Calculate the particle energy in a collision

[cnet]

Homework Statement

The particle J/Ψ can be produced in both proton-proton collisions and electron-positron collisions.
a) Consider a proton beam incident upon a fixed hydrogen target. Calculate the energy of the proton beam in the reaction
p1 + p2 → p + p + J/Ψ

b) Consider two counter propagating beams of electrons and positrons respectively. The two beams have identical energy. Calculate the value of the electron energy in the reaction
e+ + e− → J/Ψ

2. Equations:
(a)
E + mc² = 2E' + E_J
P = 2P' + P_J
(Proton energy before and after are denoted as E and E')
(b)
2E_e = m_Jc²
P_e+ + P_e- = 0

3. Attempt:

I applied the law of energy and momentum conservation.
(a) I tend to use E'² = (p'c)² + (mc²)² to get E in terms of P and P_J, but then I would have problem as I don't know about P_J.
(b) The positron and electron have equal and opposite momenta so the meson must only have rest mass energy m_Jc². So, I can try to solve for E_e.

Can anyone please tell if my approach is correct?

 

[mfb]

For (a), what can you say about the relative motion of the produced particles at the minimal energy? Higher energies can lead to the same process, but I guess the question asks about the minimal energy.

(b): Right (and that "solving or E_e" is very easy).

 

[cnet]

For (a), what can you say about the relative motion of the produced particles at the minimal energy? Higher energies can lead to the same process, but I guess the question asks about the minimal energy.

(b): Right (and that "solving or E_e" is very easy).

For (a), I think the question asks the minimum energy required to produce those output particles (p + p + J/Ψ).

For (b), yes, E_e is just the rest mass energy of the meson divided by 2.

 

[mfb]

For (a), I think the question asks the minimum energy required to produce those output particles (p + p + J/Ψ).

Yes, that is the typical question.
What do you know about the relative motion of the particles at this energy?

 

[cnet]

Yes, that is the typical question.
What do you know about the relative motion of the particles at this energy?

They share the momentum of the particles at the initial state, I think? By the conservation of 4 momentum:

P₁ + P₂ = Σ^N_i=1P_i

In this case, the initial state is a laboratory system, one of the protons (target) has zero momentum.

Would it be easier to solve it with 4-momenta?

So, if P₁ is the incoming proton, it has the 4-momentum P₁ = (E/c,p), and
P₂ is the target, its 4-momentum is P₂ = (mc,0), and the output particles have 4-momenta 2(E'/c,p') and (E'_J/c,p') for the two output protons and the J/psi meson respectively.

 

[mfb]

They share the momentum of the particles at the initial state, I think? By the conservation of 4 momentum:

P₁ + P₂ = Σ^N_i=1P_i

In this case, the initial state is a laboratory system, one of the protons (target) has zero momentum.

That is correct, but not what I asked.

Would it be easier to solve it with 4-momenta?

That is very similar.

Let's ask differently: Do you expect the three particles to fly apart at high speed relative to each other after the collision? If yes, why, if not, why not?

 

[cnet]

That is correct, but not what I asked.
That is very similar.

Let's ask differently: Do you expect the three particles to fly apart at high speed relative to each other after the collision? If yes, why, if not, why not?

No, it's a laboratory system. After the collision, the two output protons remain fixed in the target while the J/psi meson gets emitted.

 

[cnet]

From #7:

I now have:

That give two equations:
E/c + m_pc = 2m_pc + E_J/c
P = P_J

 

[mfb]

After the collision, the two output protons remain fixed in the target while the J/psi meson gets emitted.

That's not what will happen.

 

[cnet]

That's not what will happen.

After collision, it's more suitable to use the CoM system, in which the two colliding protons will have equal and opposite momentum, resulting zero momentum. And by conservation of momentum, the only particle which has a momentum is the J/psi meson and its momentum is the same as that of the incoming proton at the initial state.

The other possibility is that these three output particles will have the same momentum P'=1/3*P.

 

[mfb]

In the center of mass system, what is the minimal energy you can have? This translates to the minimal energy in every frame.

Spoiler

All three particles are at rest in the CoM system.

What does that mean for the relative velocity of the particles?

 

[cnet]

In the center of mass system, what is the minimal energy you can have? This translates to the minimal energy in every frame.

Spoiler

All three particles are at rest in the CoM system.

What does that mean for the relative velocity of the particles?

Rest mass energy. (But the law of momentum conservation should still apply, otherwise, all three particles would have zero momentum, and hence zero velocity.)

 

[mfb]

They have zero velocity in the center of mass frame.
In the lab frame they all have the same velocity.

That is the missing equation you need to solve the system.

 

[cnet]

They have zero velocity in the center of mass frame.
In the lab frame they all have the same velocity.

That is the missing equation you need to solve the system.

I will try to get that equation when I get to a nearby library in the next half hour or so.

Nevertheless, I think the initial states I got for the particles are correct. It's only that in the center of mass frame (final states) their total momentum should be zero.

 

[cnet]

They have zero velocity in the center of mass frame.
In the lab frame they all have the same velocity.

That is the missing equation you need to solve the system.

E/c + (m_p)c = E'/c + E'/c +(E_j)/c

Use E' = [(p'c)^2 + (m_p)^2c^4]^1/2

Take p'=0

And solve for E.

 

[mfb]

Take p'=0

That is not correct.
Your first equation for conservation of energy was in the lab frame. In the the lab frame, p' won't be zero.

You can either work in the lab frame or in the CoM frame, but keep it consistent. The latter is a bit shorter but both are possible.

 

[cnet]

That is not correct.
Your first equation for conservation of energy was in the lab frame. In the the lab frame, p' won't be zero.

You can either work in the lab frame or in the CoM frame, but keep it consistent. The latter is a bit shorter but both are possible.

If I work in the lab frame:
E/c + (m_p)c = E'/c + E'/c +(E_j)/c

p = p' + p' + p_j = 2(m_p)v + (m_j)v

as the output particles all have the same velocity.

 

[mfb]

You should use the relativistic momentum. Apart from that: right.

 

[cnet]

You should use the relativistic momentum. Apart from that: right.

But I cannot solve for the proton energy E.

E² = (pc)² + (m_pc²)²

p = ϒ(2m_pv + m_jv)

Substitute for p, I have:

E² = ϒ²v²c²(4m_p² + 4m_pm_j + m_j²) + m_p²c⁴

Because ϒ² = c²/(c²-v²)

E² = [v²c⁴/(c²-v²)]*(4m_p² + 4m_pm_j + m_j²) + m_p²c⁴

It still left me with the ‘v’ which I can't get rid of.

 

[mfb]

E² = (pc)² + (m_pc²)²

That is the energy of the incoming proton I guess? It is not the total energy.

You didn't consider the total energy after the collision in your last post.

 

[cnet]

That is the energy of the incoming proton I guess? It is not the total energy.

You didn't consider the total energy after the collision in your last post.

I used that because the question asks for the energy of the incoming proton. Or, I have to equate the total energy before and after collision. Get p in terms of all particle masses and put it back to the equation for the energy E of the incoming proton.

 

[mfb]

You'll have to use conservation of energy to find the answer.