What Beam Energy Achieves the Same Center of Mass Energy with a Fixed Target?

[venomxx]

1. I have to work out the centre of mass energy of a beam collider of e-e+ each beam having an energy of 45.6Gev, That was no problem, i used the formula below and got The centre of mass = (2E)^2 = 91.2Gev. My problem arises when i apply this to a beam hitting a fixed target. The question states " What energy beam is required to get the same centre of mass"

The same formula should apply and the solution iv attempted is below. I end up having to know the rest mass of the target which i wasnt given. I am working in natural units where c=1



2. Two beams ( E, Px, 0, 0 ) (E, -Px, 0, 0)
Beam and target ( E, Px, 0, 0 ) (Mt, 0, 0, 0)

Formula used -> Ecm^2 = (∑E)^2 - (∑p)^2



3. Two beams

Ecm^2 = (2E)^2 = (91.2)^2 Therefore Ecm = 91.2Gev

Fixed targert

Ecm = (E + Mt)^2 - P^2 = 2Mt(E+Mt)

Therefore = Ecm^2 = 2Mt(E+Mt)


Any help would be appriciated, cheers.

 

[anathema]

Hello there,

It seems like you are on the right track with your calculations. In order to find the energy of the beam required to get the same center of mass energy as the beam collider, you need to equate the two formulas you have used:

(2E)^2 = (∑E)^2 - (∑p)^2

and

Ecm = (E + Mt)^2 - P^2 = 2Mt(E+Mt)

This will give you a quadratic equation with two unknowns, E and Mt. You can solve for either one of them and then plug in the values to find the other. Keep in mind that the mass of the target (Mt) is not given, so you will need to use a variable for that and solve for it.

I hope this helps. Let me know if you have any other questions. Good luck with your calculations!

 

[kerimek]

I can understand your confusion and I am happy to provide some clarification on this matter. The concept of centre of mass energy is important in particle physics and it is crucial to understand it correctly in order to make accurate predictions and calculations.

First, let's review the formula you used for calculating the centre of mass energy in a collider:

Ecm^2 = (∑E)^2 - (∑p)^2

This formula is correct for two colliding beams with equal energies (E) and opposite momenta (p). However, when a beam collides with a fixed target, the situation is different. In this case, the target is not moving and therefore does not have any momentum. So the formula for calculating the centre of mass energy becomes:

Ecm^2 = (E + Mt)^2 - p^2

Where Mt is the rest mass of the target and p is the momentum of the beam. This formula is derived from the conservation of energy and momentum laws.

Now, let's apply this formula to your problem. You correctly stated that in order to get the same centre of mass energy as in the collider (91.2 GeV), the beam energy (E) and the target rest mass (Mt) must satisfy the equation:

Ecm = (E + Mt)^2 - p^2 = 91.2 GeV

However, as you mentioned, you do not have the value of the rest mass of the target (Mt). In this case, you can use the fact that in natural units, the momentum (p) is equal to the energy (E). So the equation becomes:

Ecm = (E + Mt)^2 - E^2 = 91.2 GeV

This can be simplified to:

Mt = 0.5Ecm - E

Therefore, in order to get the same centre of mass energy as in the collider, the beam energy (E) and the target rest mass (Mt) must satisfy the equation:

Mt = 0.5(91.2 GeV) - E = 45.6 GeV - E

In other words, the rest mass of the target must be half of the centre of mass energy minus the beam energy.

I hope this explanation helps you understand the concept of centre of mass energy better and how it applies to a beam colliding with a fixed target. Keep up the good work in your research!