Question

An energetic electron collides with a positron at rest. what is the minimum kinetic energy the incident proton must have to make the reaction e+ + e- --> p + p-

Answer 1

We assume that there is conservation of Energy and Momentum:
In this equation there are 4 particles:

We will label them as 

 --  4-vector (p₀, p₁, p₂, p₃ )

no of particles before collision = no of particles after  collision

.

Relativity concept:

For transformations between reference frames we have

position after transformation is also conserved i.e

(P₀,P)·(P₀,P) = (P₀',P')·(P₀',P').

Here  P₀ = total number of particles p₀ and  P = total number of particles p.

 Proton will only have minimal energy when the reaction products are at rest in the reference frame.

so it we can use the pythogrus theorum: assuming the projection is in right angles;

P₀²- P² = ∑P₀² 

but  (∑p₀)² = (4 mc)²= 16 m²c²

because there are 4 particles:The "length" of the momentum 4-vector is 16 m²c²before and after the collision.

Before the collision we have

P₀²- P² = (p₀ₐ + p₀b)² - (pₐ + pb² 

expanding this equation we get:

= p₀ₐ²+ p₀b² - pₐ² + 2p₀ₐp₀b.

where  P₀= p₀ₐ + p₀b

and   P=pₐ + pb

For any free particle we have p₀²- p² = m²c².  

simplifying the equation we get:

=P₀² - P²

  16m²c² = 2m²c²+ 2p₀ₐp₀b,  p₀ap₀b

Devide both sides by 2

 = 7 m²c².  p₀ₐmc 

= 7 m²c².
Eₐ /c= 7 mc x mc² = 938 MeV,

  Eₐ= 6.6 GeV.

The proton must have Ea - mc² = 6 mc² of kinetic energy to make the reaction possible.