How Thick Should a C12 Target Be for 1% Proton Interaction at 10 GeV?

[Krog]

Homework Statement

What target thickness d is necessary to ensure that 1 % of all protons with a laboratory momentum of 10 GeV interact in a C12 – target?

Homework Equations

I have found af formula that states

cross section = N_Reac/(N_beam * N_A * ro*d/M)

Where N_Reac is the number of reactions, N_beam is the number of beam particles, N_A is Avogado's number, ro is the density of the target, d is the thickness of the target (What i am supposed to find) and M is the molar mass of the target.

The Attempt at a Solution

I am not absolutly sure that i have to use the equation stated above - any way I still have two unknown quantities: the cross reaction and d

I guess that i am supposed to calculate the cros section between the protons and the C12-target in some way using the energy - but how?

 

[mark726]

First, let's define some terms and units that we will use in our calculation:

- N_Reac: This is the number of reactions, which we will assume to be 1% of all protons with a laboratory momentum of 10 GeV. Since we are dealing with protons, we will use the unit of particles (p).
- N_beam: This is the number of beam particles, which we will also assume to be 1% of all protons with a laboratory momentum of 10 GeV. Again, we will use the unit of particles (p).
- N_A: This is Avogadro's number, which is equal to 6.022 x 10^23 particles per mole (mol).
- ro: This is the density of the target, which we will assume to be the density of C12, which is 2.26 g/cm^3. We will convert this to kilograms per cubic meter (kg/m^3) for our calculation.
- d: This is the thickness of the target that we are trying to find, which we will measure in meters (m).
- M: This is the molar mass of the target, which for C12 is 12.01 g/mol. We will convert this to kilograms per mole (kg/mol) for our calculation.

Now, let's rearrange the equation to solve for d:

d = (N_Reac * M * N_A * ro) / (N_beam * cross section)

We can see that we still have one unknown quantity: the cross section. In order to find the cross section, we will need to use the given information that 1% of all protons with a laboratory momentum of 10 GeV interact in the C12-target.

We can rewrite this as:

cross section = (N_Reac * N_beam) / (N_A * ro * d)

Now, let's plug in our known values:

cross section = (0.01 * N_beam) / (6.022 x 10^23 * 2.26 kg/m^3 * d)

We also know that the number of beam particles (N_beam) is equal to the number of reactions (N_Reac), so we can simplify this to:

cross section = (0.01 * N_Reac) / (6.022 x 10^23 * 2.26 kg/m^3 * d)

Now, we can plug