In transfer reaction, the kinematics determines the scattering angles and momenta, but it does not tell the probability of the reaction. Not all angle have same cross section, because when a particle is being transfer to or pickup from a nucleus, the momentum has to be matched. It is like a spaceship need a proper angle and speed in order to orbit around the earth, it the incident angle or speed are not good, the spaceship will just go out or crash into the earth.

Suppose our reaction is , is incident particle, is the target, are the outgoing particles. The momentum transfer from via to or is

where is initial momentum and is final momentum.

Assume the reaction take place at the surface of nucleus . The maximum angular momentum it can create is

In QM, the angular momentum must be . Here we pause for unit conversion. In this blog, we usually using nuclear unit, that momentum in [MeV/c] and , thus, to calculate the angular momentum, simple multiple the momentum in MeV/c and nuclear radius in fm. The radius of a nucleus is roughly .

So, if the momentum , the possible angular momentum is or the l-orbit.

In the transfer reaction, the equation for and are known. It is straight forward to calculate , plot against scattering angle, and we look for the value for .

Or, we can scale the y-axis by

The new y-axis is in unit of .

We can also plot the contour of momentum matching on the versus scatering angle. The momentum matching is

where is the scattering angle of particle . Solve for

We can plot the contour plot for difference . and then overlap with the momentum and the scattering angle of the outgoing particle .

a correction, the y-axis should be , momentum of particle 2.