Chapter 17, Note 2: The Proton’s Size

If a quantum object, such as a proton, is made from wavicles, how can we make sense of and measure its size and shape? Here are two classic ways to do it.

• Scatter a diffuse beam of particles off the object, hard enough to see them bounce off, but gently enough so that the internal structure of the object, if any, plays no role.
• Strike the object hard enough that its internal structure jiggles, like the sloshing of milk in a container, or like the rattling of the contents of a box of chocolates.

The second approach is the method described in the book’s Chapter 17. But the first method requires less energy, has a longer history, and is the one that was initially used to show that the proton has a size.

The Method of Scattering

In a certain sense, the method of scattering is familiar. To learn the size of, say, a metal ball, we would naturally be inclined to shine a flashlight on it. The flashlight consists of a beam of subatomic particles — photons, the particles of light. The metal ball will reflect those photons that strike it. As a result, the light gives us two clues about the size of the object.

• The ball leaves a shadow on the wall behind it, whose size tells us the size of the ball.
• The ball scatters light in all backward directions, and from the pattern of this reflected light, we can infer the size and reflectivity of the ball.

For an object that is subatomic, the shadow is useless; the shadow’s size is ultra-microscopic, and we can’t measure it. But even from a subatomic object, the reflected light can be measured, and used to learn something about the object’s size.

The size of the nucleus was measured in just this way, as described in the book’s Chapter 6.3. The only differences from Fig. 1 are that the target was an atomic nucleus, and the nuclei of helium atoms were used in the beam in place of photons.

The proton’s size, too, was measured in this way, using an electron beam impinging on hydrogen gas. (Hydrogen gas consists only of electrons and protons, and the electrons in the hydrogen have a lesser impact on the electrons in the beam than do the protons.)

But there’s a complication. Because a proton has electric charge, it would exerts an electric attraction on an nearby electron — and so, it would deflect all electrons — even if it had no size. This electrical effect has to be accounted for. Fortunately, even though the scattering of electrons off of an electrically charged sphere is never absent, it changes with the sphere’s diameter, and so a careful measurement can reveal the sphere’s size.

This scattering experiment was done in pioneering research by McAllister and Hoftstadter. (Robert Hofstadter, who won the Nobel Prize for this work in 1961, was the father of mathematician and writer Douglas Hofstadter, author of “Gödel, Escher, Bach”.) Their studies showed that the scattering of electrons off of protons is inconsistent with a proton being a “point”, and most easily explained by the proton having a diameter about one million billionth of a meter — roughly half the diameter of a typical helium nucleus.

The Method of Excited States

Now here’s another approach, which is the one described in and around Figure 40 of Chapter 17. Again, the basic idea is simple. If I shake a box with stuff inside it, I can hear and feel the rattling of whatever makes up its interior. That rattling tells me there’s something inside.

In daily life, the rattling usually stops as soon as I stop shaking the box. But that’s just because of friction between its contents. If instead I had a box with a single spring inside it, with low friction, then shaking the box would cause the springs to vibrate for quite a while, as in Fig. 3.

If it weren’t for quantum physics, any amount of shaking would be enough to make the spring vibrate. But a quantum box is different. The quantum formula E=f[h] states that if the spring has a resonant frequency f, then the minimal amount of energy needed to make it vibrate is f[h]. More generally, to make any quantum system vibrate internally — to “excite” the system — requires a minimum amount of energy.

Atoms

Any system with complex internal structure can vibrate in a whole variety of ways, and in a quantum system, each one requires a specific amount of energy. Atoms are examples; they are made of electrons and an atomic nucleus. In atoms, the various modes of vibration of their interiors are called “excited states”, and the amounts of energy needed to excite them are called the atom’s “energy levels.”

For any particular type of atom, there is a miminal amount of energy that it may have. This occurs when it is stationary and is in its “ground state”, with the least possible amount of internal motion. (This minimal energy, divided by [c²], is quoted as the atom’s rest mass.) The atom also has a huge number of “excited” states, in which the atom remains stationary but its internal structure rattles around in one way or another. Each particular excited state has a fixed amount of energy above the ground state (and thus slightly more rest mass).

A quantum object could in principle remain in an excited state forever, if it weren’t for dissipation. But dissipation is always present. Still, as noted in Chapter 21.2, dissipation in quantum physics is more suddent and dramatic than the slow, gradual effects of friction that we observe so often in daily life. The effect of dissipation is to cause what scientists call “atomic decay”, a sudden transition.

In an atomic decay, an atom in an excited state may suddenly return to its ground state, dissipating its excess energy by emitting a photon that carries that energy away. In this case, the photon will carry off exactly the difference between the energy of the excited state and the energy of the ground state; and measuring the energies of those photons gives insight into the excited states of the atom.

Or a decay may involve multiple steps, in which one excited state transitions to another, emitting one photon, and the second excited state returns to the ground state, emitting a second photon. (Such transitions in lithium are indicated by the arrows in Fig. 4, which show how the higher excited states can take many paths back down to the ground state.)

In each case, the photons emitted carry an amount of energy that is the difference between one state’s energy and another. By observing these photons, scientists can gain clues about the excited states of an atom, and from this information they can infer its internal structure. Indeed, the photons emitted by hydrogen were used by Niels Bohr in his initial model of the hydrogen atom.

Later, the photons emitted by more complex atoms were used to verify the more accurate models of atoms that emerged from quantum physics of the 1920s and 1930s. (Click here for more examples of the photons emitted in atomic transitions; note how different they can be from one type of atom to another.)

Protons

What works for atoms made from electrons and atomic nuclei works also for the nuclei themselves, and even for the protons and neutrons made from quarks. In particular, hard collisions of other particles with protons and neutrons reveal they have excited states, thus proving that they have internal structure.

An example of such a collision between an electron and a proton is shown in Fig. 6. The collision pops the proton into an excited state, called the “Delta” for historical reasons. This excited state rapidly decays back to a proton (or neutron) plus a particle called a “pion”, made also from quarks, anti-quarks and gluons, (but with no excess quarks over gluons.) [These pions in turn decay, but we need not go into those details.]

Other excited states of a proton can decay to a proton and more than one pion. This is the reasoning behind the book’s Figure 40, where at far right it is sketched how a proton, struck hard enough, is shrunk down small and raised to an excited state which quickly decays to two or more particles. This transition from one particle to many, via an excited state, is evidence that the proton has internal structure, like an atom. It confirms that a proton is not an elementary particle.

Estimating the Object’s Size From Its Excited States

[Added April 30, 2024, after a reader pointed out I hadn’t really addressed this properly. It is more technical than the rest of the article. Note that I composed this rather quickly, so there might some small errors hidden here; please point them out if you see them.]

Once we know that that an object has excited states, how can we estimate its size? To do so precisely, one has to look at details, making careful guesses about the system’s internals and shape based on what one learns about the excited states — specifically their energies and the patterns seen in their decays.

But one can already gain a rough measure of its size using some basic reasoning, the Heisenberg uncertainty principle of quantum physics, and dimensional analysis.

Some Rough Reasoning

The reasoning goes as follows. Suppose the object has mass M and length L. Suppose there is a piece of the object (let’s call it the “subobject“) that can vibrate within the object. Then we can use the uncertainty principle to estimate how much energy this vibration will have.

Even in the ground state, the subobject has some energy (this is zero-point energy). We can estimate it first. This being quantum physics, the subobject can spread out within the box, and can do so away from the box’s center by a distance of roughly L/2. By the uncertainty principle, that means the uncertainty in its momentum, even in the ground state, will be roughly (where is called the “reduced Planck’s constant“).

Einstein’s relativity then tells us that the average energy of the vibrating subobject within the object will be given by its mass m, its average momentum-squared p², and the cosmic speed limit c, by the following formula:

Now, when the object moves to the first excited state, the subobject vibrates a little more. But here’s the point: its typical momentum will still be proportional to , only somewhat larger. The logic of dimensional analysis can be used to show that there’s simply nothing else, in this situation, to which it could be proportional.

So let’s say the average momentum in the excited state is where ; that’s a pretty safe bet for an estimate. Then

The change in energy from the ground state to the first excited state,

can be written using an approximation for the difference of two square roots:

where in the last step I further assumed that , which lies between 0 and 3, is not so different from 2.

Now, solving this last equation for L gives

Since we’ve been making guesses about x, we can’t expect this to be more accurate than a factor of 2 or 3. But if we have no idea what the size is to start with, narrowing down the options to such a small factor is big progress.

Assumptions Needed

Now, this still isn’t quite enough to give us an estimate for L. The quantity is directly measurable, as the energy emitted in the decay from the 1st excited state to the ground state. But is not directly measurable; it is the energy in the subobject within the ground state, which we don’t know. It might be a large fraction or a small fraction of the total energy of the full object.

The total energy of the full object is known, since by definition it is in terms of the mass of the full object. So now we have to make a guess; what fraction of is ?

If the subobject is all of, or a substantial fraction of, the object, then we could take

in which case

Protons vs. Atoms

For a proton,

and

which is within a factor of 2 or so of the right answer.

For an atom, however, this method would give the wrong result, because inside of an atom the vibrating subobject is an electron, whose mass is much smaller than that of the atomic nucleus and of the atom as a whole. This wasn’t recognized until the famous experiments done in Rutherford’s lab by Geiger and Marsden, mentioned in Chapter 6.

Once armed with the knowledge that the mass of an atom sits in a small core, with the electrons on the outside and free to vibrate, one would guess that for an atom,

Since (recall a GeV is a billion eV), this gives

which again is more or less in the right ballpark.

A final note

Using rough-and-dirty methods like this is a bit of an art form — a mixture of intuition, approximation, educated guesses, and cross-checks against experimental data. When used in the wild by real physicists on unsolved problems, efforts are usually made to be more precise than I’ve been here. But even when more precision is not possible based on current information, the key steps forward are clear: first, to use these theoretical methods to make definite predictions for experiments to check, and second, to rely on experiments both to test those predictions and to provide additional information that can be used to refine the theoretical approach. With parallel advances in theory and in experiment, one can hope to come to a detailed and complete understanding of the object under study. That’s a process that may take decades, but it’s one with a fantastic success record.