Systematic analysis of $t$ and ³He breakup reactions

In the four-body reaction system, the Schrödinger equation is written as

$\displaystyle\left[K_{R}+\sum_{i\in t{\rm~{}or~{}^{3}He}}U_{i}+U_{\rm C}+h-E\right]\Psi(\bm{\xi},\bm{R})=0,$

(1)

where $\bm{R}$ represents the coordinate between the target T and the center-of-mass (c.m.) of the projectile. The operator $K_{R}$ is the kinetic energy associated with $\bm{R}$, $h$ is the internal Hamiltonian of the projectile, and $\bm{\xi}$ is the intrinsic coordinate. The optical potential between T and each nucleon in the projectile is denoted by $U_{i}$. The Coulomb potential between a proton and T is represented by $U_{\rm C}$; we investigate the effect of Coulomb breakup of $t$ and ³He in this study. In E-CDCC, the scattering wave function is represented as

	$\displaystyle\Psi(\bm{\xi},\bm{R})$	$\displaystyle=$	$\displaystyle\sum_{nIm}\psi_{nIm}(b,Z)\Phi_{nIm}(\bm{\xi})e^{iK_{n}Z}$		(2)
			$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times e^{i(m_{0}-m)\varphi_{R}}\phi^{\rm C}_{n}(R),$

where $b$ is the impact parameter. The position of the $Z$-axis and the azimuthal angle of $\bm{R}$ are denoted by $Z$ and $\varphi_{R}$, respectively. $\Phi_{nIm}$ is the $n$th discretized state of the projectile with the total spin $I$ and its projection on the $z$-axis $m$, and $m_{0}$ is the $z$ component of the total spin of the ground state. We denote $\gamma=\{n,I,m\}$ in this manuscript. The wavenumber $K_{n}$ is written as

$\displaystyle K_{n}=\frac{\sqrt{2\mu(E-\varepsilon_{n})}}{\hbar},$

(3)

where $\varepsilon_{n}$ is the eigen energy of $\Phi_{\gamma}$ and $\mu$ is the reduced mass between the projectile and T. $\phi^{\rm C}_{n}$ in Eq. (2) is the incident-wave part of the Coulomb wave function given by

$\displaystyle\phi^{\rm C}_{n}(R)=e^{i\eta_{n}\ln{[K_{n}R-K_{n}Z]}}$

(4)

with

$\displaystyle\eta_{n}=\frac{Z_{\rm P}Z_{\rm T}e^{2}}{\hbar K_{n}}.$

(5)

Here, $Z_{\rm P}$ and $Z_{\rm T}$ are the atomic numbers of the projectile and T, respectively. Inserting Eq. (2) into Eq. (1), the following equation for $\psi_{\gamma}$ is obtained:

	$\displaystyle\frac{\partial}{\partial Z}\psi_{\gamma}(b,Z)=$
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{i\hbar v_{n}}\sum_{\gamma^{\prime}}\mathcal{F}_{\gamma\gamma^{\prime}}(\bm{R})\psi_{\gamma^{\prime}}(b,z)e^{i(m-m^{\prime})\varphi_{R}}\mathcal{R}_{\gamma\gamma^{\prime}}(b,z)$

with

$\displaystyle\mathcal{R}_{\gamma\gamma^{\prime}}(b,z)=\frac{(K_{n^{\prime}}R-K_{n^{\prime}}z)^{i\eta_{n^{\prime}}}}{(K_{n}R-K_{n}z)^{i\eta_{n}}}e^{i(K_{n^{\prime}}-K_{n})z}$

(7)

and

$\displaystyle\mathcal{F}_{\gamma\gamma^{\prime}}(\bm{R})=\braket{\Phi_{\gamma}}{\textstyle\sum_{i\in{\rm P}}U_{i}}{\Phi_{\gamma^{\prime}}}_{\bm{\xi}}.$

(8)

The subscript $\bm{\xi}$ of $\braket{\cdots}$ means the integral variable.

We apply the Gaussian expansion method (GEM) Hiyama et al. (2003) to obtain the ground and the dicretized-continuum states of $t$ and ³He. In GEM, a wave function of the three-body system is expanded with Gaussian basis on the Jacobi coordinate as shown in Fig. 1, and the basis are described as

	$\displaystyle\phi_{i\lambda}(\bm{x}_{c})$	$\displaystyle=$	$\displaystyle x^{\lambda}_{c}e^{-(x/x_{i})^{2}}Y_{\lambda}(\Omega_{x_{c}}),$		(9)
	$\displaystyle\varphi_{j\ell}(\bm{y}_{c})$	$\displaystyle=$	$\displaystyle y^{\ell}_{c}e^{-(y/y_{j})^{2}}Y_{\ell}(\Omega_{y_{c}})$		(10)

with

	$\displaystyle x_{i}$	$\displaystyle=$	$\displaystyle(x_{\rm max}/x_{0})^{(i-1)/i_{\rm max}},$		(11)
	$\displaystyle y_{i}$	$\displaystyle=$	$\displaystyle(y_{\rm max}/y_{0})^{(j-1)/j_{\rm max}}.$		(12)

Using the basis, we diagonalize the following Hamiltonian:

$\displaystyle h$

$\displaystyle=$

$\displaystyle K_{x}+K_{y}+V_{pn}+V_{pn}+V_{nn}$

(13)

for $t$, and

$\displaystyle h$

$\displaystyle=$

$\displaystyle K_{x}+K_{y}+V_{pn}+V_{pn}+V_{pp}+V_{\rm C}$

(14)

for ³He. Here, $K_{x}$ ($K_{y}$) means the kinetic energy operator associated with $\bm{x}$ ($\bm{y}$). The interactions for the $p$-$p$, $n$-$n$, and $p$-$n$ systems are represented as $V_{pp}$, $V_{nn}$, and $V_{pn}$, respectively. In Eq. (14), $V_{\rm C}$ is the Coulomb interaction between the two protons.

First, we obtain the ground-state wave functions of $t$ and ³He by using GEM. In this study, we adopt the nucleon-nucleon Minnesota interaction Thompson et al. (1977). We neglect the spin of each nucleon for simplicity. Thus, we use the ($S,T$) = ($0,1$) component of the Minnesota interaction for $V_{pp}$ and $V_{nn}$, where $S$ ($T$) is the total spin (isospin) of the two nucleons, whereas we use the ($S,T$) = ($1,0$) component for $V_{pn}$. To reproduce the binding energies of $t$ and ³He, a phenomenological three-body interaction

$\displaystyle V_{3b}(x,y)=V_{3}e^{-\nu(x^{2}+y^{2})}$

(15)

is added to $h$ of $t$ and ³He. In the present analysis, $V_{3}=9.7$ MeV and $\nu=0.1$ $\text{fm}^{-2}$. The parameter sets of the Gaussian basis are common in both the $t$ and ³He calculations, and summarized in TABLE 1. The spin-parity $I^{\pi}$ for the ground states is $0^{+}$ because we neglect the spin of each nucleon in $t$ and ³He. The results of the ground-state energies and root-mean-square radii are shown in TABLE 2. Our calculations reproduce well the experimental data of the ground-state energy Purcell et al. (2010). On the other hand, some deviation of the calculated root-mean-square radii from the experimental data is found. However, the difference does not affect the reaction analysis as shown below.

In order to confirm the validity of our three-body model, we analyze elastic scattering of ³He off ⁴⁰Ca, ⁵⁸Ni, and ⁹⁰Zr. In the model space, we include continuum states up to the internal energy $\varepsilon$ of 30 MeV for $I^{\pi}=$ $0^{+}$, $1^{-}$, and $2^{+}$ states of the projectile. $U_{i}$ in Eq. (1) is constructed by folding the Melbourne $g$ matrix Amos et al. (2000) with the target density Minomo et al. (2010). Figure 2 shows the elastic cross sections of ³He at 40, 70, and 150 MeV/nucleon, as a function of the transferred momentum $q$. The experimental data of the cross sections denoted by the dots are taken from Ref. Tabor et al. (1982); Willis et al. (1973); Hyakutake et al. (1980); Kamiya et al. (2003). The solid lines represent the results of the E-CDCC calculation. It is found that the E-CDCC results reproduce the experimental data in the small $q$ region, in which the cross section is large. Therefore, the three-body model for ³He adopted in this study is expected to work well.

We investigate the breakup effects of $t$ and ³He on the breakup energy spectra and compare them with those of $d$. We adopt the method proposed in Ref. Matsumoto et al. (2010) to obtain a smooth breakup spectrum. The model space is the same as in the calculation of the elastic scattering.

First, we show the breakup cross sections of the $t$ (³He) + ⁹⁰Zr reaction at 150 MeV/nucleon in Fig. 3(a) (Fig. 3(b)). $\varepsilon$ = $-2.2$ and $0$ MeV correspond to the thresholds of the $d+n$ ($d+p$) and $n+n+p$ ($p+p+n$) channels for $t$ (³He), respectively. One can see that the behaviors of the breakup cross section of $t$ and ³He are almost the same. This can be understood from the fact that the strength of the electric dipole transition, which mainly contributes to Coulomb breakup reactions, for $t$ is the same as for ³He; details are found in Appendix. The similar behavior of the cross section between $t$ and ³He is also confirmed in other reaction systems. Thus, in what follows, we will concentrate on the results of ³He.

Next, we compare systematically the breakup cross sections of ³He and $d$. For the E-CDCC calculation of the $d$ breakup reaction, the optical potentials are constructed in the same manner as of $t$ and ³He. We include continuum states of $d$ up to $\varepsilon=$ 30 MeV for $I^{\pi}$ = $0^{+}$, $1^{-}$, and $2^{+}$ states; the spin of each nucleon is neglected, as in the description of $t$ and ³He. The solid and dotted lines in Figs. 4, 5, and 6 represent the results for ³He and $d$, respectively. The dot-dashed (dashed) line corresponds to the result with only the nuclear breakup of ³He ($d$). Although the effective charge of ³He is 2/3$e$, which is larger than 1/2$e$ of $d$, the Coulomb breakup of ³He is negligible compared to that of $d$ because of the large binding energy of ³He. We show the total breakup cross section of ³He and $d$ in TABLE 3. The results for ³He are found to be about one-third of those for $d$ in all the cases.

Next, we investigate the mechanism of the ³He breakup reaction, i.e., the decomposition of the breakup channels into the following two:

	${}^{3}{\rm He}$	$\displaystyle\rightarrow$	$\displaystyle d+p,$
	${}^{3}{\rm He}$	$\displaystyle\rightarrow$	$\displaystyle p+p+n.$

For this purpose, we use the P-separation method proposed in Ref. Watanabe et al. (2021). In this method, the probability $P_{\gamma}$ of the existence of $d$ in $\Phi_{\gamma}$ is defined by

$\displaystyle P_{\gamma}=\int\braket{\Phi_{\gamma}(\bm{x},\bm{y})}{\chi_{d}(\bm{x}_{1})}_{\bm{x}_{1}}\braket{\chi_{d}(\bm{x}_{1})}{\Phi_{\gamma}(\bm{x},\bm{y})}_{\bm{x}_{1}}d\bm{y}_{1},$

where $\chi_{d}$ is the wave function of $d$. Then, by using $P_{\gamma}$, the $d+p$ and $p+p+n$ channel contributions to the total breakup cross section can be obtained as follows:

	$\displaystyle\sigma_{d+p}$	$\displaystyle\equiv$	$\displaystyle\sum_{\gamma}P_{\gamma}\sigma_{\gamma},$		(17)
	$\displaystyle\sigma_{p+p+n}$	$\displaystyle\equiv$	$\displaystyle\sum_{\gamma}(1-P_{\gamma})\sigma_{\gamma}.$		(18)

Here, $\sigma_{\gamma}$ is the breakup cross section to the discretized state $\Phi_{\gamma}$.

Figure 7 shows the results of $P_{\gamma}$ for each state of ³He. For the ground state, $P_{0}$ $\approx$ 0.9 is obtained, which is consistent with the value 90% Brida et al. (2011) obtained with the ab initio quantum Monte Carlo calculation. The left and right vertical dotted lines in Fig. 7 represent the thresholds of the $d+p$ and $p+p+n$ channels, respectively. $P_{\gamma}$ for the continuum states between the two thresholds are found to be about 0.5. For other states, $P_{\gamma}$ are mostly smaller than 0.3. This result indicates that the $p+p+n$ channel contribution is dominant in the total breakup cross section of ³He.

Figure 8 shows the results of $\sigma_{d+p}$ and $\sigma_{p+p+n}$. In this calculation, we do not include the Coulomb breakup because its contribution is negligible as mentioned above. One sees that the contribution of the $p+p+n$ three-body breakup is about five times as large as that of the $d+p$ two-body breakup in all of the three reaction systems. This behavior can be understood from the larger three-body phase volume. It should be noted, however, that this is not always the case. In Ref. Watanabe et al. (2021), the authors found with a similar approach that, for the breakup of ⁶Li, the $d+{}^{4}$He two-body channel is more important than the $p+n+{}^{4}$He one, despite that the latter has a larger phase volume. Further investigations are needed to clarify the relation between the cross sections and the sizes of the phase volume for two-body and three-body breakup processes. In any case, the results in Fig. 8 suggest that the ³He breakup reaction should be described as a four-body breakup reaction.

In the present study, we have analyzed the ³He reaction with a $p+p+n+{\rm T}$ four-body model, whereas in Ref. Iseri et al. (1986), it was investigated with a $d+p+{\rm T}$ three-body model. We investigate the difference between the two reaction models. To describe ³He as a two-body model, we use the same potential between $d$ and $p$ as in Ref. Iseri et al. (1986). While the previous study adopted a phenomenological potential for the optical potential between $d$ and T, we use the following folding-model potential:

$\displaystyle U_{d}=\braket{\chi_{d}}{U_{p}+U_{n}}{\chi_{d}}.$

(19)

It should be noted that the three-body calculation using $U_{d}$ does not include breakup of $d$. The optical potential between $p$ and T is the same as used in the four-body calculation. The solid and dotted lines in Fig. 9 represent the results of the E-CDCC calculation with the four-body and three-body reaction models with a ⁹⁰Zr target, respectively, as a function of $q$. We have included only the nuclear breakup in this calculation. Although some differences are found around the dips at low incident energy, the shapes of the oscillations are almost the same. The difference of the depth around the dips is considered to come from the effects of the $p+p+n$ channel. To discuss this in detail, we perform the four-body E-CDCC calculation including only the ground state and the $d+p$ continuum states located between the two vertical dotted lines in Fig. 7. The dot-dashed lines thus obtained are close to the results of the three-body calculation. This confirms the slight effect of the $p+p+n$ channel on the elastic scattering.

Figure 10 shows the comparison of the total breakup cross section with a ⁹⁰Zr target calculated with four- and three-body E-CDCC. The squares and circles are the same as in Fig. 8, whereas the triangles represent the cross sections calculated with three-body E-CDCC. The total breakup cross sections obtained with the four-body calculation are two times as large as those with the three-body calculation. This difference can be basically understood from the significant contribution of the $p+p+n$ channel, which is missing in three-body E-CDCC, in the ³He breakup reaction. In addition, it is suggested by the difference between the triangles and circles that the $d+p$ two-body breakup process is suppressed in the four-body calculation, probably because of the coupling between the $d+p$ and $p+p+n$ channels.