Illustrative kinematics calculations

In this section, we collect together the kinematics calculations for the sample reactions mentioned in section 1.2. They have been chosen to cover a wide range of transfers, energies and masses and represent a typical selection of reactions that could be studied. The reactions for which kinematics are presented are:

• Fig. 12 $\sim$ proton pickup with a hydrogen target:
  Kinematics for the reaction d(⁷²Kr,⁷¹Br)³He at a beam energy of 1080 MeV or 15 MeV/A. Note that in this case the kinematic solution is double-valued at forward angles. This illustrates a quite general property of pickup reactions on hydrogen targets, which is that the kinematic solution of interest is wrapped around into the forward hemisphere of laboratory angles,
• Fig. 13 $\sim$ Neutron pickup with a hydrogen target:
  Kinematics for the reaction p(¹⁷B,¹⁶B)d at a beam energy of 680 MeV or 40 MeV/A. Note that the (p,d) reaction also has the events of interest folded back into the forward laboratory angles,
• Fig. 14 $\sim$ Neutron stripping with a hydrogen target:
  Kinematics for the reaction d(²⁸Ne,²⁹Ne)p at a beam energy of 560 MeV or 20 MeV/A. Note that the kinematics for this reaction do not cause the second kinematic solution to wrap around to forward angles and the particles of interest are found at back-angles in the laboratory frame as well as in the centre of mass. This is a general feature of stripping reactions on hydrogen targets,
• Fig. 15 $\sim$ Neutron stripping with a heavy-ion target:
  Kinematics for the reaction ¹³C(²⁶Ne,²⁹Ne)¹²C at a beam energy of 390 MeV or 15 MeV/A. This is an example of a transfer reaction using a `heavy ion' target, and could be expected to show additional properties in its selectivity compared to light ion transfer, such as the $j_> \, / j_<$ selectivity introduced by angular and linear momentum matching of the nucleon at the point of transfer. The kinematic focussing is less than with a hydrogen target, and the particles of interest are found typically near 90 degrees to the beam direction.

  

Figure: Kinematics for the reaction d(⁷²Kr,⁷¹Br)³He at a beam energy of 1080 MeV or 15 MeV/A. Note that in this case the kinematic solution is double-valued at forward angles. In the table, it is the second solution that corresponds to the normal forward angles with the strongest yield in normal kinematics. Thus, the centre of mass angle of interest is actually $(180-\theta )$ where $\theta$ is the centre of mass angle quoted in the tables and plots. (The gap in the plot at large centre of mass angles is due to a numerical problem related to the folding back of the angular distribution and is not real).

  

Figure 13: Kinematics for the reaction p(¹⁷B,¹⁶B)d at a beam energy of 680 MeV or 40 MeV/A. Note that the (p,d) reaction has the events of interest folded back into the forward laboratory angles as they were for the (d,³He) examples.

  

Figure 14: Kinematics for the reaction d(²⁸Ne,²⁹Ne)p at a beam energy of 560 MeV or 20 MeV/A. Note that for this reaction there is only one solution at each laboratory angle in the kinematics. Thus, the angles of interest for strongest transfer are not folded back into the forward laboratory angles as they were for the (d,³He) examples.

  

Figure 15: Kinematics for the reaction ¹³C(²⁶Ne,²⁷Ne)¹²C at a beam energy of 390 MeV or 15 MeV/A. Note that for this reaction there is only one solution at each laboratory angle in the kinematics. Thus, the angles of interest for strongest transfer are not folded back into the forward laboratory angles as they were for the (d,³He) examples. Furthermore, the kinematic focussing is relatively small and the detectable target-like particles are found at angles much closer to 90 degrees than for the reactions with the hydrogen targets.