# The Relationship between coulomb collision, inverse square law and Rutherford scattering

I was reviewing the following question.

"A key result of the α particle scattering experiment of Geiger and Marsden was that the number of particles scattered into a given angle was proportional to the thickness of the target. Explain why this shows that the scattering is of single atoms or nuclei,and not due to multiple scattering off many atoms."

The answer was given as We know that N(θ > 90) is proportional to the number of target nuclei and proportional to t, the thickness of the target. If multiple scattering was occurring our path would consist of a series of small angle scatterings due to the Coulomb field of the nucleus. On average the scattering will be small. For multiple interactions, it should be distributed as a Gaussian with mean 0 and width σ ∝ t. Increasing the thickness will only increase the number of small scattering which will not generally add but will also cancel each other. As a result the angular distribution will be a Gaussian with width proportional to √t which cannot explain the Geiger and Marsden observation.

I was fairly satisfied with this answer however there was one particular aspect I didn't understand which is in bold above. Coulomb collision and the inverse square law don't seem to give any specific satisfactory answers as to WHY small angle scatterings occur? So my question is, why does the coulomb force give small angle scattering?

I believe there is a fundamental piece of mathematics I am missing here. Any direction would be much appreciated.

• The scattering law is in Wikipedia, could I suggest you copy and paste the details into your question, it's already mathjax friendly with only dollar symbols required on each side of the code en.wikipedia.org/wiki/Rutherford_scattering
– user167453
Aug 27, 2017 at 11:05
• Thank you, maybe I need to reword my question. In the Rutherford scattering article, it only addresses the coulomb scattering for a plum pudding model which I understand. But I do not understand the situation here, in this question where the Rutherford model is being used not the plum pudding model. Or is there a similar relationship here? Aug 28, 2017 at 22:35
• No, it's not you, it's me :) I self study and I read the wikipedia article far too quickly and misunderstood your question. My sincere apologies. So I have done what I normally do in these cases, upvoted your question in the hope that you will get an answer from a better source than I.
– user167453
Aug 28, 2017 at 23:06

I would add that the phrase "A key result of the $\alpha$ particle scattering experiment of Geiger and Marsden was that the number of particles scattered into a given angle was proportional to the thickness of the target" does not have to be true either, when considered in isolation. The phrase probably needs some qualification, like that in the answer ($\theta$>90), because for very small scattering angles one probably has multiple scattering.
The nucleus is 5 orders of magnitude smaller than the atom. This means that nuclei are very far apart from each other. If the alpha particle was a football, the atom would have the size of the football pitch and its nucleus would be another football in that pitch. The chance of shooting the first ball in the direction of the pitch and hitting the second ball is definitely too small. As the incident particle and the scattering center are most of the times far from each other, the interaction and thus the deflection are quite small. You can see this from formula for the deflection angle $\theta$ of the alpha particle, $$\tan(\theta/2)=\frac{q_1q_2}{4\pi\epsilon_0 mbv^2},$$ where $b$ is the impact parameter and $v$ is the velocity of the alpha particle. Using the data from wikipedia, $m= 6.64424×10^{−27} kg$, $q_1 = 2×(1.6×10^{−19}) C$, $q_2= 79×(1.6×10^{−19}) C$, $v = 2×10^{7} m/s$, it gives $$\tan(\theta/2)=\frac{1.37\cdot 10^{-14}}{b}.$$ Plugging an impact parameter of order $10^{-5}m$ and you get that $\theta$ is of order $10^{-9}rad$. Note that this is a classical result.