Consider a reaction with a two-body final state. The energy released in the rearrangement can be written
If Q;SPMgt;0 the reaction is exothermic; if Q;SPMlt;0 the reaction is endothermic. For an elastic collision, it is obvious that Q=0. For an endothermic reaction to occur, there must be sufficient kinetic energy T in the incident channel.
An inelastic reaction which leaves one of the products in an excited state
Measurement of the energy of the scattered particle then gives a simple method of discovering some of the excited states of a nucleus.
In general, the kinetic energies of the reacting nuclei must be taken into account in determining whether the reaction is energetically feasible. This is slightly complicated by the motion of the centre of mass of the system.
We generally consider collisions in which the target nucleus is stationary in the laboratory. After the collision the ejectile and residual nucleus move off at some angle to the initial direction. If one angle is fixed (say the angle at which the ejectile is detected in an experiment), the momenta of all the particles involved can be determined by application of the laws of conservation of momentum and energy. In particular, the kinetic enegies, and angles of motion with respect to the incident direction are often required.
The calculations of these quantities is made simpler by separating the motion in the centre of mass frame from that of the centre of mass frame. The centre of mass frame is an inertial frame in which the motion is most simply observed. The kinetic energy of the centre of mass of the system plays no rôle in the collision. In the centre of mass frame, the two colliding nuclei always have momenta which are equal in magnitude but opposite in direction. If the collision is not elastic, the magnitude of the momentum is changed in the collision.
Quantities of interest in reaction studies are always specified in term of their values in the centre of mass frame.