Direct triple-$\alpha$ process in non-adiabatic approach

The triple-$\alpha$ reaction plays an important role in nucleosynthesis heavier than ¹²C, because no stable nuclei exist in mass number $A=5$ and $A=8$ [1]. This reaction, followed by ¹²C($\alpha$,$\gamma$)¹⁶O [2], controls C/O ratio at the end of helium burning phase in stars, and it affects up to the nucleosynthesis in supernova explosion. In contrast to ¹²C($\alpha$,$\gamma$)¹⁶O, the triple-$\alpha$ reaction is currently well-understood through the experimental studies of the 0${}^{+}_{2}$ state in ¹²C ($E_{r}=0.379$ MeV) (e.g. [3, 4]), i.e., the reaction rates have been determined well with the sequential process via the narrow resonances: $\alpha+\alpha\rightarrow^{8}$Be(0${}^{+}_{1}$), $\alpha$+⁸Be$\rightarrow^{12}$C(0${}^{+}_{2}$). Pioneering works of the reaction rates have been performed by [5, 6], and their experimental upgrade has been given by NACRE [4].

Apart from the sequential process, the triple-$\alpha$ reaction from 3$\alpha$ continuum states is referred to as the direct triple-$\alpha$ process: $\alpha+\alpha+\alpha\rightarrow^{12}$C. This process is generally expected to be very slow, because three $\alpha$-particles almost simultaneously collide and fuse into a ¹²C nucleus. So, this process is neglected or is treated in some approximations. For the theoretical studies, formulae in hyper-spherical coordinates have been applied to tackle the 3$\alpha$ continuum problem (e.g. [7]), and their adiabatic approaches have paved the way for a non-adiabatic approach of [8]. The Coulomb modified Faddeev (CMF) [9] and adiabatic channel function (ACF) expansion method [10] may have also achieved the successful progress quantitatively. However, non-adiabatic quantum-mechanical description at off-resonant energies still seem to remain in unsolved problems. In the present report, the contribution of the direct 3$\alpha$ process is estimated with a non-adiabatic Faddeev hyper-spherical harmonics and $R$-matrix (HHR^∗) expansion method [11, 12]. At the same time, the difference between the non-adiabatic and adiabatic calculations is discussed.

Before discussing the calculated results, let me describe HHR^∗, briefly. The triple-$\alpha$ system satisfies the three-body Schrödinger equation, $(H_{3\alpha}-E)\Psi_{lm}=0$. $H_{3\alpha}$ is three-body Hamiltonian; $E$ is the center-of-mass energy to the 3$\alpha$ threshold in ¹²C; $l$ is spin of the states in ¹²C; $m$ is the projection of $l$. This equation is expressed as the so-called Faddeev equations, consisting of three components. Due to the symmetric 3$\alpha$ system, three identical sets of equations are found, and they are expressed in a similar form of ordinary coupled-channel (CC) equations for inelastic scattering, (e.g. [13, 14]), after translating Jacobi coordinates (${\bf{x}}_{3},{\bf{y}}_{3}$) into hyper-spherical coordinates ($\rho,\Omega_{5}$), $\Omega_{5}\equiv(\theta_{3},\hat{\bf{x}}_{3},\hat{\bf{y}}_{3})$. The deduced CC equations of the hyper-radial waves, $\chi^{l}_{\gamma}(\rho)$, are given in

$\displaystyle\left[\,T_{\gamma}+U^{l}_{\gamma\gamma}(\rho)-\epsilon\,\right]\chi^{l}_{\gamma}(\rho)$

$\displaystyle=$

$\displaystyle-\mathop{\sum}_{\gamma^{\prime}\neq\gamma}U^{l}_{\gamma^{\prime}\gamma}(\rho)\chi^{l}_{\gamma^{\prime}}(\rho),$

(1)

where $T_{\gamma}=d^{2}/d\rho^{2}-(K+3/2)(K+5/2)/\rho^{2}$; $U^{l}_{\gamma^{\prime}\gamma}$ are the coupling potentials; $\epsilon=-2m_{N}E/\hbar^{2}$; $m_{N}$ is nucleon mass; $K$ is hyper-angular momentum; $\gamma$ is a label of channels. $U^{l}_{\gamma^{\prime}\gamma}$ are calculated from $\alpha+\alpha$ and 3$\alpha$ potentials, and they are the same as [8] except that the strength of 3$\alpha$ potentials are $-20.145$ MeV ($-19.46$ MeV [8]) for 0⁺ and $-16.36$ MeV ($-15.94$ MeV [8]) for 2⁺. Using the hyper-harmonic function $\Phi^{\gamma}_{lm}(\Omega_{5})$ [11, 12], the basis functions are defined by

$\displaystyle\Psi_{lm}$

$\displaystyle=$

$\displaystyle\rho^{-5/2}\mathop{\sum}_{\gamma n}c^{n}_{\gamma}\varphi^{K}_{n}(\rho)\Phi^{\gamma}_{lm}(\Omega_{5}),$

(2)

where $\varphi^{K}_{n}(\rho)$ are harmonic oscillator wavefunctions in hyper-spherical coordinates, $\chi^{l}_{\gamma}=\mathop{\sum}_{n}c^{n}_{\gamma}\varphi^{K}_{n}$. The results are independent of $\varphi^{K}_{n}(\rho)$, if a large number of basis are used. The CC equations (1), briefly rewritten as $(\bf{T}+\bf{U})\bf{X}=\epsilon\bf{X}$, are solved by the matrix diagonalization. The matrix size of the present calculation is (8,800$\times$8,800) for 0⁺ in ¹²C. In the $R$-matrix expansion method, the continuum states with scattering boundary condition are expanded by the resultant eigenfunctions,

$\displaystyle\chi^{l\hskip 2.84526ptin}_{\alpha\beta}(k,\rho)$

$\displaystyle=$

$\displaystyle\mathop{\sum}_{i}A_{i\beta}(k)\chi^{l}_{\alpha i}(\rho),$

(3)

$$A_{i\beta}(k)=\frac{\hbar^{2}}{2m_{N}}\frac{1}{E(l^{+}_{i})-E}\mathop{\sum}_{\gamma}\chi^{l}_{\gamma i}(a_{c})\Big{[}\,H^{-\hskip 2.84526pt\prime}_{K+3/2}(\eta_{\gamma};ka_{c})\delta_{\gamma\beta}-S^{l}_{\gamma\beta}(E,a_{c})H^{+\hskip 2.84526pt\prime}_{K+3/2}(\eta_{\gamma};ka_{c})\,\Big{]},$$

(4)

where $\chi^{l}_{\alpha i}(\rho)$ and $E(l^{+}_{i})$ are the eigenfunctions and eigen-energy, respectively. $\alpha$ and $\beta$ are the channel labels. $H^{\pm}$ are the incoming ($-$) and outgoing ($+$) Coulomb wavefunctions; $\eta_{\gamma}$ is the Sommerfeld parameter; $k$ is the hyper-momentum, $k=(2m_{N}E/\hbar^{2})^{1/2}$; $a_{c}$ is the channel radius. $S^{l}_{\gamma\beta}(E,a_{c})$ is the $S$-matrix, defined by the derived $R$-matrix at $\rho=a_{c}$.

To include the long-range Coulomb couplings, the CC equations in the external region are solved numerically from $\rho=a_{c}$ to $\rho_{m}$ by using the $R$-matrix propagation technique [8]. The external wavefunctions are expanded by the resulting linearly-independent solutions $\chi^{l\hskip 2.84526ptext}_{\gamma\gamma^{\prime}}(k,\rho)$, and the coefficients of expansion $C_{\gamma^{\prime}\gamma_{0}}(k)$ are obtained by matching to the asymptotic form,

$\displaystyle\tilde{\chi}^{l}_{\gamma\gamma_{0}}(k,\rho)$

$\displaystyle\rightarrow$

$\displaystyle\frac{i}{2}\Big{[}\,I^{(\gamma_{0})}_{\gamma,K+3/2}(\eta_{\gamma};k\rho_{m})-\mathop{\sum}_{\gamma^{\prime}}S^{l}_{\gamma^{\prime}\gamma_{0}}(E)O^{(\gamma^{\prime})}_{\gamma,K+3/2}(\eta_{\gamma};k\rho_{m})\,\Big{]},$

(5)

where $O$ and $I=O^{\ast}$ are the coupled-Coulomb waves [14], $O^{(\gamma^{\prime})}_{\gamma}=a_{\gamma}^{(\gamma^{\prime})}(k,\rho)H^{+}_{\gamma}$. In the present report, the global back propagation [8] from $\rho=\rho_{m}$ to $a_{c}$ is not used. If the screening potential [8] is adopted to reduce the strength of Coulomb couplings at large $\rho$, $a_{\gamma}^{(\gamma^{\prime})}(k,\rho_{m})=\delta_{\gamma\gamma^{\prime}}$ can be used. This is effective if the off-diagonal part of coupling potentials is relatively small at $\rho_{m}$, compared with $E$, (e.g. [13]). Multiplying eq. (3) by $C_{\gamma^{\prime}\gamma_{0}}(k)$, I obtain the interior scattering waves including the long-range Coulomb couplings, $\tilde{\chi}^{l\hskip 2.84526ptin}_{\gamma\gamma_{0}}(k,\rho)=\mathop{\sum}_{\gamma^{\prime}}C_{\gamma^{\prime}\gamma_{0}}(k)\chi^{l\hskip 2.84526ptin}_{\gamma\gamma^{\prime}}(k,\rho)$.

The photodisintegration cross sections of ¹²C(2${}^{+}_{1}\rightarrow 0^{+}$) are calculated from

$\displaystyle\sigma_{g}(E)$

$\displaystyle=$

$\displaystyle\frac{2\pi^{2}}{75}\left(\frac{E_{g}}{\hbar c}\right)^{3}\frac{e^{2}}{\hbar v}\mathop{\sum}_{\gamma_{0}}\Big{|}\mathop{\sum}_{\gamma^{\prime}\gamma}\big{[}M^{2^{+}_{1}\,0^{+}\,{Int}}_{\gamma^{\prime}\gamma\gamma_{0}}(k)+M^{2^{+}_{1}\,0^{+}\,{Ext}}_{\gamma^{\prime}\gamma\gamma_{0}}(k)\big{]}F_{\gamma^{\prime}\gamma}\Big{|}^{2},$

(6)

where $E_{g}$ is the $\gamma$-ray energy, $E_{g}=E-E(2^{+}_{1})$. $c$ is speed of light, and $v=(2E/m_{N})^{1/2}$. $F_{\gamma^{\prime}\gamma}$ are the factors stemming from the hyper-angle part. $M^{2^{+}_{1}\,0^{+}\,{Int}}_{\gamma^{\prime}\gamma\gamma_{0}}$ and $M^{2^{+}_{1}\,0^{+}\,{Ext}}_{\gamma^{\prime}\gamma\gamma_{0}}$ are the internal and external components of the hyper-radial part. To execute stable calculations, quadruple precision is required. The energy-averaged reaction rates $\langle R_{3\alpha}\rangle$ are calculated from the resultant $\sigma_{g}$.

The calculated photodisintegration of ¹²C(2${}^{+}_{1}\rightarrow 0^{+}$) is shown by the solid curve in Fig. 2. The prominent narrow resonance of 0${}^{+}_{2}$ is found at $E=0.3795$ MeV, and the smoothly varying non-resonant cross sections are obtained at off-resonant energies. The calculated $\alpha$- and $\gamma$-widths are $\Gamma_{\alpha}=5.1$ eV and $\Gamma_{\gamma}=4.3$ meV, comparable to the experimental data [4]: $\Gamma_{\alpha}=8.3$ eV and $\Gamma_{\gamma}=3.7$ meV. For $0.15<E<0.35$ MeV, I find $\sigma_{g}=10^{-15}$–$10^{-3}$ pb order of cross sections. This result is almost identical to the dotted curve of [8]. The result below $E=0.15$ MeV seems similar to CMF (dashed curve) and ACF (gray curve). On the other hand, the present result above $E=0.15$ MeV is much smaller than the values predicted by CMF and ACF. CMF has been developed below the three-body threshold, e.g. low-energy p+d reactions. The internal motion of ⁸Be in break-up channels is assumed to be localized within a certain range. ACF has the feature of $\alpha$+⁸Be for $\rho<150$ fm and 3$\alpha$ for large $\rho$, and the resonance and bound states are expanded with the adiabatic basis. Judging from their theoretical approaches, most of the differences above $E=0.15$ MeV seem to stem from the internal adiabatic feature. The adiabatic approach to ⁸Be continuum states makes the assumed long resonant tail of 0${}^{+}_{2}$, leading to the sequential decay process at off-resonant energies, and it might have enhanced the photodisintegration cross sections unexpectedly. The enhancement of [8] for $E<0.15$ MeV seems to be caused by the redundant propagation.

In spite of the large difference, the derived rates are found to be consistent with NACRE for $0.08<T_{9}<3$, including the helium burning temperatures (see Fig. 2). In contrast, the present result is reduced by 10^-4 at $T_{9}=0.05$, because $\sigma_{g}$ are reduced from CMF and ACF with the sequential process at $E=0.18$ MeV. Due to the strong influence of 0${}^{+}_{2}$, the difference in $\sigma_{g}$ for $E>0.2$ MeV cannot be found in the rates.

From the present calculation, I have found that the direct triple-$\alpha$ contribution is 10^-15–10^-3 pb order in the photodisintegration cross sections of ¹²C (2${}^{+}_{1}\rightarrow 0^{+}$) for $0.15<E<0.35$ MeV. This is far below the values predicted by CMF and ACF that include the assumed long resonant tail of 0${}^{+}_{2}$, i.e. the sequential process. In spite of the large difference between the non-adiabatic and adiabatic cross sections, the derived reaction rates are concordant with NACRE at the helium burning temperatures. Due to the strong influence of 0${}^{+}_{2}$ in ¹²C, astrophysical impact of the direct triple-$\alpha$ process seems to be small. It would have been, however, important for theoretical nuclear physicists to understand the off-resonant cross sections, non-adiabatically, and to examine how slow the direct process is at the temperatures relevant to stellar evolution.

I thank M. Arnould and Y. Sakuragi for drawing my attention to the present subject and for their hospitality during my stay at Université Libre de Bruxelles and Osaka City University.