- 14.2.1 Draft: The proton
- 14.2.2 Draft: The neutron
- 14.2.3 Draft: The deuteron
- 14.2.4 Draft: Property summary

14.2 Draft: The Simplest Nuclei

This subsection introduces the simplest nuclei and their properties.

14.2.1 Draft: The proton

The simplest nucleus is the hydrogen one, just a single proton. It is
trivial. Or at least it is if you ignore the fact that that proton
really consists of a conglomerate of three quarks held together by
gluons. A proton has an electric charge

However, the proton is roughly 2 000 times heavier than the electron. On the other hand the magnetic dipole moment of a proton is roughly 700 times smaller than that of an electron. The differences in mass and magnetic dipole moment are related, chapter 13.4. In terms of classical physics, a lighter particle circles around a lot faster for given angular momentum.

Actually, the proton has quite a large magnetic moment for its mass.
The proton has the same spin and charge as the electron but is roughly
2 000 times heavier. So logically speaking the proton magnetic
moment should be roughly 2 000 times smaller than the one of the
electron, not 700 times. The explanation is that the electron is an
elementary particle, but the proton is not. The proton consists of
two up quarks, each with charge

Key Points

- The proton is the nucleus of a normal hydrogen atom.

- It really consists of three quarks, but ignore that.

- It has the opposite charge of an electron, positive.

- It has spin
.

- It is roughly 2 000 times heavier than an electron.

- It has a magnetic dipole moment. But this moment is roughly 700 times smaller than that of an electron.

14.2.2 Draft: The neutron

It is hard to call a lone neutron a nucleus, as it has no net charge to hold onto any electrons. In any case, it is somewhat academic, since a lone neutron disintegrates in on average about 10 minutes. The neutron emits an electron and an antineutrino and turns into a proton. That is an example of what is called “beta decay.” Neutrons in nuclei can be stable.

A neutron is slightly heavier than a proton. It too has spin

The reason that the neutron has a dipole moment is that the three
quarks that make up a neutron do have charge. A neutron contains one
up quark with a charge of

Key Points

- The neutron is slightly heavier than the proton.

- It too has spin
.

- It has no charge.

- Despite that, it does have a comparable magnetic dipole moment.

- Lone neutrons are unstable. They suffer beta decay.

14.2.3 Draft: The deuteron

The smallest nontrivial nucleus consists of one proton and one neutron. This nucleus is called the deuteron. (An atom with such a nucleus is called deuterium). Just like the proton-electron hydrogen atom has been critical for deducing the structure of atoms, so the proton-neutron deuteron has been very important in deducing knowledge about the internal structure of nuclei.

However, the deuteron is not by far as simple a two-particle system as the hydrogen atom. It is also much harder to analyze. For the hydrogen atom, spectroscopic analysis of its excited quantum states provided a gold mine of information. Unfortunately, it turns out that the deuteron is so weakly bound that it has no excited quantum states. If you try to excite it by adding energy, it falls apart.

The experimental binding energy of the deuteron is only about 2.22 MeV. Here a MeV is the energy that an electron would pick up in a one-million voltage difference. For an electron, that would be a gigantic energy. But for a nucleus it is ho-hum indeed. A typical stable nucleus has a binding energy on the order of 8 MeV per nucleon.

In any case, it is lucky that that 2.22 MeV of binding energy is there at all. If the deuteron would not bind, life as we know it would not exist. The formation of nuclei heavier than hydrogen, including the carbon of life, begins with the deuteron.

The lack of excited states makes it hard to understand the deuteron. In addition, spin has a major effect on the force between the proton and neutron. In the hydrogen atom, that effect exists but it is extremely small. In particular, in the true hydrogen atom ground state the electron and proton align their spins in opposite directions. That produces the so-called singlet state of zero net spin, chapter 5.5.6. However, the electron and proton can also align their spins in the same direction, at least as far as angular momentum uncertainty allows. That produces the so-called triplet state of unit net spin. For the hydrogen atom, it turns out that the triplet state has very slightly higher energy than the singlet state, {A.39}.

In case of the deuteron, however, the triplet state has the lowest energy. And the singlet state has so much more energy that it is not even bound. Almost bound maybe, but definitely not bound. For the proton and neutron to bind together at all, they must align their spins into the triplet state.

As a result, a nucleus consisting of two protons (the diproton) or of two neutrons (the dineutron) does not exist. That is despite the fact that two protons or two neutrons attract each other almost the same as the proton and neutron in the deuteron. The problem is the antisymmetrization requirement that two identical nucleons must satisfy, chapter 5.6. A spatial ground state should be symmetric. (See addendum {A.40} for more on that.) To satisfy the antisymmetrization requirement, the spin state of a diproton or dineutron must then be the antisymmetric singlet state. But only the triplet state is bound.

(You might guess that the diproton would also not exist because of the Coulomb repulsion between the two protons. But if you ballpark the Coulomb repulsion using the models of {A.41}, it is less than a third of the already small 2.22 MeV binding energy. In general, the Coulomb force is quite small for light nuclei.)

There is another qualitative difference between the hydrogen atom and
the deuteron. The hydrogen atom has zero orbital angular momentum in
its ground state. In particular, the quantum number of orbital
angular momentum

But orbital angular momentum is not conserved in the deuteron. In terms of classical physics, the forces between the proton and neutron are not exactly along the line connecting them. They deviate from the line based on the directions of the nucleon spins.

In terms of quantum mechanics, this gets phrased a bit differently.
The potential does not commute with the orbital angular momentum
operators. Therefore the ground state is not a state of definite
orbital angular momentum. The angular momentum is still limited by
the experimental observations that the deuteron has spin 1 and even
parity. That restricts the orbital angular momentum quantum number

One consequence of the nonzero orbital angular momentum is that the magnetic dipole strength of the deuteron is not exactly what would be expected based on the dipole strengths of proton and neutron. Since the charged proton has orbital angular momentum, its acts like a little electromagnet not just because of its spin, but also because of its orbital motion.

Another consequence of the nonzero orbital angular momentum is that the charge distribution of the deuteron is not exactly spherically symmetric. This asymmetric charge distribution allows the deuteron to interact with gradients in an external electric field. It is said that the deuteron has a nonzero “electric quadrupole moment.”

Roughly speaking, you may think of the charge distribution of the
deuteron as elongated in the direction of its spin. That is not quite
right, quantum-mechanically speaking, since angular momentum has
uncertainty in direction. Therefore, instead consider the quantum
state in which the deuteron spin has its maximum component,

The nonzero orbital momentum also shows up in experiments where various particles are scattered off deuterons.

To be sure, the precise probability of the

Key Points

- The deuteron consists of a proton and a neutron.

- The deuteron is the simplest nontrivial nucleus. The diproton and the dineutron do not exist.

- The deuteron has spin 1 and even parity. The binding energy is 2.225 MeV.

- There are no excited states. The ground state of lowest energy is all there is.

- The deuteron has a nonzero magnetic dipole moment.

- It also has a nonzero electric quadrupole moment.

14.2.4 Draft: Property summary

Table 14.1 gives a summary of the properties of the three simplest nuclei. The electron is also included for comparison.

The first data column gives the mass. Note that nuclei are thousands of times heavier than electrons. As far as the units are concerned, what is really listed is the energy equivalent of the masses. That means that the mass is multiplied by the square speed of light following the Einstein mass-energy relation. The resulting energies in Joules are then converted to MeV. An MeV is the energy that an electron picks up in a 1 million voltage difference. Yes it is crazy, but that is how you will almost always find masses listed in nuclear references. So you may as well get used to it.

It can be verified from the given numbers that the deuteron mass is indeed smaller than the sum of proton and neutron masses by the 2.225 MeV of binding energy. It is a tenth of a percent, but it is very accurately measurable.

The second column gives the charge. Note that all these charges are
whole multiples of the proton charge up

quarks have charge down

quarks have charge

The third column gives the charge radius. That is a measure of the spatial extent of the charge distribution. The electron is, as far as is known, a point particle with no internal structure. For the neutron, with no net charge, it is not really clear what to define as charge radius.

The fourth column shows the quantum number of net angular momentum. For the first three particles, that is simply their spin. For the deuteron, it is the nuclear spin. That includes both the spins and the orbital angular momenta of the proton and neutron that make up the deuteron.

The fifth column is parity. It is even in all cases. More complicated nuclei can have negative parity.

The sixth column is the magnetic dipole moment. It is expressed in
terms of the so-called nuclear magneton

As the table shows, nuclei have much smaller magnetic moments than electrons. That is due to their much larger masses. However, using a magnetic field of just the right frequency, nuclear magnetic moments can be observed. That is how nuclear magnetic resonance works, chapter 13.6. Therefore nuclear magnetic moments are important for many applications, including medical ones like MRI.

The last column lists the electric quadrupole strength. That is a measure for the deviation of the nuclear charge distribution from a spherically symmetric shape. It is a complicating factor in nuclear magnetic resonance. Or an additional source of information, depending on your view point. Nuclei with spin less than 1 do not have electric quadrupole moments. (That is an implied consequence of the relation between angular momentum and symmetry in quantum mechanics.)

Note that the SI length unit of femtometer works very nicely for
nuclei. So, since physicists hate perfection, they define a new
non-SI unit called the barn b. A barn is 100 f

Key Points

- The properties of the simplest nuclei are summarized in table 14.1.