Solid angles and Cross sections

In summary, to determine the total cross section for the reaction, you can use the solid angle formula and integrate over all angles to get 4π. This will give you the area of a sphere. From there, you can use the flux and number of scattering centers to calculate the total cross section in mb.
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jack7992
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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!
 
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  • #2
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

[tex]S=\int\int r^2 \sin{\theta} \, d\theta d\phi [/tex]

, therefore, the solid angle is just

[tex] d\Omega= \int \int \sin{\theta} \, d\theta d\phi [/tex]

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

1. What is a solid angle?

A solid angle is a three-dimensional concept that measures the amount of space covered by a cone-shaped section of a sphere. It is measured in steradians (sr).

2. How is solid angle different from regular angle?

Regular angle is a two-dimensional concept that measures the amount of rotation between two intersecting lines. Solid angle, on the other hand, is a three-dimensional concept that measures the amount of space covered by a cone-shaped section of a sphere.

3. How is solid angle calculated?

Solid angle is calculated by dividing the surface area of a spherical cap or cone by the square of the radius of the sphere.

4. What is a cross section?

A cross section is a 2D representation of a 3D object obtained by slicing through the object at a specific plane.

5. How is cross section related to solid angle?

A cross section can be used to calculate the solid angle of an object. The solid angle is equal to the surface area of the cross section divided by the square of the distance from the center of the object to the cross section plane.

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