14.3 Modeling the Deuteron

Addendum {A.40} explores some of the nuclear potentials that you can write down to model the deuteron. Simple potentials such as those described there give a lot of insight in the deuteron. They give ballpark values for the potential and kinetic energies. They also give an explanation for the observations that the deuteron is only weakly bound and that there is no second bound state. They show the relatively large size of the deuteron: there is a good chance that the proton and neutron can be found way apart. Simple additions to the potential can describe the spin dependence and the violation of orbital angular momentum conservation.

However, there is a problem. These potentials describe a deuteron consisting of a proton and a neutron. But the proton and neutron are not elementary particles: each consists of three quarks.

This internal structure should not be a much of a concern when the proton and neutron are relatively far apart. But if the two get really close? Surely in that case the physics should be described in terms of six quarks that interact through gluons and the Pauli exclusion principle, [5, p. 95]? Eventually the proton and neutron must lose their identity. Then there is no longer any reasonable justification for a picture of a free-space proton interacting with a free-space neutron.

A rough idea of the scales involved may be obtained by looking at charge radii. The charge radius of a particle is a measure of the size of its charge distribution. Now the proton has a charge radius of about 0.88 fm. The deuteron has a charge radius of 2.14 fm. So at least the charge radius of a proton is not that much smaller than the size of the deuteron. And a quantum description of the deuteron needs to consider all possible nucleon spacings. Clearly for the smallest nucleon spacings, the two must intrude nontrivially into each other’s space.

Consider another example of compound structures, noble gas atoms. When such atoms are relatively far apart, they can be modeled well as point particles forming an ideal gas. You might add some simple Van der Waals forces to that picture. However, when the atoms get pushed close together, the electromagnetic interactions between them become much more complex. And if you try to push the atoms even closer together, they resist that very strongly. The reason is the Pauli exclusion principle, chapter 5.10. More than two electrons cannot be pushed in the same spatial state.

The big difference is of course that the electromagnetic interactions of the electrons and nuclei that make up atoms are well understood. There is as yet no good way to describe the color force interactions between the quarks that make up nucleons. (Certainly not at the relatively low energies of importance for nuclear structure.)

Physicists have developed a model that is somewhere intermediate between that of interacting free-space nucleons and interacting quarks. In the model, the forces between nucleons are produced by the exchange of particles called “pions.” That is much like how in relativistic quantum mechanics, electromagnetic forces are produced by the exchange of photons. Or how the forces between quarks are believed to be produced by the exchange of gluons. These exchanged particles are virtual ones.

Roughly speaking, relativistic mass-energy equivalence combined with quantum uncertainty in energy allows some possibility for these particles to be found near the real particles. (See addendum {A.41} for a more quantitative description of these ideas.)

The picture is now that the proton and neutron are elementary particles but dressed in a coat of virtual pions. Pions consist of two quarks. More precisely, they consist of a quark and an antiquark. There are three pions; the positively charged $\pi^+$, the neutral $\pi^0$, and the negatively charged $\pi^-$. Pions have no spin. They have negative intrinsic parity. The charged pions have a mass of 140 MeV, while the uncharged one has a slightly smaller mass of 135 MeV.

When the neutron and proton exchange a pion, they also exchange its momentum. That produces a force between them.

There are a number of redeeming features to this model:

It explains the short range of the nuclear force. That is worked out in addendum {A.41}.

However, the same result is often derived much quicker and easier in literature. That derivation does not require any Hamiltonians to be written down, or even any mathematics above the elementary school level. It uses the popular so-called “energy-time uncertainty equality,” chapter 7.2.2,

\mbox{any energy difference you want}
\mbox{any time difference you want}
= {\textstyle\frac{1}{2}} \hbar

To apply it to pion exchange, replace “any energy difference you want” with the rest mass energy of the virtual pion that supposedly pops up, about 138 MeV on average. Replace “any time difference you want” with the time that the pion exists. (Model the pion here as a classical particle with definite values of position versus time.) From the time that the pion exists, you can compute how far it travels. That is because clearly it must travel with about half the speed of light: it cannot travel with a speed less than zero nor more than the speed of light. Put in the numbers, ignore the stupid factors $\frac12$ because you do not need to be that accurate, and it shows that the pion would travel about 1.4 fm if it had a position to begin with. That range of the pion is consistent with the experimental data on the range of the nuclear force.

(OK, someone might object the pions do most decidedly not pop up and disappear again. The ground state of a nucleus is an energy eigenstate and those are stationary, chapter 7.1.4. But why worry about such minor details?)

It gives a reasonable explanation of the anomalous magnetic moments of the proton and neutron. The magnetic moment of the proton can be written as

\mu_{\rm {p}} = g_{\rm {p}} {\textstyle\frac{1}{2}} \mu_{\...
...rac{e\hbar}{2m_{\rm p}}
\approx 5.051\;10^{-27}\mbox{ J/T}

Here $\mu_{\rm N}$ is called the nuclear magneton. It is formed with the charge and mass of the proton. Now, if the proton was an elementary particle with spin one-half, the factor $g_{\rm {p}}$ should be two, chapter 13.4. Or at least close to it. The electron is an elementary particle, and its similarly defined $g$-​factor is 2.002. (Of course that uses the electron mass instead of the proton one.)

The magnetic moment of the neutron can similarly be written

\mu_{\rm {n}} = g_{\rm {n}} {\textstyle\frac{1}{2}} \mu_{\...
...rac{e\hbar}{2m_{\rm p}}
\approx 5.051\;10^{-27}\mbox{ J/T}

Note that the proton charge and energy are used here. In fact, if you consider the neutron as an elementary particle with no charge, it should not have a magnetic moment in the first place.

Suppose however that the neutron occasionally briefly flips out a negatively charged virtual $\pi^-$. Because of charge conservation, that will leave the neutron as a positively charged proton. But the pion is much lighter than the proton. Lighter particles produce much greater magnetic moments, all else being the same, chapter 13.4. In terms of classical physics, lighter particles circle around a lot faster for the same angular momentum. To be sure, as a spinless particle the pion has no intrinsic magnetic moment. However, because of parity conservation, the pion should have at least one unit of orbital angular momentum to compensate for its intrinsic negative parity. So the neutral neutron acquires a big chunk of unexpected negative magnetic moment.

Similar ideas apply for the proton. The proton may temporarily flip out a positively charged $\pi^+$, leaving itself a neutron. Because of the mass difference, this can be expected to significantly increase the proton magnetic moment.

Apparently then, the virtual $\pi^+$ pions increase the $g$-​factor of the proton from 2 to 5.586, an increase of 3.586. So you would expect that the virtual $\pi^-$ pions decrease the $g$-​factor of the neutron from 0 to $\vphantom0\raisebox{1.5pt}{$-$}$3.586. That is roughly right, the actual $g$-​factor of the neutron is $\vphantom0\raisebox{1.5pt}{$-$}$3.826.

The slightly larger value seems logical enough too. The fact that the proton turns occasionally into a neutron should decrease its total magnetic moment. Conversely, the neutron occasionally turns into a proton. Assume that the neutron has half a unit of spin in the positive chosen $z$-​direction. That is consistent with one unit of orbital angular momentum for the $\pi^-$ and minus half a unit of spin for the proton that is left behind. So the proton spins in the opposite direction of the neutron, which means that it increases the magnitude of the negative neutron magnetic moment. (The other possibility, that the $\pi^{-1}$ has no $z$-​momentum and the proton has half a positive unit, would decrease the magnitude. But this possibility has only half the probability, according to the Clebsch-Gordan coefficients figure 12.6.)

The same ideas also provide an explanation for a problem with the magnetic moments of heavier nuclei. These do not fit theoretical data that well, figure 14.40. A closer examination of these data suggests that the intrinsic magnetic moments of nucleons are smaller inside nuclei than they are in free space. That can reasonably be explained by assuming that the proximity of other nucleons affects the coats of virtual pions.

It explains a puzzling observation when high-energy neutrons are scattered off high-energy protons going the opposite way. Because of the high energies, you would expect that only a few protons and neutrons would be significantly deflected from their path. Those would be caused by the few collisions that happen to be almost head-on. And indeed relatively few are scattered to say about 90$\POW9,{\circ}$ angles. But an unexpectedly large number seems to get scattered almost 180$\POW9,{\circ}$, straight back to where they came from. That does not make much sense.

The more reasonable explanation is that the proton catches a virtual $\pi^-$ from the neutron. Or the neutron catches a virtual $\pi^+$ from the proton. Either process turns the proton into a neutron and vice-versa. So an apparent reflected neutron is really a proton that kept going straight but changed into a neutron. And the same way for an apparent reflected proton.

It can explain why charge independence is less perfect than charge symmetry. A pair of neutrons can only exchange the neutral $\pi^0$. Exchange of a charged pion would create a proton and a nucleon with charge $\vphantom0\raisebox{1.5pt}{$-$}$1. The latter does not exist. (At least not for measurable times, nor at the energies of most interest here.) Similarly, a pair of protons can normally only exchange a $\pi^0$. But a neutron and a proton can also exchange charged pions. So pion interactions are not charge independent.

The nuclear potential that can be written down analytically based on pion exchange works very well at large nucleon spacings. This potential is called the OPEP, for One Pion Exchange Potential, {A.41}.

There are also drawbacks to the pion exchange approach.

For one, the magnetic moments of the neutron and proton can be reasonably explained by simply adding those of the constituent quarks, [30, pp. 74, 745]. To be sure, that does not directly affect the question whether the pion exchange model is useful. But it does make the dressed nucleon picture look quite contrived.

A bigger problem is nucleon spacings that are not so large. One-pion exchange is generally believed to be dominant for nucleon spacings above 3 fm, and reasonable for spacings above 2 fm, [35, p. 135, 159], [5, p. 86, 91]. However, things get much more messy for spacings shorter than that. That includes the vital range of spacings of the primary nucleon attractions and repulsions. For these, two-pion exchange must be considered. In addition, excited pion states and an excited nucleon state need to be included. That is much more complicated. See addendum {A.41} for a brief intro to some of the issues involved.

And for still shorter nucleon spacings, things get very messy indeed, including multi-pion exchanges and a zoo of other particles. Eventually the question must be at what spacing nucleons lose their distinctive character and a model of quarks exchanging gluons becomes unavoidable. Fortunately, very close spacings correspond to very high energies since the nucleons strongly repel each other at close range. So very close spacings may not be that important for most nuclear physics.

Because of the above and other issues, many physicists use a less theoretical approach. The OPEP is still used at large nucleon spacings. But at shorter spacings, relatively simple chosen potentials are used. The parameters of those “phenomenological” potentials are adjusted to match the experimental data.

It makes things a lot simpler. And it is not clear whether the theoretical models used at smaller nucleon spacings are really that much more justified. However, phenomenological potentials do require that large numbers of parameters are fit to experimental data. And they have a nasty habit of not working that well for experimental data different from that used to define their parameters, [31].

Regardless of potential used, it is difficult to come up with an unambiguous probability for the $l$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 orbital angular momentum. Estimates hover around the 5% value, but a clear value has never been established. This is not encouraging, since this probability is an integral quantity. If it varies nontrivially from one model to the next, then there is no real convergence on a single deuteron model. Of course, if the proton and neutron are modeled as interacting clouds of particles, it may not even be obvious how to define their orbital angular momentum in the first place, [Phys. Rev. C 19,20 (1979) 1473,325]. And that in turn raises questions in what sense these models are really well-defined two-particle ones.

Then there is the very big problem of generalizing all this to systems of three or more nucleons. One current hope is that closer examination of the underlying quark model may produce a more theoretically justified model in terms of nucleons and mesons, [31].