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 = ∑ m_i² = 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.

 

[PeroK]

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.

 

[Jdraper]

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.

 

[PeroK]

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 $p^2 = E^2 - m^2$

 

[PeroK]

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 $s, t, u, m$.

 

[Jdraper]

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:

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

 

[PeroK]

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.

 

[Jdraper]

Yes. I used that equation to get the result.

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

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

 

[PeroK]

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

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

Yes.

 

[PeroK]

We also know:

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

and

s + t + u = ∑ m_i² = 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.

 

[Jdraper]

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.

 

[Jdraper]

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.

 

[PeroK]

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

 

[PeroK]

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.

 

[Gref]

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