Angular Orbital Momentum and the Impact Parameters

Consider an interaction potential-energy, V(r), which tends to zero as r approaches infinity. This situation corresponds to the condition that a moving particle has positive kinetic energy at infinity. If V(r) is everywhere positive, but decreasing monotonically with r, the potential is repulsive and the radial motion of this particle in the field V(r) will have no bounds or limits in its maximum value of r. However, there is a minimum in r, the distance of closest approach, rmm, which depends on the particle s total energy and the nature of the interaction potential. 

In Fig. 3.4a, the energy curves for attractive potential-energy are presented along with an arbitrarily defined centrifugal energy curve. In Fig. 3.4b, the effective potential-energy curve, given by 

From Fig. 3.4, the distance of closest approach, rmm, is shown to be determined from the condition E = V (r), which translates to 

Using (3.25) to define the orbital angular momentum, we rewrite (3.26) in the form 

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