Atoms are all about a tenth of a billionth of a meter wide (give or take a factor of 2). What determines an atom’s size? This was on the minds of scientists at the turn of the 20th century. The particle called the “electron” had been discovered, but the rest of an atom was a mystery. Today we’ll look at how scientists realized that **quantum physics**, an idea which was still very new, plays a central role. (They did this using one of their favorite strategies: “dimensional analysis”, which I described in a recent post.)

Since atoms are electrically neutral, the small and negatively charged electrons in an atom had to be accompanied by something with the same amount of positive charge — what we now call “the nucleus”. Among many imagined visions for what atoms might be like was the 1904 model of J.J. Thompson, in which he imagined the electrons are embedded within a positively-charged sphere the size of the whole atom.

But Thompson’s former student Ernest Rutherford gradually disproved this model in 1909-1911, through experiments that showed * the nucleus is tens of thousands of times smaller (in radius) than an atom*, despite having most of the atom’s mass.

Once you know that electrons and atomic nuclei are both tiny, there’s an obvious question: **why is an atom so much larger than either one?** Here’s the logical problem”

- Negatively charged particles attract positively charged ones. If the nucleus is smaller than the atom,
?*why don’t the electrons find themselves pulled inward, thus shrinking the atom down to the size of that nucleus* - Well, the Sun and planets are tiny compared to the solar system as a whole, and gravity is an attractive force. Why aren’t the planets pulled into the Sun? It’s because they’re moving, in orbit. So
, much as planets orbit a star?**perhaps the electrons are in orbit around the nucleus** - This analogy doesn’t work. Unlike planets, electrons orbiting a nucleus would be expected to emit ample
(i.e. light, both visible and invisible), and thereby lose so much energy that they’d*electromagnetic waves***spiral into the nucleus in a fraction of a second**.

*(These statements about the radiated waves from planets and electrons can be understood with very little work, using — you guessed it — dimensional analysis! Maybe I’ll show you that in the comments if I have time.)*

So there’s a fundamental problem here.

- The tiny nucleus, with most of the atom’s mass, must be sitting in the middle of the atom.
- If the tiny electrons aren’t moving around, they’ll just fall straight into the nucleus.
- If they are moving around, they’ll radiate light and quickly spiral into the nucleus.

Either way, this would lead us to expect

- R
_{nucleus}= # R_{atom}

where # is not too, too far from 1. (This is the most naive of all dimensional analysis arguments: ** two radii in the same physical system shouldn’t be that different**.) This is in contradiction to experiment, which tells us that # is about 1/100,000! So

**it seems dimensional analysis has failed**.

Or is it we who have failed? Are we missing something, which, once included, will restore our confidence in dimensional analysis?

We are missing quantum physics, and in particular Planck’s constant **h**. When we include **h** into our dimensional analysis, a new possible size appears in our equations, and this sets the size of an atom. Details below.

### Quantum Physics Enters

The quantum revolution began in 1900 with Max Planck’s proposal that light emitted by atoms must carry energy in chunks

- E = n h f

where f is the light’s frequency, n is an integer (1,2,3,4…), and h is a previously unknown constant of nature that today we call Planck’s constant. Physicists soon found it more useful to work with the “reduced Planck constant” **ħ, which is h/(2π**)**.**

After learning this, a theoretical physicist will immediately ask, **“What are the dimensions of ħ?”** Well, frequency has units of “per time”, i.e. 1/time. Energy has units of

**mass*length**, as we saw in an earlier post. (“n”, as a number, has no dimensions.) So

^{2}/time^{2}- dimensions of
**ħ**= dimensions of E/f =**mass*length**^{2}/time

Any theoretical physicist paying attention will then notice that this is the same units as “angular momentum”, that quantity which assures that a spinning top stays upright and that spinning figure skaters spin faster when they pull in their arms. The Earth has angular momentum as it orbits the Sun — that’s why its path remains always in the same plane and at roughly the same speed — and the amount of its angular momentum L is its mass M times its speed v times its distance r from the Sun:

- L = M v r

which implies

- dimensions of L = dimensions of (M v r ) =
**mass*length**^{2}/time

So it’s not a huge step, once you know this, to imagine that maybe electrons inside atoms typically have

- L = # ħ

To assert *(incorrectly! the final section of this post fixes the problem)* that **this number # cannot ever be made small, and so angular momentum can never shrink to zero,** is bold. It’s a truly brazen guess to imagine that L is “quantized” — that it comes in steps, perhaps in integers; but Planck and then Einstein had already paved the way for the notion that certain quantities in nature come in integers, or at least in steps. So after Arthur Eric Haas first brought ħ into a model of the atom in 1910, John William Nicholson tried an atom with “quantized angular momentum” in 1911, and Niels Bohr built on these efforts in 1913 to produce the first truly successful model of an atom, complete with discrete electron orbits and jumps between them.

The Haas, Nicholson and Bohr models of the atom arose out of dimensional analysis. The details differ from one model to the next, but as it plays out in Nicholson’s and Bohr’s models, one imagines *(again, incorrectly!)* that electrons in an atom must have

- L = # n ħ

where n is never zero and # is some unknown number. Since

- L = m v r

for an object in a circular orbit, we can expect that the radius of an atom’s orbit may involve ħ, m and v, where **m is the electron’s mass, not the atom’s mass**.

Another thing we would guess is that atoms are held together by electric forces. Let’s keep things simple by considering hydrogen, which has just one electron of electric charge -e, and a nucleus of electric charge +e. The force between these two objects, by Coulomb’s law, is

- F = k e
^{2}/ r^{2}

where r is the distance between them, which should be roughly the atom’s size. So we can expect that a hydrogen atom’s size r_{H} might depend on k and e also. Since e has dimensions of charge,

**dimensions of k = dimensions of [F r**.^{2}/ e^{2}] = mass*length^{3}/time^{2}/charge^{2}

### The Estimated Size of an Atom

Now, can we guess the size of an atom? Yes! Noticing that

- dimensions of ħ
^{2}/(m k e^{2})- =
**(mass*length**divided by^{2}/time)^{2}**(mass*[mass*length**^{3}/time^{2}/charge^{2}]*charge^{2}) - =
**(mass**^{2}*length^{4}/time^{2})/(mass^{2}*length^{3}/time^{2}) - =
**length**

- =

a reasonable and consistent equation is

**r**#_{H}=**ħ**^{2}/(m k e^{2})

where again ħ is the reduced Planck constant, m and e are the electron’s mass and charge, and k is Coulomb’s constant. The number # turns out to be 1 in Bohr’s model. Plugging in numbers we find

- r
_{H}= 0.05 billionths of a meter

which agrees with experiment!! We can also get v using L = m v r = # ħ :

**v**= # ħ / (m r_{H}) = #**ke**= # * (2.187 million meters/second) = # (0.0073 c)^{2}/ħ

Since # turns out to be 1 for hydrogen, the electron’s speed is about 0.7% of the speed of light **c**. *[Notice that v depends on the strength of electromagnetism but not on the electron’s mass! Thus a stronger form of electromagnetism (i.e. with k larger than in nature) would give a smaller atom with an even faster electron.]* Meanwhile, when we estimate the energy involved in binding the electron to the nucleus

- E = # (electron’s motion energy) = # (½ m v
^{2}) = # ½**m (ke**= # 13.6 electron-volts^{2})^{2}/ħ^{2}

that comes out in agreement with experiment too!

But physicists didn’t need the detailed successes of Bohr’s model to be convinced that quantum physics has a role to play in atoms. By 1911, the work of Haas and Nicholson had already shown that **dimensional analysis using ħ correctly reproduces atomic sizes and energies. **The reason that R_{atom} >> R_{nucleus} is that ħ, the fundamental quantum parameter of nature, is nonzero.

### How Uncertainty Made It All Clear

The Bohr model, though ingenious and successful, was only partly right. What’s the modern quantum physics argument for the size of atoms? It relies on the uncertainty principle, uncovered more than a decade after this early round of atomic reasoning. This principle not only gives the right answer but explains **why** its the right answer. It says that **the more you know the position of an electron, the less you can know its velocity (more precisely, its momentum m v**), and vice versa. In particular, the uncertainty on its location Δ(r) times the uncertainty on its momentum Δ(mv) must satisfy

- Δ(r) Δ(mv) ≥
**ħ**

If an electron is squeezed down into a region of radius r, then ** the uncertainty in its position is roughly the size of the region:** Δ(r) = # r. And if it moves around with maximum speed v and changing direction, sometimes moving north, sometimes east, sometimes not at all, that means

*: Δ(m v) = # m v. And so the uncertainty formula says*

**the uncertainty on mv is comparable to mv**- Δ(r) Δ(mv) = # m v r ≥
**ħ**

*[note the product of two unknown numbers is another unknown number.]* This is essentially the same as the 1910’s scientists’ starting point — but ** they had the right formula for the wrong reasons**. The formula arises not because the electron has an orbit with angular momentum L but because, as an electron surrounds the atomic nucleus, it has a typical

**m v**and a typical

**r**which satisfies the uncertainty relation. That is,

**it’s uncertainty, not angular momentum, that can’t shrink to zero!**

*(Aha!!!!!)*All the successes with the sizes and energies of atoms are then reproduced, coming out roughly the same as in the Bohr-era models.

Incidentally, the sizes of atoms beyond hydrogen are pretty similar. The stronger pull of a nucleus with larger charge (which tends to make atoms smaller) competes with the “Pauli exclusion principle” (which tends to make atoms larger) and the two effects largely cancel. For the details, there’s no substitute for a complete theory of quantum physics and a real calculation. But I hope it’s now clear that dimensional reasoning was central on the path that led to the modern understanding of the atom, and that no atoms could exist without quantum physics!

HI Matt, I’ve really enjoyed this series of posts on dimensional analysis. It’s interesting that in fundamental physics problems there are only a few possible “natural” length scales to the problem, and most other quantities are computed in terms of those (modulo some # close to one). It reminds me a bit of systems near to phase-transitions, where the correlation length sets all the other physics (dubbed the “dictatorship principle” in Stauffer’s Introduction to Percolation Theory), and essentially all quantities of interest scale in some way with the correlation length.

I don’t know this phrasing as a “dictatorship principle”! But certainly yes, as you approach a fixed point/phase transition there tends to be a leading operator that controls everything.

Your walkthroughs are excellent because they follow an unbroken thread from initial observation to final theory and explain the motivations behind the “why”.

Including the false starts along the way is also important because that’s how students experience this material for the first time when they’re forming their cognitive models of what’s going on, so it makes it more relatable.

My differential geometry teacher followed this approach and despite it being amongst the harder subjects, I did best in it. Unlike in non-linear systems class where I’d ask things like “why is that perturbation parameter there” and get told “because it makes it work”.

I wish more of my undergraduate lecturers had used this approach and not glossed over the underlying rationale. They seemed to follow Gauss’ approach – “When the architect completes a fine building, he removes the scaffolding”.

I had the same experience; I think the problem is that there’s only so much time in these classes, and the choice of history versus technical material leans toward the latter (because of requirements for later courses) at the expense of really teaching how science is done and how it proceeds. My decades of experience in the field have helped me to see how messy discoveries are, and how close normal scientists often get to the right idea before the genius swoops in and finally sees what’s really going on.

If I take a picture of someone riding a bicycle with too long an exposure time, they will be blurred across the frame with no clear position, and with too short an exposure time, they will appear to be totally stationary even though they aren’t. This kind of uncertainty is pretty intuitive.

This is an interesting attempt at building intuition, but I’m afraid it’s quite misleading. What’s going on with an electron is very different from what you’ve described; it’s not at all about exposure time, or the blurring or freezing of an image. You should read more about the subject… it’s too long an issue for a comment, at least at this hour. What’s really going on is that waves cannot be localized without making them subsequently expand in all directions, and conversely a wave train which is clearly moving in one direction is spread out all over the place. This uncertainty is intrinsic to the waves, and not a consequence of how we try to image them.

Great analysis using dimensions. Agree with Richard here. Many of the text books leave out the dimensional analysis part so the students is presented with the equivalence of Coulomb’s force and centripetal force directly. This is fine for exam purposes but the student is left wondering how the great minds came to think of this equivalence. That Planck’s constant has the same dimensions as angular momentum was an important clue that eventually led the theoreticians in the right direction.

If the proton radius can’t be predicted/calculated by the theory, would you trust the theory?