frac{\partial^{2}N}{\partial\lambda\partial\Delta}=-\int_{-\Lambda}^{\Lambda}d\tilde{\epsilon}\rho(\tilde{\epsilon}+\lambda)\frac{\partial^{2}g(\tilde{\epsilon},\Delta)}{\partial\tilde{\epsilon}\partial\Delta}=-\frac{1}{2\Delta}\int_{-\Lambda/\Delta}^{\Lambda/\Delta}\rho(\lambda+x\Delta)\frac{1-2x^{2}}{(x^{2}+1)^{5/2}}dx.$$

(35)

In the energy interval of $2\Lambda$, we approximate the level density in the linear order as

$$\rho(\lambda+x\Delta)\approx\rho(\lambda)+x\Delta\rho^{\prime}(\lambda),$$

(36)

which is inserted into Eq. (35) to obtain

$$\frac{\partial\mathscr{I}_{\rm pair}}{\partial\Delta}=-\frac{\rho(\lambda)}{\Delta}I(\Lambda/\Delta),$$

(37)

and

$$I(\eta)\equiv\int_{0}^{\eta}dx\frac{1-2x^{2}}{(x^{2}+1)^{5/2}}=\frac{\eta}{(\eta^{2}+1)^{3/2}}.$$

(38)

For $\eta=\Lambda/\Delta>0$, it is trivial to have $I(\eta)>0$. Thus, Eq. (31) is proved.

Using the ASCC method, we have microscopically derived a collective subspace (reaction path) for the sub-barrier fusion reaction. In addition, the method provides the potential and the inertial functions, which leads to the collective Hamiltonian of the reaction model. In this paper, we particularly focus our discussion on properties of the inertial functions.

The ASCC inertial functions perfectly agree with the correct values in the asymptotic region, such as the total mass for the translational motion, the reduced mass for the relative motion at large $r$, and the reduced moments of inertia at large $r$. On the other hand, the cranking formulae fail to reproduce these asymptotic values. This indicates the superiority of the ASCC method over the cranking formula.

The moment of inertia at the ground state is approximately equal to that of the rigid-body value, but surprisingly, it decreases as the deformation increases. We may expect that the isotropic velocity distribution at the equilibrium is distorted to be anisotropic, leading to a transition from the “rigid body” to the “irrotational flow.” Since the moment of inertia for the irrotational fluid vanishes for spherical systems, the two spherical nuclei, whose mass numbers $A_{1}$ and $A_{2}$, at distance $r$, produce the moment of inertia, $\mu_{\rm red}r^{2}$. This is well reproduced by ASCC moments of inertia, but not for the cranking formula.

Further studies on the collective masses are desired, particularly effects of pairing and beyond the mean field. We have shown that the recovery of the Galilean invariance violated in the KS potential is essential to produce the collective inertia. Since the pair field is known to break the Galilean invariance BM75 , it is of great interest to investigate residual effects of the pair fields.

We have also studied properties of the pair rotation in the gauge space, then, have found that the moment of inertia also decreases as the pair deformation increases. This property can be explained by the BCS pair model. This may indicate a prominent difference in the effect of the deformation between the spatial rotation and the pair rotation. Further investigation is under progress.

This work is supported in part by JSPS KAKENHI Grant Nos. 18H01209, 20K14458, 20K03964, 23H01167, 23K25864, and by JST ERATO Grant No. JPMJER2304. This research in part used computational resources provided by the Multidisciplinary Cooperative Research Program in the Center for Computational Sciences, University of Tsukuba.