Microscopic collective inertial masses for nuclear reaction in the presence of nucleonic effective mass

The ASCC method provides a collective subspace which describes nuclear reaction. The generators of the collective variables $(q,p)$ are locally given by $(\hat{Q}(q),\hat{P}(q))$ and determined by the ASCC equations. In the asymptotic region on the reaction path (collective subspace) at $p=0$, these should correspond to the relative distance and its conjugate momentum. Thus, we expect that

where the proportional constants are arbitrary as far as they satisfy the weak canonicity condition

In this paper, we study systems that preserve the axial symmetry on the reaction path. The one-body operator $\hat{R}$ for the relative distance between projectile and target nuclei can be defined in the following way. The symmetry axis is chosen as the $z$ axis, assuming the projectile on the right ($z>0$) and target on the left ($z<0$). We introduce a separation plane at $z=z_{s}$, so that

where $\rho(\mathbf{r})$ is the nucleon density and $\theta(x)$ is the step function. The operator $\hat{R}$ is given by $\hat{R}\equiv\hat{Z}_{P}-\hat{Z}_{T}$, with

where the upper sign is adopted for $\hat{Z}_{P}$ and the lower for $\hat{Z}_{T}$.

The operator $\hat{R}$ is artificially defined as a convenient measure for the relative distance. Its meaning is clear and well defined in the asymptotic region, though it is no longer meaningful when the projectile and the target are merged into a single nucleus. Nevertheless, this choice of $\hat{R}$ does not affect the final result as far as there is a one-to-one correspondence between $q$ and $R$, because the reaction path $|{\Psi(q)}\rangle$ and its canonical variables $(q,p)$ are self-consistently determined without any assumption. After determining all the quantities in the collective Hamiltonian, we perform the coordinate transformation from $q$ to $R$, by using the relation $R(q)=\langle{\Psi(q)}|\hat{R}|{\Psi(q)}\rangle$. This is merely a change in the scale (and the dimension) of the coordinate.

In the ASCC method, the reaction path $|{\Psi(q)}\rangle$ at $p=0$ and the local generators $(\hat{Q}(q),\hat{P}(q))$ are determined by solving the ASCC equations

where curvature terms are neglected for simplicity Matsuo et al. (2000). The Hamiltonian in the moving frame $\hat{H}_{\rm mv}$ and the potential $V(q)$ are respectively defined as

$V(q)$ in Eq. (9) is used as $V(R)$ in Eq. (1) at $R=\langle{\Psi(q)}|\hat{R}|{\Psi(q)}\rangle$.

Equation (6) which depends on $\hat{Q}(q)$ determines the reaction path $|{\Psi(q)}\rangle$. The local generators, $\hat{P}(q)$ and $\hat{Q}(q)$, are given by a solution of Eqs. (7), and (8). However, since solutions of Eqs. (7) and (8) are not unique, we need to select the one corresponding to the reaction path we study. Normally, we choose a solution with the lowest frequency, $\omega^{2}=(\partial^{2}V/\partial q^{2})M^{-1}(q)$, except for the Nambu-Goldstone (NG) modes. The generators should satisfy the asymptotic condition of Eq. (2).

In Eq. (8) there appears the inertial mass parameter $M(q)$ with respect to the coordinate $q$. The magnitude of $M(q)$ depends on the scale of the coordinate $q$ which is arbitrary. We can choose to set the mass $M(q)$ in Eq. (8) to be a constant, for instance $M_{q}=1$ MeV^-1$[q]^{-2}$ without losing generality Wen and Nakatsukasa (2016), where $[q]$ represents the unit of $q$.

In order to obtain an intuitive picture of the collective dynamics, we map the collective coordinate $q$ to the relative distance $R$ between projectile and target nuclei. The inertial mass should be transformed as

where the derivative $dR/dq$ can be obtained as

with the local generator $\hat{P}(q)$. $R_{nj}(q)$ and $P_{nj}(q)$ are the particle-hole (ph) matrix elements of $\hat{R}(q)$ and $\hat{P}(q)$, obtained by solving ASCC equations (7) and (8) Wen and Nakatsukasa (2016). The calculated mass $M(R)$ will replace the constant mass $\mu_{\rm red}$ in the first term of Eq. (1).

Apart from the the relative motion of our current interest, Eqs. (7) and (8) provide solutions for the NG modes, such as the translation and the rotation. These modes have natural and global generators, namely, the total linear momentum $\hat{\mathbf{P}}$ for the translation and the total angular momentum $\hat{\mathbf{J}}$ for the rotation. For example, the rotational motion around the $x$ axis can be generated by $\hat{J}_{x}$ without any cost of energy ($\omega^{2}=0$), leading to the Thouless-Valatin equations Thouless and Valatin (1962); Ring and Schuck (1980),

where $\hat{\Theta}$ is the angle variable conjugate to $\hat{J}_{x}$, and ${\cal J}(q)$ is the moment of inertia around $x$ axis. Equations (12) and (13) indicate that $(\hat{Q}(q),\hat{P}(q))=(\hat{\Theta}(q),\hat{J}_{x})$ correspond to a solution of the ASCC equations (7) and (8) with $\partial^{2}V/\partial q^{2}=0$ and $M(q)={\cal J}(q)$. The calculated rotational moments of inertia can be used to replace $\mu_{\rm red}R^{2}$ in the second term of Eq. (1).

To calculate the inertial masses for the NG modes, an efficient method was proposed in Ref. Hinohara (2015). The inertial mass of the NG modes $M(q)$ are given by the zero-frequency limit of the strength function in the linear response for the momentum operator $\hat{P}$. We apply this method to the calculations of the translational and rotational inertial masses.

A simple method based on the cranking formula to calculate the inertial mass is widely used for nuclear collective motion. The collective path (reaction path) is usually produced by the constrained-Hartree-Fock (CHF) calculation with a constraining operator $\hat{C}$ given by hand. The CHF states are given by the variation

where $\lambda$ is the Lagrange multiplier. Upon the CHF states $|{\Psi_{0}(R)}\rangle=|{\Psi_{0}(\lambda)}\rangle$ with $R=\langle{\Psi_{0}(\lambda)}|\hat{R}|{\Psi_{0}(\lambda)}\rangle$, the inertial mass with respect to the coordinate $R$ is calculated using the cranking formula Baran et al. (2011):

where the single-particle states $|{\varphi_{\mu}(R)}\rangle$ and energies $e_{\mu}(R)$ are defined with respect to the CHF single-particle Hamiltonian, $\hat{h}_{\lambda}=\hat{h}_{\rm HF}[\rho]-\lambda\hat{C}$, as

Note that the Lagrange multiplier $\lambda$ is a function of $R$ given by the condition, $R=\langle{\Psi_{0}(\lambda)}|\hat{R}|{\Psi_{0}(\lambda)}\rangle$. The constraining operator $\hat{C}$ can be in general different from the relative-distance operator $\hat{R}$. In the following calculations in Sec. III, we use the mass quadrupole and octupole operators as $\hat{C}$.

Another cranking formula, which is even more frequently used in many applications, can be derived by neglecting the rearranged fields induced by the change of $\lambda$, namely, $\delta h_{\rm HF}/\delta\rho\cdot d\rho/d\lambda=0$. Normally, one chooses the constraining operator as the collective coordinate, $\hat{C}=\hat{R}$, for the CHF calculation. However, in the present calculation, since we need to find the separation plane $z=z_{s}$ to determine the operator $\hat{R}$, it is convenient to adopt a different one-body operator $\hat{C}$. Let us derive the formula for this case. From Eq. (18) and the orthonormality condition $\langle\varphi_{\mu}(R)|\varphi_{\nu}(R)\rangle=\delta_{\mu\nu}$, it is easy to find

Neglecting the rearrangement, $\hat{h}_{\lambda+\Delta\lambda}\approx\hat{h}_{\lambda}-\Delta\lambda\hat{C}$, the derivative $dR/d\lambda$ is estimated with the first-order perturbation as ${dR}/{d\lambda}=2S_{1}(\hat{R},\hat{C})$, where

Thus, the formula reads

According to Ref. Baran et al. (2011), we call the former one in Eq. (17) “non-perturbative” cranking inertia and the latter in Eq. (21) “perturbative” one. In contrast to the ASCC mass, the cranking masses of Eqs. (17) and (21) have a drawback that they both neglect residual effect of the time-odd density fluctuation. As we shall see in Sec. III, when the nucleonic effective masses are present, the cranking formulae produce wrong masses for the translation and for the relative motion at $R\rightarrow\infty$.

For the NG modes such as the translation and the rotation, we know the generator of the collective coordinate. For instance, in the translational case, the center-of-mass (COM) coordinates $X_{k}$ ($k=x,y,z$) are the trivial collective coordinates. Then, the non-perturbative cranking formula (17) can be transformed into

where $\hat{p}_{k}$ is the single-particle linear momentum. In the second equation, we take advantage of the fact that the single-particle energies and the wave functions relative to the COM are invariant with respect to its position $X_{k}$. For the rotation, we may replace $X_{k}$ by an angle $\phi_{k}$ and $\hat{p}_{k}$ by the angular momentum $\hat{j}_{k}$. This leads to nothing but the cranking formula for the moment of inertia, originally proposed by Inglis Inglis (1954),

It is instructive to investigate properties of the cranking mass formula for the NG modes Ring and Schuck (1980); Bohr and Mottelson (1975); Thouless and Valatin (1962). Here, let us examine the formula of Eq. (23) for the translation, following the argument given in Ref. Bohr and Mottelson (1975). The summation with respect to the index $n$ in Eq. (23) is restricted to the particle states. This restriction can be removed, because the summation with respect to the hole-hole components, $\sum_{n\in h}\sum_{j\in h}$, identically vanishes. When the mean-field (Hartree-Fock) Hamiltonian $\hat{h}_{\rm HF}[\rho]$ conserves the Galilean invariance,

the translational mass becomes

Therefore, the Galilean invariance of the mean-field Hamiltonian guarantees that the cranking formula reproduces the total mass $Am$ for the translation. However, the Galilean invariance is violated by velocity dependence of the one-body mean-field potential in most of nuclear energy density functionals. This results in a wrong mass $M_{\rm cr}\neq Am$ in the cranking formula. This violation should be corrected by the residual fields that depend on the velocity of the translational motion Bohr and Mottelson (1975). Indeed, the ASCC mass reproduces the exact total mass even when the mean-field potential violates the Galilean invariance, which will be shown in Sec. III.

In order to investigate the effect of this time-odd mean-field potential on the collective inertial masses, we adopt an energy density functional of the simplest choice, namely, the BKN energy density functional Bonche et al. (1976) with the minimum extension.

The original BKN functional assumes the spin-isospin symmetry without the spin-orbit interaction, which is a functional of the isoscalar kinetic and local densities, $\tau(\mathbf{r})$ and $\rho(\mathbf{r})$, only. The mean-field potential is local and has no velocity dependence. Thus, the nucleon’s effective mass is identical to the bare nucleon mass. However, in most of realistic energy density functionals, the effective mass is smaller than the bare mass, typically $m^{*}/m\sim 0.7$. In order to introduce the effective mass, we extend the energy density by adding terms $\rho\tau-\mathbf{j}^{2}$ where $\mathbf{j}(\mathbf{r})$ is the isoscalar current density. The appearance of the current density is necessary to preserve the Galilean invariance.

The modified BKN energy density functional reads,

where $\rho(\mathbf{r})$, $\tau(\mathbf{r})$, and $\mathbf{j}(\mathbf{r})$ are calculated as

In Eq. (27), $v(\mathbf{r})$ is the sum of the Yukawa and the Coulomb potentials (Bonche et al., 1976),

The new parameter $B_{3}$ controls the effective mass and the velocity dependence of the mean-field potential.

The variation of the total energy with respect to the density (or equivalently single-particle wave functions) defines the mean-field (Hartree-Fock) Hamiltonian,

where the effective mass is now deviated from bare nucleon mass,

For $B_{3}\neq 0$, Eq. (28) indicates the velocity (momentum) dependence of the mean-field potential and the presence of the time-odd mean fields, $iB_{3}(\nabla\cdot\mathbf{j}+2\mathbf{j}\cdot\nabla)$. For the time-even states, such as the ground state of even-even nuclei, the current density disappears, $\mathbf{j}(\mathbf{r})=0$. Nevertheless, as will be shown later, the terms associated with $\mathbf{j}(\mathbf{r})$ play an important role in the collective inertial mass.

The parameters, $t_{0}$, $t_{3}$, $V_{0}$, and $a$ are adopted from Ref. (Bonche et al., 1976), and we vary $B_{3}$ to change the effective mass and the time-odd mean fields.

First, we demonstrate the importance of the time-odd mean fields, taking the translational total mass as a trivial example. We adopt the simplest case, namely, the single alpha particle.

In Fig. 1, we show the translational mass of a single alpha particle as a function of $B_{3}$. Increasing $B_{3}$, the effective mass $m^{*}$ decreases, as we see in Eq. (29). In the present case, we use the center-of-mass coordinate $\mathbf{R}_{\rm cm}$ as the collective coordinate $R$ in Sec. II.1 and II.2. Since the system is isotropic, we use its $z$ component $Z_{\rm cm}$ in the numerical calculation. In the present model, neutrons and protons have the identical mass $m$. Therefore, the total inertial mass of an alpha particle should be equal to $4m$.

It is clearly demonstrated that the ASCC always reproduces the correct total mass $Am$, irrespective of values of the parameter $B_{3}$. The non-perturbative cranking mass reproduces the total mass, $M^{\rm np}_{\rm cr}=4m$, at $B_{3}=0$, which agree with Eq. (26). This is due to the velocity-independent mean-field potential, for which the mean-field Hamiltonian $\hat{h}_{\rm HF}$ preserves the Galilean invariance. However, at $B_{3}>0$, the effective mass becomes smaller than the bare nucleon mass, $m^{*}<m$, inside the nucleus with $\rho(\mathbf{r})>0$. This leads to the velocity-dependent potential, and the violation of the Galilean invariance in the mean-field level. Figure 1 indicates $M_{\rm cr}^{\rm np}<4m$ at $B_{3}>0$, and shows a monotonic decrease with respect to $B_{3}$. We calculate the average effective mass defined by

It turns out that the cranking mass for the translation approximately coincides with $Am\times(m^{*}/m)_{\rm av}$ as shown in Fig. 1.

For the translational motion, the ASCC inertial mass becomes identical to the Thouless-Valatin moments of inertia Thouless and Valatin (1962) in Eqs. (12) and (13), with $(\hat{J}_{x},\hat{\Theta})$ being replaced by $(\hat{P}_{z},\hat{Z}_{\rm cm})$. The fact that the ASCC inertial mass agrees with the total mass $Am$ means that the Galilean invariance, violated in the mean-field level, is properly restored by the residual induced fields in the leading order. This shows a clear advantageous feature of the ASCC mass over the cranking formulae.

Next let us present results for the moment of inertia which is another important inertial mass for the collective Hamiltonian in Eq. (1). The rotational motion is a NG mode whose ASCC inertial mass can be calculated with the strength function for the angular momentum operator at zero frequency Hinohara (2015), instead of solving the Thouless-Valatin equation (13). For this purpose, we perform the i-FAM calculation Nakatsukasa et al. (2007); Inakura et al. (2009); Avogadro and Nakatsukasa (2011) based on the reaction path self-consistently constructed by the ASCC method. For comparison, we also apply the cranking formula to the moments of inertia, Eq. (24), based on the reaction path calculated with the CHF method.

In Eq. (1), the point-particle approximation was adopted, leading to the moment of inertia, $\mu_{\rm red}R^{2}$. There is another classical limit, namely, the rigid-body moment of inertia. In the present case, the system preserves the axial symmetry on the reaction path. Choosing the axis of the symmetry as the $z$ axis, we have ${\cal J}_{x}={\cal J}_{y}$ whose rigid-body value is given by

The rigid-body moment of inertia about the $z$ axis is non-zero, ${\cal J}_{z}\neq 0$, which contradicts with the trivial quantum mechanical requirement that there exists no rotation around the symmetry axis. On the other hand, it is known that the cranking moments of inertia, ${\cal J}_{x}$ and ${\cal J}_{y}$, for axially deformed nuclei in the harmonic-oscillator-potential model are given by the rigid-body value at the equilibrium Bohr and Mottelson (1975). The cranking formula also satisfies the quantum mechanical condition, producing ${\cal J}_{z}=0$ in the axially symmetric case.

When two nuclei are far away, we expect the point-particle approximation is good. Therefore, we expect that the moments of inertia, ${\cal J}_{x}$ and ${\cal J}_{y}$, on the reaction path changes from the point-particle value ${\cal J}(R)=\mu_{\rm red}R^{2}$ at large $R$ to the rigid-body value ${\cal J}(R)={\cal J}_{\rm rig}$ near the equilibrium (ground) state. It is of significant interest and of importance to examine where and how this transition takes place during the reaction.

In Ref. Wen and Nakatsukasa (2020), we have published the result for $\alpha+\alpha\rightarrow^{8}$Be. Thus, in this paper, we present the results for $\alpha$+¹⁶O $\rightarrow^{20}$Ne and ¹⁶O+¹⁶O $\rightarrow^{32}$S. First, let us show results for the velocity-independent potential with $B_{3}=0$. Figure 6 shows the moments of inertia calculated for the two reaction systems with $B_{3}=0$. The ground state of ²⁰Ne corresponds to $R=3.8$ fm in the upper panel of Fig. 6, and the superdeformed minimum of ³²S is located at $R=5$ fm in the lower panel. We can see that the ASCC and the (non-perturbative) cranking formulae give results very similar to each other. This is the consequence of the local Galilean invariance of the mean-field potential under the momentum transformation $\mathbf{p}\rightarrow\mathbf{p}-m(\bm{\omega}\times\mathbf{r})$ Bohr and Mottelson (1975). The calculated moments of inertia are close to their rigid-body values near these equilibrium states, ${\cal J}(R_{e})\approx{\cal J}_{\rm rig}(R_{e})$, where $R_{e}$ represents the value of $R$ at the potential equilibrium ($dV/dR=0$). However, when the nucleus is more elongated along the reaction path, ${\cal J}(R)$ decreases as $R$ increases. Since the rigid-body value is a monotonically increasing function of $R$, they become smaller than the rigid-body values, ${\cal J}(R)<{\cal J}_{\rm rig}(R)$, then, quickly approach the point-particle values, $\mu_{\rm red}R^{2}$. Beyond the scission point $R=R_{s}$ where the system splits into two separated nuclei, we have ${\cal J}(R)\approx\mu_{\rm red}R^{2}$ ($R\geq R_{s}$). The calculated moments of inertia give parabolic convex lines as functions of $R$, showing their minima located around the midpoint between $R_{e}$ and $R_{s}$.

It is surprising that the moments of inertia decrease as the deformation develops near the equilibrium. This contradicts with our naive expectation based on the classical model. For instance, the rigid-body moments of inertia linearly increases with the deformation $\delta$, ${\cal J}_{\rm rig}\approx{\cal J}_{0}(1+\delta/3)$. Another classical model, irrotational fluid model suggests that it increases quadratically with the deformation as ${\cal J}_{\rm irrot}\approx{\cal J}_{\rm rig}\delta^{2}$ Bohr and Mottelson (1975). In any case, a single classical model cannot explain the reduction in ${\cal J}(R)$ as a function of $R$. A hint to understand this behavior may come from the harmonic oscillator model in the quantum mechanics which reproduces both the rigid-body value and the irrotational-fluid value depending on the configuration of nucleus Bohr and Mottelson (1975). During the evolution of deformation as a function of $R$, different configurations appear, which may lead to moments of inertia corresponding to different classical models. This suggests the importance of the quantum mechanical calculation.

Another striking feature in Fig. 6 is ${\cal J}(R)\approx\mu_{\rm red}R^{2}$ at $R\geq R_{s}$, which is significantly smaller than ${\cal J}_{\rm rig}(R)$. When the two nuclei are separated, the structure of the projectile and the target nuclei are almost invariant with respect to $R$. Let us denote the rigid-body moments of inertia of the projectile and the target with respect to their own center-of-mass coordinates as ${\cal J}_{P}$ and ${\cal J}_{T}$, respectively. The total rigid-body moments of inertia can be written as

This equation is easy to prove in the rigid-body case. In the quantum mechanical treatment of the present cases, there is another trivial result, namely, ${\cal J}_{P}={\cal J}_{T}=0$. This is because both the projectile and the target nuclei ($\alpha$ and ¹⁶O) are spherical in the ground state, thus, the quantum mechanics requires the vanishing moments of inertia (See also Eq. (24)). Assuming that Eq. (32) is valid for the quantum mechanical systems, we find ${\cal J}(R)\approx\mu_{\rm red}R^{2}$ in the region where the projectile and the target become spherical. This is also a consequence of the quantum mechanics.

Next, let us show how the $B_{3}$ term influences the moments of inertia. Figure 7 shows the results. For $B_{3}\neq 0$, the velocity dependence in the mean-field potential violates the local Galilean invariance. The calculated cranking moments of inertia are significantly smaller than the rigid-body value ${\cal J}_{\rm rig}$ at $R\approx R_{e}$. They are also smaller than the point-particle value $\mu_{\rm red}R^{2}$ at $R>R_{s}$. On the other hand, the ASCC calculation includes the residual effects of the time-odd mean fields which restore the local Galilean invariance, then, nicely reproduces both rigid-body and point-particle values at $R\approx R_{e}$ and at $R>R_{s}$, respectively. In fact, for these reaction systems, the ASCC moments of inertia are insensitive to the $B_{3}$ value over the entire region.

In this section, we present the calculation of the astrophysical $S$ factor for fusion reactions, using the potentials and the inertial masses obtained in Sec. III.3 and III.4. The ASCC calculation provides us with the collective Hamiltonian with the optimal reaction paths for ¹⁶O+$\alpha$ $\rightarrow$ ²⁰Ne and ¹⁶O+¹⁶O$\rightarrow^{32}$S. The Hamiltonian for the reaction is given as that of Eq. (1) in which the moment of inertia $\mu_{\rm red}R^{2}$ is replaced by ${\cal J}(R)$, and the reduced mass $\mu_{\rm red}$ by $M(R)$. We investigate the dependence of the sub-barrier fusion cross sections on these inertial masses.

The sub-barrier fusion cross section is calculated with the WKB approximation, following the procedure in Ref. Reinhard et al. (1984). The total sub-barrier fusion cross section is given by the sum of all the partial waves with angular moment $\ell$. Under the WKB approximation, the transmission coefficient for the partial wave $\ell$ at incident energy $E_{\rm c.m.}$ is given by

with

where $a$ and $b$ are the classical turning points on the inner and outer sides of the barrier respectively. In addition to the potential $V(R)$, the coordinate-dependent inertial mass for the relative motion $M(R)$ and for the rotational moment of inertia ${\cal J}(R)$ are necessary for the calculation. The term $\ell(\ell+1)/2{\cal J}(R)$ represents the centrifugal potential.

The fusion cross section is given by

For identical incident nuclei, ¹⁶O+¹⁶O, Eq. (35) must be modified according to the proper symmetrization. Only the partial wave with even $L$ contribute to the cross section as

Instead of $\sigma(E_{\rm c.m.})$, we show the astrophysical $S$ factor defined by

where $v$ is the relative velocity at $R\rightarrow\infty$. The astrophysical $S$ factor is preferred for sub-barrier fusion because it removes the change by tens of orders of magnitude present in the cross section due to the penetration through the Coulomb barrier. The $S$ factor can reveal in a more transparent way the influence of the nuclear structure and dynamics.

Figure 8 shows the calculated $S$ factor with $B_{3}=0$ for the scattering of $\alpha+^{16}$O (upper panel) and ¹⁶O+¹⁶O (lower panel), respectively. In order to clarify the effect of the inertial masses, we use the same potential $V(R)$ for all the curves in each panel of Fig. 8, which is the ASCC potential obtained for $B_{3}=0$ (solid lines in the bottom panels of Fig. 4 and 5). Different curves show the calculations with different inertia masses. For the case of $B_{3}=0$, the ASCC and the cranking inertial masses are similar to each other. Thus, the calculation with the cranking inertial masses produces the astrophysical $S$ factor similar to the ASCC result. Generally speaking, the larger the inertial mass $M(R)$ for the relative motion is, the smaller the astrophysical $S$ factor is. The opposite effect can be seen for the rotational moments of inertia ${\cal J}(R)$. The larger moments of inertia give the larger $S$ factor. The replacement of the constant mass $\mu_{\rm red}$ by ASCC $M(R)$ gives a significant effect for the fusion cross section. For the reaction system ¹⁶O+¹⁶O, we find a strong suppression of the fusion cross section at $E_{\rm c.m.}<4$ MeV compared to the calculation with $\mu_{\rm red}$, which is an order of magnitude or even larger.

Figure 9 shows the same $S$ factors with inertial masses calculated with $B_{3}=25$ MeV fm⁵ for the scattering of $\alpha+^{16}$O (upper panel) and ¹⁶O+¹⁶O (lower panel), respectively. In order to see the effect of inertial masses, we use the potential $V(R)$ obtained for $B_{3}=0$. Thus, the differences between Figs. 8 and 9 come from the change in inertial masses. Again, the change from the reduced mass $\mu_{\rm red}$ into $M(R)$ gives the largest impact on the $S$ factor. The suppression effect of $M(R)$ is even more significant than the $B_{3}=0$ case, because the $M(R)$ is larger for finite values of $B_{3}$, as we can find in Fig. 4 and 5.

Although the BKN density functional provides us only with qualitative results, these calculations reveal the important roles of the inertial masses, and suggest significant influence of the inertial masses on the sub-barrier fusion cross sections.