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In the derivation of Rutherford's scattering formula, for example, this one here or this one here, we conclude that: $$\frac{db}{d\theta} < 0,$$ i.e., as the collision parameter increases, the scattering angle decreases.

The thing that bothers me is that the minus sign is merely dropped in what follows. In the first derivation I linked, it says (page 4):

We omit the minus sign in the following, because it has no physical meaning.

And in the second derivation, it says (page 5):

The minus sign has been dropped as it merely indicates that as b increases, the scattering angle decreases - N must be positive.

I understand the reasons... but I just feel extremely uncomfortable dropping the minus sign there, why is it acceptable, why should the "mathematically correct" way give an incorrect result in this case?

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1 Answers1

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These derivations are just being a bit sloppy; they are missing absolute values in certain locations.

For example, in the second derivation there should be an absolute value on the $db$ in equation (2.2.5) because $dN$ is the flux times the area of the ring whose thickness is $|db|$ not $db$. You can see this by considering the case $db<0$.

If you include the absolute values in the correct places, then it can be shown (Derivation of differential scattering cross-section) that the differential scattering cross-section formula becomes $$ D(\theta) = \frac{b(\theta)}{\sin\theta}\left|b'(\theta)\right| $$ where the absolute value on the right hand side takes care of the sign in the correct way.

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joshphysics
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