Breakup of three particles within the adiabatic expansion method

General expressions for the breakup cross sections in the laboratory frame for $1+2$ reactions are given in terms of the hyperspherical adiabatic basis. The three-body wave function is expanded in this basis and the corresponding hyperradial functions are obtained by solving a set of second order differential equations. The $\mathcal{S}$ matrix is computed by using two recently derived integral relations. Even though the method is shown to be well suited to describe $1+2$ processes, there are particular configurations in the breakup channel (for example, those in which two particles move away close to each other in a relative zero-energy state) that need a huge number of basis states. This pathology manifests itself in the extremely slow convergence of the breakup amplitude in terms of the hyperspherical harmonic basis used to construct the adiabatic channels. To overcome this difficulty the breakup amplitude is extracted from an integral relation as well. For the sake of illustration, we consider neutron-deuteron scattering. The results are compared to the available benchmark calculations.

Figure 1

Wave function ${\tilde{u}}^{s_x=1}\left(r_{np}\right)$ obtained from the lowest breakup angular eigenfunction ${\Phi }_2^{J=1/2}(\rho ,{\Omega }_{\rho })$ for the $n$-$d$ case with total angular momentum $J=3/2$ (see text) and for three fixed values of the Jacobi coordinate $y$: 57 fm (dot-dashed line), 85 fm (dashed line), and 114 fm (solid line). The thin solid line is the zero-energy neutron-proton wave function $u^{s_x=1}$ introduced in Eq. (32).

Figure 2

Typical integrand of Eq. (51) obtained for the $n$-$d$ reaction and for the followingcases: (a) as a function of $x$ for two fixed values of $y$; (b) as a function of $y$ ($x$ coordinate integrated away), with $n$ being a $1+2$ channel for the neutron-neutron potential (solid curves) and for the neutron-proton potential with $s_x=0$ (dashed curves); (c) as a function of $y$ ($x$ coordinate integrated away), with $n$ being a $1+2$ channel for the neutron-proton potential and $s_x=1$; and (d) as a function of $y$ ($x$ coordinate integrated away), with $n$ being a breakup channel. In (c) and (d) dashed curves are the asymptotic matching given by Eqs. (54) and (59), respectively.

Figure 3

Breakup cross sections, Eq. (14), as a function of the arclength $S$ for $n$-$d$ scattering in the quartet case ($J=3/2$) for a laboratory incident energy of 14.1 MeV. Cross sections are given in mb/(MeV sr${}^2$). The value of $K_{\max }$ refers to the grand-angular quantum number $K$ associated with the last adiabatic term included in the expansion in Eq. (30). The four cases shown correspond to the four directions of the outgoing neutrons specified in the text. The filled-circle curves are the result of the benchmark calculations given in Ref. [18].

Figure 4

Top: Total cross sections for cases 1 and 2 in the upper panels in Fig. 3. They are obtained by adding the quartet ($J=3/2$) contribution in Fig. 3 and the doublet ($J=1/2$) contribution shown by the dashed curves. Bottom: For the same two cases, the corresponding values of ${\alpha }_i$ ($i=1,2,3$) as a function of the arclength $S$.

Figure 5

The same as Fig. 4, for cases 3 and 4.

Figure 6

Total cross sections [in mb/(MeV sr${}^2$)] for cases 1 and 2 shown in the upper panels in Fig. 4 [$E_{\mathrm{in}}^{\left(\mathrm{lab}\right)}=14.1$ MeV]. Filled circles are the benchmark calculations from Ref. [18]. Dashed curves are the results shown in Fig. 4, which were computed using the expansion in Eq. (30). Cross sections obtained by means of Eq. (46) are shown by the filled-circle, thin solid, and thick solid curves, which correspond, respectively, to calculations including 1200, 1800, and 3000 hyperspherical harmonics when computing the integral in Eq. (51).

Figure 7

Top: Total cross sections [in mb/(MeV sr${}^2$)] for cases 5 and 6 and $E_{\mathrm{in}}^{\left(\mathrm{lab}\right)}=42.0$ MeV (see text). Filled circles are the benchmark calculations from Ref. [18]. Dot-dashed and dashed curves are the quartet ($J=3/2$) and doublet ($J=1/2$) contributions, respectively. Total cross sections are given by solid curves. For case 6 (right panel) the calculation using the expansion in Eq. (30) is accurate enough. For case 5 (left panel) the expansion in Eq. (46) is required in order to obtain sufficient accuracy in the vicinity of the peaks. Bottom: For the same two cases, the corresponding values of ${\alpha }_i$ ($i=1,2,3$) as a function of the arclength $S$.