Longitudinal momentum distributions of the reaction residues following fast two-nucleon knockout reactions

The longitudinal momentum distributions of the cross sections of heavy projectile-like residues after fast, direct two-like-nucleon knockout reactions are discussed. Both the two-nucleon inelastic breakup (stripping) and the stripping-diffraction removal events are considered. We show that, because the two mechanisms have a very similar nuclear surface localization, they generate essentially identical longitudinal momentum distributions. The approach used combines reaction dynamics, using the sudden, eikonal and spectator-core approximations, with structure wave functions from the many-body shell model. The sensitivities of the resulting longitudinal momentum distributions to the orbital angular momenta, the separation energies, and the angular momentum coupling of the two removed nucleons are clarified. In particular, the widths of these distributions are shown to provide a very clear signal of the total angular momentum of the removed-nucleon pair—pairs coupled to larger total angular momentum giving broader distributions. These now complete distributions, from correlated wave functions, are significantly different from earlier uncorrelated estimates. Confirmation of these theoretical expectations is presented, based on very recent intermediate-energy, residue final-state inclusive and exclusive two-nucleon removal measurements.

Figure 1

Diagram showing the laboratory momenta of the projectile ${\overrightarrow{K}}_{A+2}$ and the reaction residue ${\overrightarrow{K}}_c$ (the mass $A$ core $c$). The momenta ${\overrightarrow{\kappa }}_c$, ${\overrightarrow{\kappa }}_1$, and ${\overrightarrow{\kappa }}_2$ are the core and nucleon momenta in the rest frame of the projectile with ${\overrightarrow{\kappa }}_1+{\overrightarrow{\kappa }}_2+{\overrightarrow{\kappa }}_c=0$.

Figure 2

The particle coordinates used in this article. The vectors ${\overrightarrow{s}}_i$ are the components, in the plane perpendicular to the beam direction (the $z$ axis), of the position vectors ${\overrightarrow{r}}_i$ of the knocked-out nucleons relative to the core of nucleons to which they are initially bound. The two nucleons have impact parameters $\overrightarrow{b_i}=\overrightarrow{b}+{\overrightarrow{s}}_i$ relative to the target nucleus. The target $T$ overlaps the wave function of the two nucleons at and near the surface of the projectile.

Figure 3

Schematic of the angular momentum and isospin coupling schemes used here for the two-nucleon knockout reaction.

Figure 4

The differential stripping cross section with respect to $s_1$, the component of the nucleon position vector in the plane perpendicular to the beam direction. The calculation is for $\pi [0d_{5/2}]{}^2$ two-proton removal from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. The localization for $I^{\pi }=0^{+}$ (solid), $2^{+}$ (dotted), and $4^{+}$ (dashed) states are shown, the peak moving marginally inward as $I$ increases. Also shown is the associated $z$-integrated ${}^{26}\mathrm{Ne}$ core density as a function of the radial distance $s_1$ (symbols).

Figure 5

The variation of the stripping longitudinal momentum distribution with $I$ for $\pi [0d_{5/2}]{}^2$ two-proton removal from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. The curves are for $I^{\pi }=0^{+}$ (solid), $2^{+}$ (dotted), and $4^{+}$ (dashed). The solid line with open circles shows the corresponding $\pi [0d_{5/2}]$ one-proton knockout distribution for the same single-particle orbital and separation energy, which is significantly wider than the $I^{\pi }=0^{+}$ two-proton distribution. The solid line with open squares shows the distribution from removal of two uncorrelated protons, discussed in Sec. 3c. All distributions have been normalized to the same peak value.

Figure 6

The sensitivity of the calculated stripping longitudinal momentum distributions to $\left|\mu \right|$ for $\pi [0d_{5/2}]{}^2$ two-proton removal from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. The cases of $I^{\pi }=4^{+}$ with $\left|\mu \right|=0$ (solid), 1 (dashed), 2 (dotted), 3 (dot-dashed), and 4 (dot-dot-dashed) are shown. The solid curve with open circles is the $I^{\pi }=4^{+}$ distribution summed over all μ, as is shown in Fig. 5. The variation with $\left|\mu \right|$ is clear, the narrower distributions arising from the largest projections (see text).

Figure 7

Calculated $I^{\pi }=0^{+},\pi [0d_{5/2}]{}^2$ two-proton stripping longitudinal momentum distributions from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. The curves result from protons initially bound by 1 (solid), 2, 5, 10, 15, and 20 MeV. All distributions have been normalized to the same peak value.

Figure 8

Comparison of the calculated longitudinal momentum distributions from the diffraction-stripping (dashed) and the stripping (solid) mechanisms for $\pi [0d_{5/2}]{}^2$ two-proton removal from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. Curves are for the $I^{\pi }=0^{+}$ (top), $2^{+}$ (middle) and $4^{+}$ (bottom) final states.

Figure 9

Theoretical inclusive longitudinal momentum distributions (solid) with the $0^{+}$ (dashed) and $2^{+}$ (dash-dot) components. All curves have Lorentz boost and beam profile adjustments applied. The $0^{+}$ and $2^{+}$ distributions have been scaled to match the experimental $\gamma$-ray branching ratios of 84 and 16%, respectively. Applying the theoretical (shell model) branching ratios (of 64 and 36%) gives the dotted curve, normalized to the same peak value, which is slightly broader than the experimental distribution off-peak.

Figure 10

Measured (symbols) and theoretical (solid curves) inclusive momentum distributions of ${}^{36}\mathrm{Mg}$ residues after two-proton knockout from ${}^{38}\mathrm{Si}$ at 83 MeV/u. The theoretical curves for the $0^{+}$ (dashed) and $2^{+}$ (dotted) states are weighted according to the $\gamma$-ray data, being 58 and 42%, respectively. All calculations take account of the beam profile and differential momentum loss in the target.

Figure 11

Measured (symbols) and theoretical (curves) inclusive and exclusive longitudinal momentum distributions after two-proton knockout from ${}^{28}\mathrm{Mg}$ at 82 MeV/u. Panel (a) shows the measured inclusive distribution and that constructed from the weighted $0^{+}$, $2_1^{+}$, $4_1^{+}$, and $2_2^{+}$ distributions [22]. Panel (b) shows the ${}^{26}\mathrm{Ne}$($0^{+}$, g.s.) distribution, panel (c) the ${}^{26}\mathrm{Ne}$($4_1^{+}$, 3.50 MeV) final state distribution, and panel (d) shows all distributions on a logarithmic scale. All the calculated shapes take account of the beam profile (0.1 GeV/$c$) and differential energy loss in the target (0.24 GeV/$c$) broadening.

Figure 12

Heavy residue longitudinal momentum distributions after $\pi [0h_{11/2}]{}^2$ two-proton knockout from ${}^{208}\mathrm{Pb}$ for $I^{\pi }=0^{+}$ (solid), $2^{+}$ (dotted), $4^{+}$ (dashed), $6^{+}$ (solid triangles), $8^{+}$ (open squares), and ${10}^{+}$ (solid circles). The distribution for $I^{\pi }=0^{+},\pi [2s_{1/2}]{}^2$ two-proton knockout is also shown (solid line with open diamonds). The difference in the $0^{+}$ distributions for $\pi [0h_{11/2}]{}^2$ and $\pi [2s_{1/2}]{}^2$ knockout is relatively small when compared to the changes when $I$ is increased.

Figure 13

Heavy residue longitudinal momentum distributions for $\pi [0h_{11/2}]{}^2$ two-proton knockout from ${}^{208}\mathrm{Pb}$ with $I^{\pi }={10}^{+}$, showing the different magnetic substate distributions. Shown are $\left|\mu \right|=0$ (solid), 1 (open circles), 2 (dashed), 3 (open squares), 4 (dotted), 5 (open triangles), 6 (dot-dashed), 7 (open diamonds), 8 (dot-dot-dashed), 9 (inverted solid triangles), and 10 (crosses). The shaded region shows a possible cut in the heavy residue momentum that would very significantly alter the relative population of the large $\left|\mu \right|$ substates.