# I must be discomprehending the Rutherford scattering cross section.

I do not understand the interpretation of the http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html" [Broken].

To me, that equation says:
(1) for a given θ, the proportion of particles exiting at θ does NOT depend on mass or charge;
(2) if you integrated over all possible scattering angles, you would not get 1, but would get a number that depends on mass and charge.

What do these symbols really mean, in this context?

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## Answers and Replies

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
That's right - the fraction of particles inside a solid angle Ω doesn't depend on mass (actually kinetic energy) or charge, but the total number scattered does.

And you don't see why I have a problem with that?
I found another source that suggested that the scattering formula is only true for some angles (above some minimum). Is that correct? Do you know what the cutoff might e?

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
No, why would you have a problem with that? (Or, put another way, it's difficult to guess the problem - this will help understand where the difficulty is)

This formula, like most formulas, is an idealization, just like massless pulleys and massless ropes. Where it breaks down depends on how accurate an answer you need.

Ken G
Gold Member
Perhaps the problem is you are not placing enough importance on the word "fraction". If I say I spend a certain fraction of my money on food, and another fraction on entertainment, then you still have no idea how much money I spend on either. You might spend similar fractions of your money on those things-- but you might have a totally different amount of money to spend. So the amounts we spend might depend on how much we have, but the fractions might not. So it is with the scattering formula.

But clearly if you integrate the formula over all solid angles, (1) the answer is infinite, (2) the total changes with mass and charge.
Yet, presumably the total number of particles going out should equal the number going in.
I'm re-reading Vanadium's first answer, which suggests that only some of the particles get scattered at all, which might explain the problem.
Is there a source on what range of θ the rule applies to?

Ken G
Gold Member
But clearly if you integrate the formula over all solid angles, (1) the answer is infinite, (2) the total changes with mass and charge.
Apparently the formula cannot be taken seriously in the forward direction where theta is small, as it does indeed diverge in that direction. So there's no easy way to know the integrated number of alpha particles that get deflected in total, but you can still find the rate of alpha particles entering any theta bin when theta is not small. That number depends on the KE (explicitly) and the charge (implicitly) of the alpha particle.
Yet, presumably the total number of particles going out should equal the number going in.
Not necessarily, the formula does not count the alpha particles that pass right through. To know how many scatter at all, we'd need to be able to believe the formula near theta=0 enough to be able to do the integral over solid angle, but as you point out that is not the case here. Note that we are definitely in the limit where most of the alpha particles pass right through, or else the scattered fraction would not be proportional to L.
Is there a source on what range of θ the rule applies to?
I can't answer that, but you are right this is crucial to knowing the total number of alpha particles that get scattered-- the formula is highly forward scattering.