What Determines the Center of Mass Energy in Particle Collisions?

[nastassja]

Homework Statement

a. The Hermes experiment is a fixed target experiment. The HERA accelerator creates circulating beams of electrons of energy 27 GeV; these are repeatedly crashed into a target of protons which are at rest. What is the center of mass energy of the electron-proton collisions?

Use 1 GeV/c² as the mass of the proton. Treat the electrons as massless.
b.
c. (Both of these are based on the answer of the first, and I think I can figure them out if I understand part a.)

d. Consider two beams of identical particles with rest mass m; each beam has energy 4mc². What is the center of mass energy E_CM of the two beam system in the following cases:
--beams 1 and 2 are antiparallel (head on collision)
--beams 1 and 2 are directed in the +x and +y directions respectively (90[tex]\circ[/tex] collision)
--beams 1 and 2 are parallel: both momenta are along the +x direction.

Homework Equations

E_CM=[tex]\sqrt{E^{2}_{total}-(p_{total})^{2}c^{2}}[/tex]
E=pc for massless particles.

The Attempt at a Solution

I only have a couple questions, and I should be able to figure out the rest on my own.

a. For the total energy, I'm guessing we just add the energy of the protons to the energy of the electrons, but how would you determine the energy of the protons? Since their momentum is zero, should it just be E=mc²?

d. I'm completely lost on how to do the second one. The first and last seem fairly simple as long as I keep track of my signs, but I'm not sure how to express the momentum in terms of m and c.

 

[beman]

(a) E_cm = [ (E_1+E_2)^2 - ( p_1_vector + p_2_vector)^2];

I think yes. E_2 is just rest mass energy of proton.

(b) E_cm = [ E_1^2 + 2*E_1*E_2 + E_2^2 - p_1^2 - p_2^2 -2*p_1*p_2* cos (90 degree) ]^(1/2)
= [ ( E_1^2 - p_1^2) + ( E_2^2 - p_2^2) + 2* E_1 * E_2 ]^(1/2)
= [ m^2 + m^2 + 2 * (4*m) * (4*m) ]^(1/2)
= [ 34 m^2 ]^(1/2)
I have confusion about beam energy. Is this right to put E_1=E_2= 4m ? Please correct me if I am wrong.

I have another question. What is the physical meaning of center of mass energy?