How to Determine the Total Cross Section for an Isotropic Reaction?

[jack7992]

Homework Statement

A thin (1mg/cm^2) target of 48-Ca is bombarded with a 10-nA bean of alpha particles. A detector, subtending a solid angle of 2E-3 steradians, records 15 protons per second. If the angular distribution is isotropic, determine the total cross section, in mb, for the reaction. You can take the atomic mass of 48-Ca to be A= 48.

Homework Equations

The Attempt at a Solution

I know basically how to do this problem. I can calculate flux, number of scattering centers, etc. The only thing I am getting hung up on is the solid angle idea. From my understanding, this is basically saying that a detector with a solid angle of 2E-3 str detects 15 protons/sec., which is only a portion of the total deflected. I can get the differential cross section. My question is how do you get the total cross section. Do you just multiply by a str factor or something? Thanks a lot!

 

[Oerg]

I am not sure what cross section you are trying to find but...

an angle is the ratio of the length of the arc to the radius, similarly, the solid angle is the ratio of the area subtended to its radius squared. The area of a part of a sphere is

$$S=\int\int r^2 \sin{\theta} \, d\theta d\phi$$

, therefore, the solid angle is just

$$d\Omega= \int \int \sin{\theta} \, d\theta d\phi$$

and if you do the integration over all $$\phi$$ and all $$\theta$$ you will get $$4 \pi$$ which is the total solid angle of a sphere. Of course if you multiply this by $$r^2$$ you will get the area of a sphere.