## III. CAPTURE PROCESSES WITH THREE OR MORE COLLISIONS 

## A. Preliminaries 

In order for a collision sequence to lead to capture, a few simple conditions must be met: 

1. Consecutive collisions must be distinct since two isolated particles cannot collide more than once. 

2. The first collision cannot be (1) because particle 1 initiates the capture process and the last collision must not be (3) because if 1 and 2 collide at the end of the process they cannot emerge with the same velocity. 

3. The times between collisions must be positive. 

Using these conditions, we prove the following useful lemma in Appendix A. 

Lemma 1 When three distinct collisions between three particles occur consecutively, a fourth collision must be of the same type as the second. Furthermore, sequences of the form (31323) in which the first, third and fifth collisions are of the same type, but the second and fourth are of a different type, cannot occur. 

Since capture begins with 2 and 3 having the same velocity and ends with 1 and 2 having the same velocity, the collision sequence can be considered to begin with the collision (1) and to end with the collision (3). If three distinct collisions were to occur consecutively between the virtual (1) and virtual (3), our lemma would be violated. Therefore a capture process must only consist of two types of collisions. There are only five such processes, namely: (32) $^{n}$, (2)(32) $^{n}$, (21) $^{n}$, (21) $^{n}$(2) , and (31) $^{n}$. This is fortuitous because the classical scattering matrices for such processes are computed simply by raising to a power the scattering matrix of the repeated pair. 

We will first consider (32) n and (2)(32) $^{n}$, which have allowed regions which are within the forbidden region of the Lieber diagram. Because each collision in these processes involves particle 1, the whole problem can be done in the rest frame of 1. The scattering matrices are easily found for such a frame, but since particle 1 accelerates when it collides in an inertial frame and in the frame we are describing, it is at rest, this frame is noninertial. Thus energy and momentum will not be conserved in collisions (2) and (3) (the ones in which particle 1 is involved). In this scheme, one simply applies the scattering matrices to the velocity vector and then subtracts from each of the three velocities, the velocity of particle 1. If we 

