content_image/368217.jpg content_image/368216.jpg content_image/368219.jpg content_image/368218.jpg content_image/368215.jpg content_image/368213.jpg content_image/368214.jpg content_image/368212.jpg content_image/368211.jpg
Using these expressions and the above for ϵ$_{+}$ and ϵ − , we find that 

<img src='content_image/368217.jpg'>

<img src='content_image/368216.jpg'>

<img src='content_image/368219.jpg'>

<img src='content_image/368218.jpg'>

The process is completed when 1 has zero velocity so the condition for capture is: 

<img src='content_image/368215.jpg'>

<img src='content_image/368213.jpg'>

<img src='content_image/368214.jpg'>

When this equation is combined with the definition of cos( θ ) , the simple result: sin(( n + 1) θ ) = sin( nθ ) , emerges. Aside from the trivial solution θ = 0 , which corresponds to no collisions at all, there are n solutions: 

<img src='content_image/368212.jpg'>

The allowed region is now readily found by insisting that α be real, which is to say, − 1 ≤ cos( α ) ≤ 1 . If we solve Eq. (90) for cos( α ) and insist that it be real and between -1 and 1, we are left with: 

<img src='content_image/368211.jpg'>

From this equation, it follows that θ cannot be π for that would yield the condition: 1 + a + b ≤ 0 . However, the remaining n possible choices for θ yield allowed regions which are bounded by the hyperbola: 1 + a + b = cot $^{2}$( θ/ 2) . The hyperbola is symmetric about a = b and has asymptotes at a = tan $^{2}$( θ/ 2) and b = tan $^{2}$( θ/ 2) . For example, if n=1, we are dealing with (32) which is the symmetric curve of the Lieber diagram. In that case, 

