Collision Theory

                                    

The impact parameter, b, is the off-set distance of the centers as they approach one another. The velocity component along the lines of centers, U_LC, can be obtained by resolving the approach velocity into components.

            At the point of collision the center of B is within the distance _AB

                                            

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The energy along the line of centers can be developed by a simple geometry arguments

                                                                                                                    (1)

The component of velocity along the line of centers

                                                         U_LC = U_R cos                                                        (2)

The kinetic energy along the line of centers is

                                                                          (3)

                                                                                (4)

The minimum energy along the line of centers necessary for a reaction to take place, E_A, corresponds to a critical value of the impact parameter, b_crit. In fact, this is a way of defining the impact parameter and corresponding reaction cross section

                                                                                                                       (5)

Substituting for E_A and b_crit in Equation (4).

                                                                                                         (6)

Solving for

                                                                                                        (7)

The reaction cross section for energies of approach E > E_A, is

                                                                                         (8)

The complete reaction cross section for all energies E is