Centre-of-mass Energy of Colliding protons

[joakley]

Homework Statement

LHC can accelerate proton beams to energies of 8Tev.
1. What is the centre-of-mass energy of two colliding protons?
2. What would be the beam energy needed if one wanted to reach the same centre-of-mass energy when colliding beam protons with protons in a target at rest?

Homework Equations

E² = (pc)²+(E₀)²

KE = E - E₀

The Attempt at a Solution

Common sense tells me that two particles of equal mass and energy colliding head on results in all the energy of the particle transferring to "collision energy". So I assume an E_tot = (8+8)TeV = 16TeV sounds reasonable unless I need to include rest mass energy, which i would have though is negligible. Does this E_tot relate simply to E_cm?

I am having trouble understanding the loss of collision energy in the rest target case, surely all energy supplied to the particle for momentum must be conserved/transferred to the rest proton and there is no center-of-mass energy? I guess my understanding of what E_cm (center-of-mass energy) may be throwing me off slightly?

 

[mfb]

LHC can accelerate proton beams to energies of 8Tev.

This is wrong, the achieved value is 4 TeV and the design value is just 7 TeV. Anyway, does not matter here.

Common sense tells me that two particles of equal mass and energy colliding head on results in all the energy of the particle transferring to "collision energy". So I assume an E_tot = (8+8)TeV = 16TeV sounds reasonable unless I need to include rest mass energy, which i would have though is negligible. Does this E_tot relate simply to E_cm?

Right.
The rest energy is already included in the 8 TeV (negligible of course...).

I am having trouble understanding the loss of collision energy in the rest target case, surely all energy supplied to the particle for momentum must be conserved/transferred to the rest proton and there is no center-of-mass energy? I guess my understanding of what E_cm (center-of-mass energy) may be throwing me off slightly?

The center-of-mass system is moving here, there is always a reference frame where both protons have an opposite velocity of equal magnitude. In the lab, you need much more energy as you have to provide the CMS energy PLUS the energy for the motion of the center of mass.