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)

Make things easy

@Johnjbarton: I don't want to offend you but your way of explaining things is hard to understand. I went to pains to lay out all the steps to make things easy to understand for a high school student. We don't have to be faithful to Beiser as long as we produce something that is correct. Kurzon (talk) 08:10, 18 July 2024 (UTC)

I also do not want to offend, but your version was incorrect and also not easier to understand. Consequently we do need to be faithful to Beiser unless we can agree. My suggestion is that you restore my Beiser based version and then let's discuss what parts you think are difficult to follow and find better ways to explain them. Johnjbarton (talk) 15:25, 18 July 2024 (UTC)

Formatting of math blocks.

Unfortunately math formatting has issues. As far as I understand it, the best compromise for web and mobile is to use

• <math display="block">...</math>

This adds the correct space above and below the math if placed in a paragraph. If extra blank lines are added, extra vertical space appears in the article. I assume that the extra blank lines are to make the math stand out in the editor? Maybe a format like

• <math display="block">
• ...
• </math>

with no extra lines would be useful? Johnjbarton (talk) 22:12, 18 July 2024 (UTC)

NeE

@Johnjbarton: I'm confused about Rutherford use of NeE whereas I used kqQ, with k being the Coulomb constant and the charges expressed in Coulombs. How would you rewrite Rutherford's equation to use modern conventional variables? Kurzon (talk) 18:13, 19 July 2024 (UTC)

I added a paragraph to address this, please take a look.

Unfortunately "modern conventional" for electrostatics depends on where you look and what is "modern". For a long time cgs held the field, then MKS. The SI system was changed as recently as 2019. And these are mostly application or engineering-focused works. Physics theory usually adopts natural units. Rutherford's use of the variable 'b' is similar: by expressing lengths in units of 'b', the formula are simpler and the units only come in one time. Johnjbarton (talk) 21:47, 19 July 2024 (UTC)

If we take e = 4.65×10⁻¹⁰ esu, then NeE with a gold atom is 3.41×10⁻¹⁷. But kqQ, where q and Q are in Coulombs, is 1.82×10⁻²⁶. I don't understand. Kurzon (talk) 22:31, 19 July 2024 (UTC)

I guess that 3.4×10⁻¹⁷ will be in dyne-cm², in CGS units. 1 dyne is 1×10⁻⁵ Newtons (per wikipedia anyway) and 1 m is 1×10² cm so I get 3.4×10⁻²⁶N*m²

For kqQ, where q and Q are in Coulombs, 8.987×10⁹ N·m²/C² * 1.26×10⁻¹⁷ C * 3.20×10⁻¹⁹ C so about 3.7×10⁻²⁶N*m² (numbers copied from article)

Did I mention how great it is to have units only come in one time ;-) Johnjbarton (talk) 23:19, 19 July 2024 (UTC)

Units.

@Kurzon I deliberately used Rutherford's formulas as presented in his paper to ensure verifiability. As we discussed in other topics here, the units for electromagnetism are not standardized universally; adopting any one convention makes understanding the sources harder. I'm not against changing the formulas to one consistent approach if we can do it in a way that addresses this concern.

Things that I think would help address this concern would include:

• explicit discussion of units and their appearance in historical work.
• footnotes on each conversion (I generally disagree with footnotes but this is one case where I think they make sense.
• a specific reliable reference or references as the standard agreed, called out.
• limited use of specific values to avoid clutter.
• consistency throughout.

Most modern physicists use natural units because all the extra k's and $1/4\pi \epsilon _{0}$ stuff is not physics. But I understand that textbooks are fascinated with units so I'm ok with picking one. Johnjbarton (talk) 15:49, 20 July 2024 (UTC)

Possible references:

• Kibble 5th ed uses SI units, $F={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}r_{12}^{2}}}$ (not k)
  • Kibble, T. W. B.; Berkshire, F. H. (2004). Classical mechanics (5 ed.). London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub. ISBN 978-1-86094-424-6. OCLC 54415965. Page 8.
• Hand and Finch use natural units.
  • "It is common practice in physics to chose units to simplify the formula..." page 85.
  • Hand, L. N., & Finch, J. D. (1998). Analytical Mechanics. Cambridge: Cambridge University Press.
• Goldstein 3rd edition uses cgs $F={\frac {ZZ'e^{2}}{r^{2}}}$ Page 109
  • Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. ISBN 978-0-201-65702-9.

By the way, the use of k is common force and potential problems as meaning "whatever constants". So it's not a standard notation for Coulomb's constant AFAIK. Johnjbarton (talk) 16:38, 20 July 2024 (UTC)

I proposed to use Kibble as the reference for units. Two options however:

1. Put $\left({\frac {q_{Au}q_{\alpha }}{4\pi \epsilon _{0}}}\right)$ in front of most equations or
2. define $F={\frac {k}{r^{2}}},\ k=\left({\frac {q_{Au}q_{\alpha }}{4\pi \epsilon _{0}}}\right)$ and use k everywhere.

Johnjbarton (talk) 22:07, 20 July 2024 (UTC)

Legacy + Reception.

An edit by @22merlin made me realize that we did not properly integrate the Legacy section during the merge. I think we want the Summary to have a wrap up eg Legacy, but the Summary now needs to include the scattering topic a bit more. So I will move some of the Reception content towards Legacy rather than the other way around. Johnjbarton (talk) 21:07, 20 July 2024 (UTC)

Ok I think this is mostly fixed up. The Reception section is merged into Legacy. The Legacy section needs a few more references. Also

• The astronomer Arthur Eddington called Rutherford's discovery the most important scientific achievement since Democritus proposed the atom ages earlier.

is unclear: Which achievement? Johnjbarton (talk) 21:59, 20 July 2024 (UTC)

Rutherford's scattering experiments with light nuclei.

Around 1917 Rutherford did more more experiments with alpha particle scattering, ultimately leading to the discovery of the proton. I guess these are covered in Proton but we need to mention this aspect in Legacy. A good ref is already used in the article:

• Barrette, J. (2021). Nucleus-nucleus scattering and the Rutherford experiment. Journal of the Royal Society of New Zealand, 51(3–4), 434–443. https://doi-org.wikipedialibrary.idm.oclc.org/10.1080/03036758.2021.1962368 Johnjbarton (talk) 22:48, 20 July 2024 (UTC)

Cloud chamber image.

The cloud chamber image of Rutherford scattering was removed in an edit summarized: This image screws up the layout on desktop Instead of removing something like this for a minor reason, please post the issue and we can find a way to fix it.

In general wikipedia pages do not render perfectly. Removing content to fix layout will on be a temporary fix and it won't impact most readers.

I added the image back with more concise text. The rendering is fine for the moment. Johnjbarton (talk) 00:53, 24 July 2024 (UTC)

Is that cloud chamber image even useful? Does it tell us anything the diagram above it can't? Kurzon (talk) 11:52, 24 July 2024 (UTC)

Yes, the image does three things:

1. Historically significant independent experimental evidence
2. Direct visual experimental demonstration of scattering angle
3. Introduction of alternative particle physics arenas beyond gold foil scattering.

I thought about also adding an image related to particle accelerators. The primary legacy of Rutherford's scattering experiments was in the physics of scattering. Johnjbarton (talk) 16:47, 24 July 2024 (UTC)

If you can pad out the Legacy section a bit, that would solve the layout issues. Kurzon (talk) 17:56, 24 July 2024 (UTC)

The Geiger Marsden section in Summary

The section "Rutherford, Geiger, and Marsden" is, as I understand it, intended to explain the the gold-foil backscattering result of 1909. Due to recent changes it no longer does this. It ends with a comment about uranium sources. Johnjbarton (talk) 16:41, 24 July 2024 (UTC)

Johnjbarton's writeup of Rutherford's math

@Johnjbarton: I'm confused. In a diagram you use $\varphi$ as one of the polar co-ordinates of the alpha particle but later on you use $\phi$ for the scaterring angle. Your diagram also uses the Greek letter rho for the impact parameter, not p. Double-check Rutherford's paper. Kurzon (talk) 13:02, 27 July 2024 (UTC)

Rutherford uses $\phi$ for the "angle of deviation". He never uses polar coordinates for the particle because he jumps directly to "the eccentricity is $\sec \theta$". I used another source as cited to try to explain this jump, and that source used polar coordinates with $\varphi$. To further distinguish these I used $\varphi '$.

To me Rutherford's diagram looked like a rho but once I went through the math detail I realized it was p. I'll fix that. Johnjbarton (talk) 03:26, 28 July 2024 (UTC)

Oh Christ, this terrible and I'm fed up. The notation you use is inconsistent with the rest of the article and inconsistent with modern convention. I suggest you translate all your equations from Rutherford's notation to modern notation. We don't have to be that faithful to Rutherford's paper as long as we get the physics right. Use $\theta$ for the scattering angle and $\varphi$ for the angle between $r$ and $r_{A}$ and b for the impact parameter. Fix this yourself or I'll do it myself and you might not like how I butcher things. Kurzon (talk) 11:07, 28 July 2024 (UTC)

I changed the URLs for Rutherford's 1911 paper to a PDF scan of the original. The re-typed versions we previously used have some copy errors. Kurzon (talk) 17:03, 27 July 2024 (UTC)

Thanks, good move. Johnjbarton (talk) 03:11, 28 July 2024 (UTC)

Newtonian model for scattering from Thomson atomic model.

Once again I removed the Newtonian model for scattering from the Thomson atomic model.

1. The content appears in the article on Thomson's model, Plum pudding model.
2. Historically the case against Thomson was not made using these approaches.
3. Any discussion of a Newtonian mechanics approach to scattering is in my opinion bad physics. Force based models have not been used in atomic physics since before the time of Rutherford; energy based models are used. Force based methods are more complex (because the energy approaches already include the integrals) and do not work in quantum problems.
4. The description is lengthy and detracts from the topic of the article, which is scattering from Rutherford's model.

I think the material works well in Plum pudding model because Thomson did not provide a detailed account of scattering from the positive sphere. In addition, the source for this material, Beiser, presents it as Thomson-model scattering. A short WP:SUMMARY section could be included or we could expand the discussion of the historic case against Thomson's model.

I was originally against any presentation of a force based scattering. I did not oppose the presentation of the force-based model of Rutherford scattering as a compromise, but now I wonder if that was a good choice. Johnjbarton (talk) 03:59, 28 July 2024 (UTC)

I am fine with deleting that stuff once we fix the notation issues in your stuff. Kurzon (talk) 16:45, 28 July 2024 (UTC)

Request for comment: presenting Rutherford's 1911 paper in Rutherford's way.

As of this version the section Atomic model in Rutherford's crucial 1911 paper presented the 1911 paper using Rutherford's notation. @Kurzon has changed the formulas to arbitrary and confusing notations. I proposed to revert these changes and leave the original notation. I am asking for the input of other editors so we don't go back and forth on this issue. Johnjbarton (talk) 22:29, 28 July 2024 (UTC)

Use Rutherford 1) Any notation is arbitrary, 2) This paper is a classic, 3) Using Rutherford's notation allows a simple narrative, 4) Using Rutherford's notation makes verification straightforward. Johnjbarton (talk) 00:57, 29 July 2024 (UTC)

How many readers are going to even read the original paper? If they make that effort, it shouldn't be too hard for them to translate the notation. Kurzon (talk) 06:29, 29 July 2024 (UTC)

 Done I concede the issue. Johnjbarton (talk) 22:16, 30 July 2024 (UTC)

@Headbomb: Could you comment on this? Rutherford used p for the impact parameter, but p is more often used for momentum these days. Kurzon (talk) 11:54, 29 July 2024 (UTC)

I don't get this insistence on following Rutherford's paper. I find the article has become an overall downgrade from what it used to be just a month or so ago. And I find impact parameter b to be much clearer (and modern) because it can't be confused with momentum. Headbomb {t · c · p · b} 12:09, 29 July 2024 (UTC)

Although I respect Johnjbarton's expertise, his sense of presentation leaves something to be desired. Kurzon (talk) 12:55, 29 July 2024 (UTC)

Per the Wikipedia code of conduct WP:CIVIL keep your personal comments to yourself and discuss the article content here. Johnjbarton (talk) 15:51, 29 July 2024 (UTC)

There is nothing incivil about Kurzon's comments. Headbomb {t · c · p · b} 16:05, 29 July 2024 (UTC)

I disagree. I have ignored his insults in the past but seems to have been a mistake. There is no reason to post an insulting personal opinion about my abilities. Johnjbarton (talk) 16:50, 29 July 2024 (UTC)

You're a little sensitive, I meant no offense. Kurzon (talk) 20:18, 29 July 2024 (UTC)

Rather than more unfounded claims about my character, the traditional response is an apology. Johnjbarton (talk) 21:30, 29 July 2024 (UTC)

Alright I'm sorry. Kurzon (talk) 03:05, 30 July 2024 (UTC)

Thanks. Johnjbarton (talk) 16:02, 30 July 2024 (UTC)

A month ago the article was factually incorrect and based on WP:OR. If you have specific concerns, please be specific. Johnjbarton (talk) 15:53, 29 July 2024 (UTC)

I've changed p to b, swapped phi and theta to be consistent with other diagrams in this article and others, and made a few more changes. Tell me what still remains incorrect. Once we've sorted this all, I'll be happy to delete the stuff I adapted from Hyperphysics. Your historically-relevant stuff is better (it also gave me a reason to brush up on hyperbolic geometry, thanks for that). Kurzon (talk) 15:05, 30 July 2024 (UTC)

Using b for impact parameter is common in my experience, and using p for momentum is of course a widespread convention. Physicists are also more inclined to use $\theta$ for the angle from the axis, and $\phi$ for the azimuthal angle (as noted in our Spherical coordinate system article). I think it makes sense to follow the modern conventions; anyone taking the time to look up Rutherford's original paper will be competent enough to make a few letter substitutions. XOR'easter (talk) 21:46, 30 July 2024 (UTC)

r min

@Johnjbarton: In one part of the article, $r_{\text{min}}$ is defined as

$r_{\text{min}}={\frac {kq_{a}q_{g}}{{\tfrac {1}{2}}mv_{0}^{2}}}$

but in another part it's

$r_{\text{min}}=-{\frac {kq_{a}q_{g}}{{\tfrac {1}{2}}mv_{0}^{2}}}$

Which is it? Kurzon (talk) 16:25, 31 July 2024 (UTC)

Well r_min can't be negative. I removed the minus. But this copyedit fail means we may have a sign error in the other equations. Johnjbarton (talk) 16:31, 31 July 2024 (UTC)

Did RUTHERFORD make mistake? Look, I'll retype what he put in his own paper:

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}-{\frac {NeE}{SA}}$

$v^{2}=V^{2}(1-{\frac {b}{SA}})$

and on a following page $b={\frac {2NeE}{mu^{2}}}$

That's his stuff. Now I'm going to do my own rearranging:

${\frac {1}{2}}mV^{2}+{\frac {NeE}{SA}}={\frac {1}{2}}mv^{2}$

$V^{2}+{\frac {NeE}{{\frac {1}{2}}mSA}}=v^{2}$

Help me I'm going mad. Kurzon (talk) 17:06, 31 July 2024 (UTC)

@Johnjbarton, Headbomb, and XOR'easter: Did Rutherford make a mistake in his conservation of energy equation? This is what he wrote in his 1911 paper:

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}-{\frac {NeE}{SA}}$

V is the velocity of the alpha particle at the start and v is it's velocity at the point of closest approach A. I figure that as the alpha particle approaches A, it should LOSE energy, not gain it as the equation suggests. Should it instead be like this?

${\frac {1}{2}}mV^{2}={\frac {1}{2}}mv^{2}+{\frac {NeE}{SA}}$

Kurzon (talk) 19:37, 31 July 2024 (UTC)

I agree with your reasoning if we assume $NeE>0$ and SA positive. Rutherford did not now the charge on the nucleus and the value of SA could depend on the branch of the hyperbola. Earlier in the paper he has head-on energy balance (where SA is b)

$${\frac {1}{2}}mu^{2}=NeE\cdot \left({\frac {1}{b}}-{\frac {3}{2R}}+{\frac {b^{2}}{2R^{3}}}\right)$$

If we neglect the 2nd and 3rd terms as he does, then the potential energy term is positive:

$${\frac {1}{2}}mu^{2}={\frac {NeE}{b}}$$

BTW we should either use $u$ (initial) in our form of this equation or $V$ (approaching nucleus from afar) but not $v$ (apse velocity) as currently set. Johnjbarton (talk) 19:53, 31 July 2024 (UTC)

OK, I chose to change the energy equation rather than redefine r_min. Kurzon (talk) 20:19, 31 July 2024 (UTC)

I think this change is fine, but Rutherford was not mistaken. If you use the other sign and follow through the derivation you get an equation relating $b/r_{\text{min}}$ to $-\theta$, which is just the other half of the angles in the table or the bottom of the diagram. Both sets of angles are observed in the experiment: the two results are indistinguishable. This is what Rutherford meant when he said "The deductions of the experiment so far considered are independent of the sign of the central charge..." Johnjbarton (talk) 23:51, 31 July 2024 (UTC)

The eccentricity of a hyperbola can be calculated

Two or three times I have tried to fix this. Each time @Kurzon removes it.

We have a paragraph that starts

• The eccentricity of a hyperbola can be calculated ...

My immediate reaction is "wait what"? What is this "eccentricity" thing? Where did it come from? How is this all related to the problem?

My solution was simply to start the paragraph with a sentence about eccentricity, hyperbola, and the geometry of the problem:

---

Any hyperbola can be written in polar coordinates $(r,\varphi )$ with the origin at the center as

$$r={\frac {1}{\sqrt {e^{2}\cos ^{2}\varphi -1}}}.\,$$

where $e$ is the eccentricity. Comparing Fig. 1 to the geometry of a hyperbola shows that SO is the focal distance and OA is the length of the semi-major axis. The eccentricity, e, is the focal distance divided by the length of the semi-major axis or ⁠SO/OA⁠.

---

I think something like this is essential. Johnjbarton (talk) 22:44, 31 July 2024 (UTC)

But you don't use that formula for anything here, it serves no purpose. All we need is ⁠SO/OA⁠ = sec Phi. Kurzon (talk) 01:47, 1 August 2024 (UTC)

A historical narrative for why the scattering result surprised Rutherford

@Johnjbarton: You rejected the stuff I previously wrote for Why the Thomson model was wrong. Let's talk about a more historically accurate one, then.

I looked through the 1968 paper by Heilbron you shared with me and on page 270 he addresses beta particle scattering in the Thomson model, offering two equations:

$\tan(\varphi /2)=(e_{1}e_{2})/(m'v^{2}p)$

and

$\sin ^{2}(\varphi /2)=(1+p^{2}m'^{2}v^{4})/e^{4})^{-1}$

The first one is pretty much what we produced in the article, in your write-up of Rutherford's equations and my write-up of the Hyperphysics/Beiser stuff. The second one confuses me. I'm not sure where to go from here. I found a 1906 paper by Thomson which Heilbron says is important.

https://gilles.montambaux.com/files/histoire-physique/Thomson-1906.pdf

Kurzon (talk) 10:37, 1 August 2024 (UTC)

@Kurzon I'm confused by your questions.

First "why the scattering result surprised Rutherford". Rutherford worked on scattering for probably 30 years so I'm unsure what result you have in mind, but I suppose it relates to the invented-history around the often repeated cannon ball quote and Geiger-Mardsen's results.

The 1909 Geiger-Mardsen result obviously did not surprise Rutherford: he proposed the experiment. Most experiments confirm existing ideas and so it was in this case per Heilbron and other references. However, the cannonball comment (many years after the fact) does make sense of the events between 1906 and 1909, during which time Rutherford and Geiger did many scattering experiments under the influence of Thomson's atom model. Their results did not make sense given the model. The results in around 1908 were to demonstrate that alpha particles scattered at all: that was surprising given Thomson's model. Hence the title of Geiger paper "On the Scattering of α-Particles by Matter". So yes Rutherford was overall "surprised" by the scattering results, but not by the Geiger-Mardsen result per se.

As discussed in Plum pudding model, Thomson's scattering model had three parts: 1) electron-electron Coulomb scattering (your formula above) 2) positive sphere (largely ignored by Thomson), 3) multiple scattering. The third one was essential. Let me repeat it: the multiple scattering in Thomson model was essential. Without multiple scattering Geiger's 1908 results invalidate Thomson's model. Crowther's work was exactly on this issue. Geiger's thickness dependence experiments aimed at testing the multiple scattering. That is why Rutherford's 1911 paper focuses on multiple scattering as discussed in the section "Comparison to JJ Thomson's results". He invalidates multiple scattering most of all.

But this kind of invalidation in 1911 is very weak. The experiments are crude. The sources are diffuse and poorly understood. They don't know what alpha and beta particles really are. They don't know what atoms or solids really are. One bit of new evidence against the only atomic model existing does not instantly elevate an alternative.

Most especially Thomson had established that his model had potential to explain chemistry via arrangement of the electrons. In comparison, Rutherford's model explained some experiments Geiger did which no one else performed and few understood. Rutherford's model had no electrons and the only existing model for electrons with a compact positive core was the already dismissed Saturnian model. So the 1911 paper was really only of interest to physicists like Wilson interested in scattering.

Thomson's model was not "wrong" until Bohr added new the electron model to Rutherford's nucleus. During this time it was not like Thomson's model was widely used. The very existence of atoms was still being debated.

The scattering formula in Thomson's "On the number of corpuscles in an atom" is a minor step in his analysis of absorption. Historically absorption, related to Geiger's thickness experiments, was critical in understanding the physics. However modern texts make more of Coulomb scattering because it is simpler. I don't know what trig relations connect the tangent and sin forms in Heilbron's paper on 270, but as far as I can make out Heilbron is just trying to explain where the sin formula in Thomson's paper could have come from. I don't think it is otherwise significant. Johnjbarton (talk) 17:21, 1 August 2024 (UTC)

I think I'm getting what you say. On the plum pudding article, and in Beiser, its says that the odds of a particle being deflected by more than 90 degrees after 10,000 collisions is negligible. But what about ten million collisions? If an atom contains many thousands of electrons instead of just a hundred or fewer, then the plum pudding model could produce strong deflection of alpha particles. Kurzon (talk) 16:07, 2 August 2024 (UTC)

Yes, that is one example of the many poorly known issues at the time. Geiger spent a lot of time on thickness dependence experiments for this reason. By doubling the thickness you double the number of collisions and can create a graph. Crowther had done this for beta scattering and used it to support Thomson, but his data was inaccurate for a critical range, affecting the conclusion (this from Heilbron) around p279. Johnjbarton (talk) 17:19, 2 August 2024 (UTC)

OK if I wanted to rewrite the stuff in the plum pudding model article to make it more historically accurate, what should I do? Kurzon (talk) 18:06, 2 August 2024 (UTC)

I think the plum pudding model is pretty good now. We could add a paragraph on how the beta scattering results only apparently supported Thomson's model. Beta scattering was much less sensitive to the effects Rutherford saw with alpha particles, beta scattering turns out to be quite complex, and Crowther's experiments were inaccurate in just the range that showed important effects. If you think this would be valuable I can find the refs.

Heilbron spends much of his 1968 article discussing how Rutherford had to address Crowther's results after 1906. That is, the dramatic reduction in the number of electrons discovered by Thomson did not eliminate the multiple scattering concept. They did not have good data on scattering by angle as implied by our scattering model presentations. They had a narrow cone of angles and variation with material and thickness. Within this range of data Thomson's model works. It was only the large angle scattering and careful work with thicknesses done by Geiger that showed the model fails. The quality data for this came out in Geiger's 1913 paper. Johnjbarton (talk) 16:19, 3 August 2024 (UTC)

The first three paragraphs of the Heilbron 1968 paper summarize the role of multiple scattering. Johnjbarton (talk) 18:18, 3 August 2024 (UTC)

Non-scientific use of significant figures.

The section on the impulse model uses an inappropriate number of significant figures creating the impression that 1) the model is somehow amazing accurate and 2) that the mass of the atom is significant. Neither of these are true. Rutherford's own paper only uses 2 figures.

The current text directly contradicts physics: it attempts to show that the mass effect of gold is significant, but it was not historically and the impulse model assumes infinite mass for the target.

To account for finite mass, the reduced mass can be used (with a reference) and if Aluminum is used for the example a significant effect will be shown, as was noted by Rutherford on page 385. Johnjbarton (talk) 14:28, 1 June 2024 (UTC)

I am not even sure what calculations Rutherford used. I think it's enough that it be consistent with the history, not a perfect reflection. Kurzon (talk) 14:50, 1 June 2024 (UTC)

Rutherford's calculation are in his paper, and in particular he considers the effect of finite mass on page 384 in the section "Alteration of velocity in an atomic encounter". The content is not consistent with history, that is exactly my point. Johnjbarton (talk) 15:02, 1 June 2024 (UTC)

My changes to the article to correct this error have been reverted by @Kurzon. I want to know why.

A scientific model typically start with conceptual ideas that approximate a physical scenario. The ideas generate a mathematical model which leads to calculations. The results of the calculation, being numerical can be exact as far as mathematicians are concerned. But this article is about physics and the model is an approximation to an immensely complex physic system. It is inappropriate and incorrect to write three significant figures for a result that is this crude. The model prediction is about 0.02 degrees, not 0.0186 degrees. Johnjbarton (talk) 18:09, 18 June 2024 (UTC)

@Kurzon Once again I had to fix the article for this error. Please stop. Johnjbarton (talk) 22:17, 5 August 2024 (UTC)

@Johnjbarton: I looked through some history books and it seems that while Perrin and Rutherford knew each other they didn't collaborate much. Still, I'd like to know how Rutherford knew that atomic radii were on the order of 10^-10. It's a pity that you had to delete that information because I think it's interesting for readers to know where scientists got these measurements and how, though perhaps you are correct that they belong in another article, perhaps History of atomic theory. Kurzon (talk) 12:25, 2 June 2024 (UTC)

@Kurzon Just want to point out that this reply belongs on the topic "Historical measurements of the variables".

You have reverted may change to the significant figures without discussion. I continue to disagree with the content. Johnjbarton (talk) 14:01, 2 June 2024 (UTC)

Considering I'm working with an incomplete picture of what happened, I think it's OK for me to fill in a few gaps in the mathematics with educated guesses, so long as I get the maths and the physics right. Kurzon (talk) 16:19, 2 June 2024 (UTC)

I disagree, it is not OK. Your "fill in" is incorrect. Using an approximate model, adjusting it for an insignificant factor, and then using lots of significant figures to make it look significant is not correct.

And it so unnecessary! The recoil effect was discussed by Rutherford using Aluminum. Just put my text back and change your formula to use Al mass in the second case.

If your picture is incomplete, then omit it. Don't add stuff that is wrong. Johnjbarton (talk) 00:26, 3 June 2024 (UTC)

Ok Kurzon (talk) 00:28, 3 June 2024 (UTC)

Explicit interjection of link to (trivial) trig page

Several times @Kurzon adds something like:

• This next part makes several references to Wikipedia's list of trigonometric identities.

Why? This is among the most trivial bits of related material, not a major issue that needs to be called out. Each and every use of a trig identity can be link inline in the text. This is out of character for the article or article in general. I will keep removing this. Johnjbarton (talk) 17:49, 1 August 2024 (UTC)

I think it's useful to students to have this resource pointed out. You take this knowledge for granted. Kurzon (talk) 18:47, 1 August 2024 (UTC)

First this is not a textbook. Second an injected explicit link like this is lazy and not helpful to any readers, especially students. There are hundreds of formula on that page.

Any step in the article which is not clear to a typical reader should be clarified. An explicit link followed later by a series of obscure steps does not clarify.

To this point, you removed my description of secant which, in my opinion, is obscure for readers. Most readers will understand sin, cos, tan, maybe cot, but the rest are rarely used. My original version had links for steps. When you add new trig steps, add the appropriate links. An explicit statement that "reader, you need to learn a hundred or so trig identities to follow this" is not helpful. Johnjbarton (talk) 19:01, 1 August 2024 (UTC)

In fact I can't figure out the section that discusses the OA SA $\Phi$ geometry now. Johnjbarton (talk) 19:14, 1 August 2024 (UTC)

Writing this article (and I wrote most of it) was a learning process for me, I didn't take physics in college. I think that makes me well equipped to empathize with the laymen. I remember the parts where I stumbled. So maybe I should write the explanation and you should check it to see if I got the physics right. I wasn't aware that Wikipedia even had a list of trigonometric identities until I started studying Rutherford's work, and that was one of many stumbling blocks I had to overcome. I want to make that block easier for those who come after me. Kurzon (talk) 20:08, 1 August 2024 (UTC)

I don't see the connection. The explicit link to the overall page does not aid in understanding. If you don't know that trig identities exist, the explicit link sentence is a completely mysterious interjection. If you do know the exist, the list of 100 does not help.

The links like cotangent double angle formula lead you directly to the appropriate section.

I don't understand why you would be against that. And the OA SA geometry would be much clearer if we use sin/cos/tan and triangle sides. People who take trig learn the "opposite over hypotenuse" rules but not the reciprocal functions like secant/cosecant. Johnjbarton (talk) 21:10, 1 August 2024 (UTC)

What do you think of it now? Kurzon (talk) 16:43, 7 August 2024 (UTC)

I'm unsure which aspect you are asking about. The wikilinks for trig are fine now. I think the derivation of the SO SA bits could be a bit clearer with a dedicated diagram showing the right triangles, but I don't think this is essential.

I think overall the article is in excellent condition. Perhaps neither one of us is happy with the process but we can be proud of the net result. Johnjbarton (talk) 16:56, 7 August 2024 (UTC)

Oh come on, don't you have an appetite for argument? Kurzon (talk) 19:55, 7 August 2024 (UTC)

Feedback on current state

@Headbomb and Materialscientist: Johnjbarton is satisfied with this article, how about you guys? Kurzon (talk) 08:13, 11 August 2024 (UTC)