# Four momentum proton-proton scattering question

• Jdraper

## Homework Statement

In a fixed target experiment a particle of mass M and kinetic energy T strikes a stationary particle of mass M. By evaluating s, t and u in the laboratory frame and using the above relation, or otherwise, show that the kinetic energy T' of the particle scattered elastically at an angle θ is given by

T' = T cos^2 θ / (1 + ((T/2M)*sin^2 θ))

## Homework Equations

We know that:
We define P1 as the four momentum of the incoming proton:
P1 = (T+M,0,0,T)
P2 as the stationary proton:
P2 = (M,0,0,0)
P3 as the initially incoming proton after the scattering event:
P3 = (T'+M, 0, T'sinθ, T'cosθ)
P4 as the initially stationary proton after the scattering event:
P4 = (E4+M, P4) where P4 in the bracket is a 3 vector not 4.

We also know:

s = (P1 + P2)2 , t = (P1 − P3)2 and u = (P1 − P4)2

and

s + t + u = ∑ mi2 = 4m2

## The Attempt at a Solution

I have attempted this multiple times over a few days, I realize that P4 and therefore u need to be eliminated as they aren't in the solution but I am unsure how.

Any help/ suggestions would be appreciated. Thanks, John.

## Homework Statement

In a fixed target experiment a particle of mass M and kinetic energy T strikes a stationary particle of mass M. By evaluating s, t and u in the laboratory frame and using the above relation, or otherwise, show that the kinetic energy T' of the particle scattered elastically at an angle θ is given by

T' = T cos^2 θ / (1 + ((T/2M)*sin^2 θ))

## Homework Equations

We know that:
We define P1 as the four momentum of the incoming proton:
P1 = (T+M,0,0,T)
.

You seem to be using the KE instead of the momentum in your four-vector.

You seem to be using the KE instead of the momentum in your four-vector.
Hey, forgot to mention that I'm using the convention that c=1.

Hey, forgot to mention that I'm using the convention that c=1.

Okay, but you still don't have ##T = p##:

##T = E - M## and ##p^2 = E^2 - m^2##

PS The problem comes out from conservation of energy and momentum, although I haven't looked at how to do it using your equations for ##s, t, u, m##.

Okay, but you still don't have ##T = p##:

##T = E - M## and ##p^2 = E^2 - m^2##

You've opened up a can of worms in my understanding here. My professor uses this convention and I have never really questioned it.

So you are saying:

##P=(E^2 -M^2)^{0.5}##
therefore as ##E=T+M## we can rewrite this as:

##P=((T+M)^2 + M^2)^{0.5} = (T^2 +2TM)^{0.5}##

You've opened up a can of worms in my understanding here. My professor uses this convention and I have never really questioned it.

So you are saying:

##P=(E^2 -M^2)^{0.5}##
therefore as ##E=T+M## we can rewrite this as:

##P=((T+M)^2 + M^2)^{0.5} = (T^2 +2TM)^{0.5}##

Yes. I used that equation to get the result.

Yes. I used that equation to get the result.
So, just to check, the first four momentum vector would be:

##P_1 = (T+M, 0,0, (T^2 + 2TM)^{0.5})##

So, just to check, the first four momentum vector would be:

##P_1 = (T+M, 0,0, (T^2 + 2TM)^{0.5})##

Yes.

We also know:

s = (P1 + P2)2 , t = (P1 − P3)2 and u = (P1 − P4)2

and

s + t + u = ∑ mi2 = 4m2

I've had a quick look at using this equation. I must be missing something but I don't see the trick. It looks harder to me than starting from the equation for conservation of momentum using the cosine rule. Perhaps I'm missing a trick but the term in ##P_1P_4## is a nuisance.

I've had a quick look at using this equation. I must be missing something but I don't see the trick. It looks harder to me than starting from the equation for conservation of momentum using the cosine rule. Perhaps I'm missing a trick but the term in ##P_1P_4## is a nuisance.

Currently trying to do it your way by conservation of momentum:

## (P_1 + P_2)^2 = (P_3 + P_4)^2 ##

If you want to read further into them, s,u and t are called the mandelstam variables. Yes, it's very difficult to cancel.The algebra becomes very messy.

Here is what my lecturer said when I emailed him a few days ago regarding this question:

For q1b in 2014 you first need to write down explicitly the expressions for s, t, and u in terms of m, T, T’ and theta. Then there are several options. One way is to take the t variable and, using s + t + u = 4m^2, express it via s and u in terms of their explicit expressions obtained in the first step. You then equate this expression to the explicit expression of t.

After that it’s just a tedious and longish algebra and you come up with the answer.

Here is what my lecturer said when I emailed him a few days ago regarding this question:

For q1b in 2014 you first need to write down explicitly the expressions for s, t, and u in terms of m, T, T’ and theta. Then there are several options. One way is to take the t variable and, using s + t + u = 4m^2, express it via s and u in terms of their explicit expressions obtained in the first step. You then equate this expression to the explicit expression of t.

After that it’s just a tedious and longish algebra and you come up with the answer.

Maybe my way is better! Here's a summary of the first steps. I kept it in terms of ##E_i## and ##p_i## to begin with and then used ##E = T + M## half way through.

1) Get an equation for ##E_4^2## using conservation of energy.

2) Get an equation for ##p_4^2## using conservation of momentum and the cosine rule. Substitute ##p_i = E_i - M##, which gives you another equation for ##E_4^2##.

3) Use 1) and 2) to eliminate ##E_4^2##.

(This is a fairly standard approach for collisions, so it's worth being familiar with). In this case, you should get:

##E_1E_3 - M(E_1 - E_3) - M^2 = p_1p_3 cos\theta##

• Jdraper
Maybe my way is better! Here's a summary of the first steps. I kept it in terms of ##E_i## and ##p_i## to begin with and then used ##E = T + M## half way through.

1) Get an equation for ##E_4^2## using conservation of energy.

2) Get an equation for ##p_4^2## using conservation of momentum and the cosine rule. Substitute ##p_i = E_i - M##, which gives you another equation for ##E_4^2##.

3) Use 1) and 2) to eliminate ##E_4^2##.

(This is a fairly standard approach for collisions, so it's worth being familiar with). In this case, you should get:

##E_1E_3 - M(E_1 - E_3) - M^2 = p_1p_3 cos\theta##

Thank you, think I've followed your method by eliminating ##E_4 and P_4 ## but I've become bogged down in the algebra. Will have another attempt and let you know

Thank you, think I've followed your method by eliminating ##E_4 and P_4 ## but I've become bogged down in the algebra. Will have another attempt and let you know

I substituted ##E = T + M## in the left-hand side and simplified.

Just got it solved. I used conservation of 4 momenta to derive P4. After that work out s, u, t and use s+t+u= 4m^2

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