Inverse kinematics

When dealing with radioactive beams, transfer reactions such as (p,d), (d,p) and (d,³He) need to be studied using hydrogen targets. In general, the beam particle is many times heavier than the target, and the kinematics are massively inverse. Transfer reactions initiated by heavy ions (as the target nucleus) lead to problems if the target-like recoil is to be observed except in special cases (since, except for the lightest heavy ions, the recoil will usually be too low in energy to escape from the target) and will not be discussed here. A key result of the highly inverse kinematics is that the energy-angle systematics of the target-like particles (the d, p or ³He in the above examples) are very similar for all reactions of a given type. That is, the mass or energy of the incident beam, and to a large extent the Q-value, all have relatively little effect. The over-riding factor is the change in mass of the target-like particle: from 1 to 2, 2 to 1, or 2 to 3 in the examples. To see this, it is perhaps easiest to consider the velocity addition diagrams for the two-body kinematics. Even though these are not relativistically accurate, they give a good qualitative indication of the results and are even rather good quantitatively in the energy regime of interest. For elastic scattering (see fig. 1) the target-like particles emerge close to 90 degrees in the laboratory frame, for small centre of mass scattering angles. Their energies also start at zero and rise rapidly with increasing c.m. scattering angle. For the beam-like particle, the velocity in the c.m. frame is very much lower (by the ratio of the target to the beam mass) and the deflection angle is also very small. These results are clearly independent of the precise mass or velocity of the incident beam particle, so long as it is heavy compared to the target.

  

Figure 1: For elastic scattering, the velocity of the c.m. in the laboratory is equal and opposite to the initial velocity of the target in the c.m. frame, and small c.m. scattering angles give laboratory angles near 90 degrees.

Now, considering transfer reactions, the key factor is the change in mass for the target-like particle, since this dramatically affects its c.m. velocity. In the c.m. frame, the kinetic energy of the the heavy (beam-like) particle is smaller than that of the light particle by a factor of order $1/m_{{\rm beam}}$, and is negligible for $m_{{\rm beam}} \gg m_{{\rm ejectile}}$. Considering (p,d) as an example, and ignoring Q-value effects for the moment, the kinetic energy of the d in the c.m. frame is thus about the same as that of the p in elastic scattering. However, its mass is doubled and therefore since $K \approx p^2 /2m$ its momentum is increased by $\sqrt{2}$, and its velocity v=p/m in the c.m. frame is multiplied by a factor of $\sqrt{2}/2$. In the general case where the mass of the target M_T and the ejectile M_e are in the ratio f=M_T /M_e then the factor is $\approx \sqrt{f}$. The consequences for the velocity diagrams are illustrated in fig. 2. If the masses of the projectile and beam-like `recoil' are denoted as M_P and M_R respectively, then [7]

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

where $v_{{\rm c.m.}}$ is the velocity of the c.m. in the laboratory frame 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: $q \approx 1 + Q_{{\rm tot}}/(E/A)_{{\rm beam}}$.

  

Figure 2: As for the previous figure, except for the situation when transfer changes the mass of the light particle (and recoil): (a) for reactions such as (p,d) or (d,³He) where the target-like mass increases, (b) for reactions such as (d,p) where the mass is decreased.

Thus, for a reaction such as (p,d) the light ejectiles are confined (cf. fig. 2) 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. These general results are illustrated in fig. 3 using two examples with substantially different projectile masses and velocities. The figures are labelled with the c.m. scattering angles (according to the traditional `normal kinematics' convention, where the light particle is the projectile) from 0 to 30 degrees, where the differential cross section is the greatest. The figures confirm that the yield for particle removal from the projectile, for elastic scattering, and for particle addition are concentrated at forward angles, near 90 degrees, and at backward angles respectively. An important additional result is that the energies of the scattered light particles are similar for the various reactions, which means that an all-purpose charged particle array can be considered for transfer studies with radioactive beams. It is also apparent that, if the Z of the beam-like particle is measured, then the energy and angle of the light particle largely serve to identify its Z and A.

  

Figure 3: Typical energy-angle systematics for transfer reactions in inverse kinematics: (a) for ¹⁶C at 35 MeV/u, and (b) for ⁷⁴Kr at 12.16 MeV/u, each on proton and deuteron targets.

An interesting further feature of the inverse kinematics is the effect of the jacobian which relates the differential cross section measured in the laboratory frame to that in the c.m. frame (see, for example, ref. [8]). The nature of the kinematic focussing of angles can be inferred from the labelling of the c.m. angles in the kinematics of fig. 3. In some angular ranges, the experimental angular resolution becomes critical. A particularly interesting result is the defocussing that occurs for (d,p) at the small scattering angles in the `light ion' convention (near 180 degrees in the laboratory frame). This is illustrated in fig. 4 which refers to a proposed study of Sr isotopes [9].

  

Figure: Calculation of the differential cross section for (d,p) on ⁹⁵Sr, using zero-range DWBA, plotted as (a) solid line, d$\sigma$/d$\Omega$ vs. $\theta _{{\rm c.m.}}$ in the traditional light ion convention, (b)dashed, d$\sigma$/d$\Omega$ vs. $\theta _{{\rm lab}}$ in the laboratory frame, see text.

In fig. 4 the dashed line would be the mirror image of the solid line, around 90 degrees, if it were not for the jacobian effects. Due to the kinematic focussing, the region of largest differential cross section is no longer the region of smallest scattering angles. However, it is still true in general that the smallest scattering angles are the ones best modelled by the reaction codes and therefore most important to measure. The difference compared to traditional measurements is that it becomes relatively easy to extend the experiment to include larger scattering angles as well.