Quasifree ($p$,$2p$) and ($p$,$pn$) reactions with unstable nuclei

We study ($p,2p$) and ($p,pn$) reactions at proton energies in the range of 100 MeV–1 GeV. Our purpose is to explore the most sensitive observables in unpolarized reactions with inverse kinematics involving radioactive nuclei. We formulate a model based on the eikonal theory to describe total cross sections and momentum distributions of the recoiled residual nucleus. The model is similar to the one adopted for knockout reactions with heavy ions. We show that momentum distributions are sensitive to the angular momentum of the ejected nucleon which can be used as an spectroscopic tool. The total cross sections are sensitive to the nucleon separation energies and to multiple scattering effects. Our calculations also indicate that a beam energy around 500 MeV/nucleon has a smaller dependence on the anisotropy of the nucleon-nucleon elastic scattering.

Figure 1

The coordinates used in the text are shown.

Figure 2

Cross sections of ${}^{12}$C($p,2p$)${}^{11}$B fragmentation on hydrogen targets. The data are from Refs. [76, 77]. The dashed-dotted curve shows the cross section calculated according to Eq. (34), with ${\langle d{\sigma }_{pp}/d\Omega \rangle }_{o.s.}={\sigma }_{pp}^{\mathrm{tot}}/4\pi$ where ${\sigma }_{pp}^{\mathrm{tot}}$ is total $p$-$p$ free cross section. The full curve shows the same calculation multiplied by a factor 0.77. The dashed curve uses ${\langle d{\sigma }_{pp}/d\Omega \rangle }_{o.s.}$ given by the angular average of the elastic $pp$ cross section satisfying the energy constraint of Eq. (7).

Figure 3

Cross sections of ${}^{12}$C($p,pn$)${}^{11}$C fragmentation on hydrogen targets. The data are from Refs. [76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. The dashed-dotted curve shows the cross section calculated according to Eq. (34), with ${\langle d{\sigma }_{pn}/d\Omega \rangle }_{o.s.}={\sigma }_{pn}^{\mathrm{tot}}/4\pi$ where ${\sigma }_{pn}^{\mathrm{tot}}$ is total $p$-$n$ free cross section. The full curve shows the same calculation multiplied by a factor 0.77. The dashed curve uses ${\langle d{\sigma }_{pn}/d\Omega \rangle }_{o.s.}$ given by the angular average of the elastic $p$-$n$ cross section satisfying the energy constraint of Eq. (7).

Figure 4

Free $p$-$p$ and $p$-$n$ total cross sections (solid curves) as compared to the constrained angle averaged elastic $p$-$p$ and $p$-$n$ cross sections (dashed curves) according to Eq. (35) and for ${}^{12}$C($p,pN$), with $N=p$ or $n$.

Figure 5

Proton and neutron knockout probabilities from $p$ states in ${}^{12}$C at 500 MeV/nucleon, according to Eq. (36). The solid curves are for ($p,pn$) and dashed curves for ($p,2p$). We also show the results obtained with ${\langle \mathcal{S}\left(b\right)\rangle }_{o.s.}=1$ in Eq. (36) (no rescattering).

Figure 6

Longitudinal (upper panel), transverse (middle panel), and total momentum distributions (lower panel) for the missing momentum in ${}^{12}$C($p,2p$)${}^{11}$B fragmentation on hydrogen targets. The full curves are for $p$ states, the dashed curves are for $s$ states. The dotted curves $s^{\prime }$ correspond to the calculations where the energy of the $s$ states are taken as the $p$-state energy. The curves are rescaled to be shown in the same plot.

Figure 7

Cross sections for neutron removal in ($p,pn$) reactions on ${}^{23}$O from ${\left[0d_{5/2}\right]}^6$ and $\left[1s_{1/2}\right]$ orbitals as a function of the separation energy. The cross sections for neutron removal from the ${\left[0d_{5/2}\right]}^6$ orbital is divided by the number of the neutrons in the orbital (6). The separation energies are varied artificially. The dashed and dotted curves are to guide the eyes.

Figure 8

Neutron skin of even-even Ni isotopes obtained with a HFB calculation (upper panel). The lower panel displays the ($p,pn$) and ($p,2p$) cross sections on Ni isotopes at 500 MeV/nucleon assuming a fixed binding energy for the knocked out nucleon (see text for explanation). The dashed curves are to guide the eyes.

Figure 9

Upper panel: The dashed curve shows the probability for removal of a neutron in the reaction ${}^{12}$C(${}^{68}$Ni,${}^{67}$Ni) at 500 MeV/nucleon as a function of the distance to the c.m. of ${}^{68}$Ni. The dotted curve represents the removal probability in a ${}^{68}$Ni($p,pn$) reaction at the same energy. For comparison the square of the radial wave function $u\left(r\right)$ is also shown (solid curve). Middle panel: Same as the upper panel, but for the reactions ${}^{12}$C(${}^{11}$Be,${}^{10}$Be) and ${}^{11}$Be($p,pn$) at 500 MeV/nucleon. Lower panel: Same as upper panel, but for the reactions ${}^{12}$C(${}^8$B,${}^7$Be) and ${}^8$B($p,2p$) at 500 MeV/nucleon.

Figure 10

Full width at half-maximum (FWHM) of the transverse momentum distributions in ($p,2p$) and ($p,pn$) reactions of ${}^{20}$O at 500 MeV/nucleon as a function of the proton and neutron separation energies. The upper curve is for ($p,pn$) and lower curve for ($p,2p$) reactions. The separation energies are varied artificially (see text for explanation). The dashed and dotted curves are to guide the eyes.

Figure 11

Longitudinal momentum distributions for ${}^{11}$Be($p,n$) and ${}^{68}$Ni($p,n$) reactions at 500 MeV/nucleon. For ${}^{11}$Be we assume neutron removal from the $1s_{1/2}$ states with 0.502 MeV separation energy, while for ${}^{68}$Ni we assume the removal from 0f${}_{3/2}$ orbital with separation energy of 15.68 MeV. PWIA results are shown as dashed lines, whereas the solid lines are for the full DWIA.

Figure 12

Same as in Fig. 11, but for transverse momentum distributions.