Talk:Rutherford scattering experiments/Archive 2

Archive 1

Archive 2

Maximum deflection in collision with an electron

@Johnjbarton and Headbomb: What do you think of this stuff?

In his treatment of beta particle scattering, Thomson provided the following equation for how a beta particle might be scattered by a single atomic electron:

$\tan {\frac {\theta }{2}}={\frac {kq_{\beta }q_{\beta }}{bm_{\beta }v^{2}}}$

where m_β and q_β are the mass and charge of an electron or beta particle. We will replace m_β and q_β with m_a and q_a and, in not assuming the atomic electron has infinite mass due to atomic binding, we account for conservation of momentum:

$\theta =2\arctan \left({\frac {kq_{a}q_{e}}{bm_{a}v^{2}}}\cdot {\frac {m_{e}}{m_{a}+m_{e}}}\right)$

Kurzon (talk) 18:58, 1 July 2024 (UTC)

In his 1911 paper Rutherford writes:

$$\cot {\frac {\phi }{2}}={\frac {2p}{b}}.$$

Inverting this formula and replacing Rutherford's variable for the impact parameter ($p$) with yours, $b$, while substituting for Rutherford's b:

$$b=-{\frac {NeE}{{\tfrac {1}{2}}m_{\alpha }v_{0}^{2}}}.$$

gives

$$\tan {\frac {\phi }{2}}={\frac {NeE}{bm_{\alpha }v^{2}}}.$$

This looks like your first formula, but there may be a factor of 2 for the reduced mass in electron-beta collisions:

$$\mu ={\frac {m_{\beta }m_{e}}{m_{\beta }+m_{e}}}=2m_{e}$$

For collisions between alpha and electron, $m_{\alpha }$ is replaced by:

$$\mu ={\frac {m_{\alpha }m_{e}}{m_{\alpha }+m_{e}}}\approx m_{e}$$

so your second equation seems incorrect to me. Johnjbarton (talk) 19:28, 1 July 2024 (UTC)

@Johnjbarton: So it really should be

$\theta =2\arctan \left({\frac {kq_{a}q_{e}}{bm'v^{2}}}\right)$

where $m'={\frac {m_{a}m_{e}}{m_{a}+m_{e}}}$

That's what I think it says in Heilbron's paper on page 270, but I get weird results when I punch that formula and values into Desmos. I get a scattering angle of 179° (when b = 7×10⁻¹⁵ m).

Kurzon (talk) 19:45, 1 July 2024 (UTC)

Rutherford writes his equation as a ratio of impact parameter (his $p$) to the minimum approach distance (his $b$):

$$\cot {\frac {\phi }{2}}={\frac {2p}{b}}.$$

That choice was not an accident. The ratio amounts to measuring the impact parameter in units of the minimum approach distance, so much easier to think about.

For electron + alpha, from the formula

$$r_{\text{min}}={\frac {1}{4\pi \epsilon _{0}}}\cdot {\frac {2q_{1}q_{2}}{mv^{2}}}$$

the minimum approach will be 7200 times larger for the ratio of alpha and electron mass but 79 times smaller for the charge ratio.

His minimum approach was 3.4 x 10^-14, so the new minimum is about 300 x 10^-14. If your impact parameter is 0.7 x 10^-14, the ratio is very small, and thus you get 179 degrees (see the Rutherford's table). You basically hit the bullseye and got direct backscatter.

The kinetic energy of an electron at the same velocity as an alpha particle is 7200 times less, and the potential energy due to charge difference is only 79 times less. So the electron can't get as close to the alpha particle as the alpha particle can get to the nucleus. Another way to say this is that the cross section for the electron is large. The difference in mass means that the electron recoil is huge, the alpha particle basically plows through and the electron gets blasted off. Johnjbarton (talk) 21:51, 1 July 2024 (UTC)

That kinda sounds like what I put in the article that you criticized. The electrons are so light compared to the alpha particle that they get blasted out of the way and therefore have negligible impact.

OK, so what should I go with? Kurzon (talk) 00:34, 2 July 2024 (UTC)

I suggest putting the Thomson scattering discussion in the plum pudding model article.

Use Thomson/Heilbron for beta-electron scattering. Use Beiser/hyperphysics for alpha scattering from positive sphere since Thomson evidently is silent on this subject. That directly eliminates many of my complaints on this article. Johnjbarton (talk) 01:55, 2 July 2024 (UTC)

But what about alpha scattering by the atomic electrons? Kurzon (talk) 02:09, 2 July 2024 (UTC)

Rutherford explicitly ignores this effect on the alpha particle scattering, citing Thomson's work that any single encounter results in small angle scattering. Thomson's results were for beta particles with even less momentum than alpha particles. Rutherford's assumption is ultimately justified by his success in explaining the small but not insignificant large angle scattering. This is the key to Rutherford's paper -- large angle scattering is not insignificant as assumed by Thomson -- and that is why the Geiger-Mardsen experiment is so much the focus of modern explanations.

That is core to my complaint with the use of the Thomson model in an article on Rutherford scattering. The fact that the Thomson model gives only small angle scattering is only in support of ignoring the electrons: a big deal is made of that part of the model that Rutherford completely ignores.

I did add a section to Rutherford scattering based on your question here. Johnjbarton (talk) 15:37, 2 July 2024 (UTC)

So for this article I should say "Here is a scattering of a beta particle by a single encounter with an electron. It is trivially small. Since alpha particles have thousands times more momentum, alpha particle scattering by electron collisions will be even smaller, and there is no need to go into the math for that". Kurzon (talk) 16:50, 2 July 2024 (UTC)

I suppose I should go with the conservation of momentum approach in the Beiser textbook. Kurzon (talk) 00:53, 2 July 2024 (UTC)

"integral above has three unknown variables"

In the integral above the subject phrase, dt is not an unknown. using capital R for a variable is not standard notation. the integral would be much clearer if you write the radius and angle as functions of time. the steps which follow convert to a polar coordinate form, which is where standard treatments start. Johnjbarton (talk) 18:57, 2 July 2024 (UTC)

I have made several attempts to fix the math content in the scattering sections to match the textbook reference that this derivation seems to be based on:

• Beiser (1969). Perspectives of Modern Physics, p. 109

Note that this is the ref that was used by the Hyperphysics site. However that site attempts to condense the entire derivation down to one slide. The missing parts have been filled in incorrectly.

The content is still not correct but @Kurzon keeps reverting my changes. I'm done with this. Johnjbarton (talk) 15:08, 14 July 2024 (UTC)

Ok I will take a closer look at the Beiser book. Kurzon (talk) 18:51, 14 July 2024 (UTC)

The only thing I reverted was you writing R as R(t). I don't feel it's necessary and Beiser doesn't do it. I understand it's frustrating to see your edits reverted but this is overreacting. Kurzon (talk) 19:53, 14 July 2024 (UTC)

The presentation was incorrect about exactly the integration variable. Making the functional dependence explicit is the best way to avoid this. Johnjbarton (talk) 21:34, 14 July 2024 (UTC)

$$\int \limits {\frac {kq_{a}q_{g}}{R^{2}}}\cdot \cos \varphi \cdot \mathrm {\mathrm {d} } t$$

So you're saying that unless you make it clear that R and phi are functions of t, a reader might mistakenly resolve the integral to

$${\frac {kq_{a}q_{g}}{R^{2}}}\cdot \cos \varphi \cdot t+C$$

Is that your complaint? Kurzon (talk) 22:10, 14 July 2024 (UTC)

No, I am saying that an editor may create a version like this one with limits in angles and integration in time. Johnjbarton (talk) 22:14, 14 July 2024 (UTC)

Ah, now I understand. Well spotted. Kurzon (talk) 22:29, 14 July 2024 (UTC)

OK, is it better now? Kurzon (talk) 22:40, 14 July 2024 (UTC)