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An important class of two particle scattering is when the forces are central, {\it i.e.} the potential depends on the separation \(\vec r_1-\vec r_2\) between the particles: \[ V(\vec r_1, \vec r_2)= V(\vec r_1-\vec r_2).\] The two particle elastic scattering problem becomes equivalent to scattering of a particle, of reduced mass, from a potential \(V(\vec r)\), with \(\vec r= \vec r_1-\vec r_2,\) in the centre of mass frame.
We shall discuss elastic scattering \( a+b \Longrightarrow a+b \) of two particles interacting via a central potential.
If the forces are central, the scattering, two body problem, in the centre of mass frame is equivalent to one particle motion in potential $V(\bf r)$, if we take the mass of the particle in the equivalent problem to be the reduced mass $\mu$ given by
\[\mu = \frac{m M}{m+M}.\] where \(m\) and\(M\) are the masses of the incident particle and the target.
The reduction of two particle problem to potential scattering can also viewed in an alternate fashion. When the mass of the target is very large compared to the incident particle, the motion of the target can be neglected. In this limit \(\mu \to m\) and two particle, of mass \(m\), scattering reduces to scattering of a particle from a fixed target.
We shall assume that the potential $V(\bf r)$ is spherically symmetric and goes to zero for large $r$ (How fast? faster than $\frac{1}{r^2}$).
It is, therefore, sufficient to consider potential scattering. The results of the potential scattering are to be interpreted as being valid in the centre of mass frame. In order to compare theoretical predictions with experiments, the potential scattering expressions must be transformed to the laboratory frame.
