# The Downward Spiral

In 1909, Ernest Rutherford (actually his grad students) shot subatomic particles at gold atoms to try to probe the insides of those atoms. To his surprise, he found that instead of being one continuous glob, atoms actually had most of their mass concentrated in a small nucleus at the center of the atom. This quickly developed into the idea that the atom was like a tiny solar system: massive heavy thing at the center, orbited by electrons. Like this, except replace the planets with electrons and erase all that messy stuff like asteroids and the Kupier belt.

Now the parts of an atom are far too light to be held together by gravity, but the electrical attraction of the negative electrons to the positive nucleus works just as well.

The model is wrong. Atoms aren’t actually like that. Now it’s not a useless model. It’s closer than older models, and even today it’s not bad as a rough schematic way of thinking about atoms (especially in the context of intro chemistry). But it’s wrong. One reason is that if it were true, we would all die instantly. No kidding. Here’s why.

Yesterday we talked about how accelerating charges radiate energy. As you know, acceleration need not involve a change in speed, it can also involve a change in direction. Anything orbiting in a circle is doing that all the time. Let me write down two expressions: one for the force exerted by the nucleus on the electron, one for the force required to make the electron move in a circle. They have to be equal, and to keep things simple let’s do the hydrogen atom where the nucleus and the electron have the same charge.

You can solve that for v, and then plug the result into the equation for kinetic energy, K = 1/2 mv2 in order to find the kinetic energy of that orbiting electron. I get:

I can also plug the velocity back into the acceleration equation above and find the actual value for the acceleration:

Holy zark that’s a pretty huge acceleration. While electrons falling under the comparitively sedate force of earth’s gravity might not radiate much, an electron with that much acceleration might radiate quite a bit. Plug into the Larmor formula to find out how much:

But we just saw above that the electron only has something like 2.18 x 10-18 J of energy in the first place. All the energy would be radiated away in a fraction of a nanosecond, and the electron will spiral down into the nucleus. Every atom in the universe would almost instantly collapse into a neutral glob of electron/proton/neutron soup.

It’s a fact that such a thing does not happen. Therefore something is wrong with our theory of the atom. It turns out that in fact electrons do not orbit like little planets. Instead, they exhibit wave-like behavior. Each possible configuration of the wave must satisfy particular properties of the Schrodinger equation, and it turns out that there’s one particular minimum energy that the electron simply can’t get below. It will radiate (in a quantum mechanical fashion) until it’s reached that energy and then it will simply stay put.

Cool, huh?

1. #1 Alex M
August 7, 2008

So are electrons actually moving around their wacky orbitals, or kind of smeared out across the orbitals, according to the probability density?

2. #2 Matt Springer
August 7, 2008

At the level of the orbitals, the particle description of electrons breaks down completely. Describing them as smeared out is much better. The probability density pretty much is the electron.

3. #3 _Arthur
August 7, 2008

4. #4 Uncle Al
August 7, 2008

A muon can replace an orbital electron. Being 206.77 times heavier it has 1/206.77 the orbital radius. In heavy elements its lowest energy orbit is inside the nucleus, density 2×10^14 g/cm^3. The common perception of stuff and its mutual interaction is a subset of reality – leptons ignore the strong force.

5. #5 IBY
August 7, 2008

That is one cool finding.

6. #6 DG
August 7, 2008

Just learned the Larmor formula in E&M II this past Spring. They start telling you your Freshman year all about accelerating charges radiating energy and the problem it poses for the planetary model of the atom, but you don’t actually get to derive the formula for another 2 or 3 years. Talk about building suspense!

7. #7 WithoutSol
August 7, 2008

Excellent read. Thank you for blogging. I am just getting into math and physics, I have not had any advanced courses yet but your postings are making them seem even more appealing. I can’t wait.

8. #8 Luke O'Dell
August 7, 2008

Love the blog. I did my PhD in physics (nuclear magnetic resonance) but am now postdoc-ing in a chemistry department, so this blog is a great way for me to stay in touch with physics. Keep it up!

9. #9 DataJack
August 7, 2008

Excellent post, and excellent blog. I am very glad to see you here on SB. Thanks for sharing.