Nucleon transfer in inverse kinematics

As mentioned above, transfer reactions induced by radioactive beams on p and d targets have great potential for giving new nuclear structure information. Some have already been reported (e.g. [11,4,12]). It is a general result that pickup reactions such as (p,d) and (d,t) result in light ejectiles that are focussed into the forward hemisphere of laboratory angles (see fig. 1a). The light ejectiles in stripping reactions such as (d,p) are confined to the backward hemisphere, if we consider just the small scattering angles ( $\theta _{{\rm c.m.}} < 30^{\circ }$ say) with highest yield. Elastic scattering is similarly concentrated at angles near $90^\circ$ in the laboratory frame (fig. 1b). These results have a significant impact on experimental design as highlighted by Hardy [13].

  

Figure: Velocity diagrams for inverse kinematics: (a) generic diagram with lengths as in (p,d) reactions, (b) for elastic scattering. Here, $v_{{\sf c.m.}}$ is the velocity of the centre of mass in the laboratory frame; $v_{{\sf e}}$ and $v_{{\sf R}}$ are velocities of the light ejectile and the heavier recoil in the c.m. frame. This presentation using classical, rather than relativistic, velocity addition helps to show most clearly the general results.

 

 

The kinematic focussing is dependent principally on the target and ejectile masses, and only residually on the projectile mass and the reaction Q-value. For elastic scattering, note that in the centre of mass frame the target velocity (before the reaction) and after the reaction (as the lighter ejectile) are equal in magnitude. This velocity also equals the speed of the c.m. frame relative to the laboratory. The highest yield of elastic scattering, corresponding to the most weakly deviating particles in the c.m. frame, then occurs close to $90^\circ$ in the laboratory frame (fig. 1b) since $v_e ^{{\sf lab}}$ is at $90^\circ$ for the case of $v_{{\rm c.m.}}$ and v_e nearly anti-parallel. In the case of pickup such as (p,d), where the ejectile mass exceeds the target mass, the length of the vector v_e is reduced relative to v_R according to momentum conservation and this confines the lighter ejectiles to within a forward cone labelled POQ (fig. 1a). In contrast, the weakly deflected ejectiles in (d,p) will come at backward angles in the laboratory frame. If the masses of the target, projectile, ejectile and recoil are denoted as M_T, M_P, M_e and M_R respectively, then [14]

$$\frac{v_e}{v_{{\rm c.m.}}} ~~=~~ \left( ~q~f~\frac{M_R}{M_P} ... ...^{1/2} ~~\approx ~~ \sqrt{~qf~} {\rm ~~if~~} M_R \approx M_P ,$$

where f is given by f=M_T / M_e and $q=1+Q_{{\rm tot}}/E_{{\rm c.m.}}$, with $Q_{{\rm tot}}=(Q_{{\rm g.s.}} - E_x )$ being the Q-value for a state at energy E_x in the recoil, and $E_{{\rm c.m.}}$ the collision energy in the c.m. frame. Typically q differs from unity by less than 10%, getting closer as the beam energy per nucleon (E/A) is increased: $q \approx 1 + Q_{{\rm tot}}/(E/A)_{{\rm beam}}$. Thus, for a reaction such as (p,d) the light ejectiles are confined (cf. fig. 1a) to within a cone of half-angle POZ given by where f=1/2 for (p,d) and f=2/3 for (d,t). This gives about $50^\circ$ in each case, but the extra focussing for (p,d) can be significant experimentally. Application of the cosine and sine rules shows that, for a (d,p) reaction, the scattering angles $\theta _{{\rm c.m.}} < 30^{\circ }$ are focussed to laboratory angles backward of about $110^\circ$. Note that the beam mass and bombarding energy have no effect on these results, within the q=1 approximation.

  

Figure: Kinematics for transfer reactions induced by 30Mg ions on p and d targets.

As an example, the kinematics for d(30Mg,31Mg)p and associated reactions are shown in fig. 2. The equivalent c.m. scattering angles for `normal' kinematics are indicated. Note that the reactions (p,d), (d,t) and (d,3He) are double-valued in energy at the forward angles, but the higher energy solution corresponds to large c.m. scattering angles. A case when the largest scattering angles would be of interest is sub-barrier transfer, which offers the possibility to measure directly the nucleon density in the tails of wave functions [15].