Multistep processes in heavy-ion-induced single-nucleon transfer reactions

It was first noted during the 1970s that finite-range distorted wave Born approximation (FR-DWBA) calculations were unable satisfactorily to describe the shape of the angular distributions of many single-proton (and some single-neutron) transfer reactions induced by heavy ions, with calculations shifted to larger angles by up to $\sim 4^{\circ }$ compared with the data. These reactions exhibited a significant mismatch, either of the reaction $Q$ value or the grazing angular momentum of the entrance and exit channels, and it was speculated that the inclusion of multistep transfer paths via excited state(s) of the projectile and/or ejectile could compensate for the effect of this mismatch and yield good descriptions of the data by shifting the calculated peaks to smaller angles. However, to date this has not been explicitly demonstrated for many reactions. In this work we show that inclusion of the two-step transfer path via the 4.44-MeV $2^{+}$ excited state of the ${}^{12}\mathrm{C}$ projectile in coupled channel Born approximation calculations enables a good description of the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{11}\mathrm{B}){}^{209}\mathrm{Bi}$ single-proton stripping data at four incident energies which could not be described by the FR-DWBA. We also show that inclusion of a similar reaction path for the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{13}\mathrm{C}){}^{207}\mathrm{Pb}$ single-neutron pickup reaction has a relatively minor influence, slightly improving the already good description obtained with the FR-DWBA.

Figure 1

Calculations for the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{11}\mathrm{B}){}^{209}\mathrm{Bi}$ reaction at bombarding energies of 77.4 MeV (left) and 97.9 MeV (right) compared with the data of Ref. [1]; (a) to (e) denote stripping to the 0.0-MeV $9/2^{-}$, 0.90-MeV $7/2^{-}$, 1.61-MeV $13/2^{+}$, 2.84-MeV $5/2^{-}$, and 3.12-MeV $3/2^{-}$ states of ${}^{209}\mathrm{Bi}$ at a bombarding energy of 77.4 MeV and (f) to (j) at a bombarding energy of 97.9 MeV. The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopic amplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBA calculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBA calculations using the projectile-overlap spectroscopic amplitudes of Ref. [9].

Figure 2

Calculations for the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{11}\mathrm{B}){}^{209}\mathrm{Bi}$ reaction at bombarding energies of 101 MeV (left) and 116.4 MeV (right) compared with the data of Refs. [2] and [1], respectively; (a) to (e) denote stripping to the 0.0-MeV $9/2^{-}$, 0.90-MeV $7/2^{-}$, 1.61-MeV $13/2^{+}$, 2.84-MeV $5/2^{-}$, and 3.12-MeV $3/2^{-}$ states of ${}^{209}\mathrm{Bi}$ at a bombarding energy of 101 MeV and (f) to (i) at a bombarding energy of 116.4 MeV. The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopic amplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBA calculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBA calculations using the projectile-overlap spectroscopic amplitudes of Ref. [9].

Figure 3

Calculations for the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{13}\mathrm{C}){}^{207}\mathrm{Pb}$ reaction at bombarding energies of 77.4 MeV (left) and 97.9 MeV (right) compared with the data of Ref. [1]; (a) to (e) denote pickup to the 0.0-MeV $1/2^{-}$, 0.57-MeV $5/2^{-}$, 0.90-MeV $3/2^{-}$, 1.63-MeV $13/2^{+}$, and 2.34-MeV $7/2^{-}$ states of ${}^{207}\mathrm{Pb}$ at a bombarding energy of 77.4 MeV and (f) to (j) at a bombarding energy of 97.9 MeV. The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopic amplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBA calculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBA calculations using the projectile-overlap spectroscopic amplitudes of Ref. [13].

Figure 4

Calculations for the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{13}\mathrm{C}){}^{207}\mathrm{Pb}$ reaction at bombarding energies of 101 MeV (left) and 116.4 MeV (right) compared with the data of Refs. [2] and [1], respectively; (a) to (e) denote pickup to the 0.0-MeV $1/2^{-}$, 0.57-MeV $5/2^{-}$, 0.90-MeV $3/2^{-}$, 1.63-MeV $13/2^{+}$, and 2.34-MeV $7/2^{-}$ states of ${}^{207}\mathrm{Pb}$ at a bombarding energy of 101 MeV and (f) to (h) the 0.0-MeV $1/2^{-}$, 0.90-MeV $3/2^{-}$, and 2.34-MeV $7/2^{-}$ states of ${}^{207}\mathrm{Pb}$ at a bombarding energy of 116.4 MeV. The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopic amplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBA calculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBA calculations using the projectile-overlap spectroscopic amplitudes of Ref. [13].

Figure 5

For the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{11}\mathrm{B}){}^{209}\mathrm{Bi}$ reaction: In (a) to (e) the solid curves plot the optimum angular momentum transfer ($L_{\mathrm{opt}}$) values as a function of ${}^{12}\mathrm{C}$ bombarding energy for stripping from the ${}^{12}\mathrm{C}$ ground state and population of the 0.0-MeV $9/2^{-}$, 0.90-MeV $7/2^{-}$, 1.61-MeV $13/2^{+}$, 2.84-MeV $5/2^{-}$, and 3.12-MeV $3/2^{-}$ states of ${}^{209}\mathrm{Bi}$, respectively. The hatched bands denote the range of allowed $L$ transfers for each transition. In (f) to (j) the dashed curves plot the $L_{\mathrm{opt}}$ values as a function of ${}^{12}\mathrm{C}$ bombarding energy for stripping from the ${}^{12}\mathrm{C}$ 4.44-MeV $2^{+}$ excited state. The hatched bands denote the range of allowed $L$ transfers when the stripped proton is in the $1p_{3/2}$ level (as for stripping from the ${}^{12}\mathrm{C}$ ground state) and the dotted bands when the stripped proton is in the $1p_{1/2}$ level. In (k) to (o) the solid curves plot $Q-Q_{\mathrm{opt}}$ for stripping from the ground state of ${}^{12}\mathrm{C}$ and population of the 0.0-MeV $9/2^{-}$, 0.90-MeV $7/2^{-}$, 1.61-MeV $13/2^{+}$, 2.84-MeV $5/2^{-}$, and 3.12-MeV $3/2^{-}$ states of ${}^{209}\mathrm{Bi}$, respectively, while the dashed curves plot $Q-Q_{\mathrm{opt}}$ for stripping from the ${}^{12}\mathrm{C}$ 4.44-MeV $2^{+}$ excited state.

Figure 6

For the ${}^{208}\mathrm{Pb}({}^{12}\mathrm{C},{}^{13}\mathrm{C}){}^{207}\mathrm{Pb}$ reaction: In (a) to (e) the solid curves plot the optimum angular momentum transfer ($L_{\mathrm{opt}}$) values as a function of ${}^{12}\mathrm{C}$ bombarding energy for pickup to the ${}^{12}\mathrm{C}$ ground state and population of the 0.0-MeV $1/2^{-}$, 0.57-MeV $5/2^{-}$, 0.90-MeV $3/2^{-}$, 1.63-MeV $13/2^{+}$, and 2.34-MeV $7/2^{-}$ states of ${}^{207}\mathrm{Pb}$, respectively. The dotted bands denote the range of allowed $L$ transfers for each transition. In (f) to (j) the dashed curves plot the $L_{\mathrm{opt}}$ values as a function of ${}^{12}\mathrm{C}$ bombarding energy for pickup to the ${}^{12}\mathrm{C}$ 4.44-MeV $2^{+}$ excited state. The hatched bands denote the range of allowed $L$ transfers for each transition. In (k) to (o) the solid curves plot $Q-Q_{\mathrm{opt}}$ for pickup to the ground state of ${}^{12}\mathrm{C}$ and population of the 0.0-MeV $1/2^{-}$, 0.57-MeV $5/2^{-}$, 0.90-MeV $3/2^{-}$, 1.63-MeV $13/2^{+}$, and 2.34-MeV $7/2^{-}$ states of ${}^{207}\mathrm{Pb}$, respectively, while the dashed curves plot $Q-Q_{\mathrm{opt}}$ for pickup to the ${}^{12}\mathrm{C}$ 4.44-MeV $2^{+}$ excited state.