14.1 Fundamental Concepts

This section describes the most basic facts about nuclei. These facts will be taken for granted in the rest of this chapter.

Nuclei consist of protons and neutrons. Protons and neutrons are therefore called “nucleons.” Neutrons are electri­cally neutral, but protons are positively charged. That allows the protons in a nucleus to attract electrons. The electrons have much larger quantum mechanical uncertainty in position than the much heavier nucleus. So they form a “cloud” around the nucleus, producing an atom.

Protons mutually repel each other because of their electric charge. That is due to the same Coulomb force that allows them to attract electrons. By itself, it would cause a nucleus to fly apart. But nucleons also attract each other through another force, the “nuclear force.” It is this force that keeps a nucleus together.

The nuclear force is strong, but it is also very short range, extending over no more than a few femtometers. (A femtometer, or fm, equals 10$\POW9,{-15}$ m. It is sometimes called a fermi after famous nuclear physicist Enrico Fermi.)

The strength of the nuclear force is about the same regardless of the type of nucleons involved, protons or neutrons. That is called “charge independence.”

More restrictively, but even more accurately, the nuclear force is the same if you swap the nucleon types. In other words, the nuclear force is the same if you replace all protons by neutrons and vice-versa. That is called “charge symmetry.” For example, the nuclear force between a pair of protons is very accurately the same as the one between a pair of neutrons. (The Coulomb force is additional and is of course not the same.) The nuclear force between a pair of protons is also approximately equal to the force between a proton and a neutron, but less accurately so. If you swap the nucleon type of a pair of protons, you get a pair of neutrons. You do not get a proton and a neutron.

The nuclear force is not a fundamental one. It is just a residual of the “color force” or “strong force” between the “quarks” of which protons and neutrons consist. That is why the nuclear force is also often called the “residual strong force.” It is much like how the Van der Waals force between molecules is not a fundamental one; that force is a residual of the electro­magnetic force between the electrons and nuclei of which molecules exist, {A.33}.

However, the theory of the color force,“quantum chrome­dynamics,” is well beyond the scope of this book. It is also not really important for nano­technology. In fact, it is not all that important for nuclear engineering either because the details of the theory are uncertain, and numerical solution is intractable, [18].

Despite the fact that the nuclear force is poorly understood, physicists can say some things with confidence. First of all,

Nuclei are normally in the ground state.

The “ground state” is the quantum state of lowest energy $E$. Nuclei can also be in “excited” states of higher energy. However, a bit of thermal energy is not going to excite a nucleus. Differen­ces between nuclear energy levels are extremely large on a microscopic scale. Still, nuclear reactions will typically leave nuclei in excited states. Usually such states decay back to the ground state very quickly. (In special cases, it may take forever.)

It should be noted that if a nuclear state is not stable, it implies that it has a slight uncertainty in energy, compare chapter 7.4.1. This uncertainty in energy is commonly called the “width” $\Gamma$ of the state. The discussion here will ignore the uncertainty in energy.

A second general property of nuclei is:

Nuclear states have definite nuclear mass $m_{\rm {N}}$.

You may be surprised by this statement. It seems trivial. You would expect that the nuclear mass is simply the sum of the masses of the protons and neutrons that make up the nucleus. But Einstein’s famous relation $E$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mc^2$ relates energy and mass. The nuclear mass is slightly less than the combined mass of the protons and neutrons. The difference is the binding energy that keeps the nucleus together, expressed in mass units. (In other words, divided by the square speed of light $c^2$.) Sure, even for nuclear energies the changes in nuclear mass due to binding energy are tiny. But physicists can measure nuclear masses to very great accuracy. Different nuclear states have different binding energies. So they have slightly different nuclear masses.

It may be noted that binding energies are almost never expressed in mass units in nuclear physics. Instead masses are expressed in energy units! And not in Joule either. The energy units used are almost invariably “electron volts” (eV). Never use an SI unit when talking to nuclear physicists. They will immediately know that you are one of those despised non­experts. Just call it a “blah.” In the unlikely case that they ask, tell them “That is what Fermi called it.”

Next,

Nuclear states have definite nuclear spin $j_{\rm {N}}$.

Here the “nuclear spin” $j_{\rm {N}}$ is the quantum number of the net nuclear angular momentum. The magnitude of the net nuclear angular momentum itself is

$$J = \sqrt{j_{\rm {N}}(j_{\rm {N}}+1)}\hbar$$

Nuclei in excited energy states usually have different angular momentum than in the ground state.

The name nuclear “spin” may seem inappro­priate since net nuclear angular momentum includes not just the spin of the nucleons but also their orbital angular momentum. But since nuclear energies are so large, in many cases nuclei act much like elementary particles do. Externally applied electro­magnetic fields are not by far strong enough to break up the inter­nal nuclear structure. And the angular momentum of an elementary particle is appro­priately called spin. However, the fact that “nuclear spin” is 2 words and “azimuthal quantum number of the nuclear angular momentum” is 8 might conceivably also have something to do with the terminology.

According to quantum mechanics, $j_{\rm {N}}$ must be integer or half-integer. In particular, it must be an integer if the number of nucleons is even. If the number of nucleons is odd, $j_{\rm {N}}$ must be half an odd integer.

The fact that nuclei have definite angular momentum does not depend on the details of the nuclear force. It is a consequence of the very fundamental observ­ation that empty space has no “build-in” preferred direction. That issue was explored in more detail in chapter 7.3.

(Many references use the symbol $J$ also for $j_{\rm {N}}$ for that spicy extra bit of confusion. So one reference tells you that the eigenvalue [singular] of $J^2$ is $J(J+1)$, leaving the $\hbar^2$ away from concise­ness. No kidding. One popular book uses $I$ instead of $j_{\rm {N}}$ and reserves $J$ for electro­nic angular momentum. At least this reference uses a bold face $I$ to indicate the angular momentum itself, as a vector.)

Finally,

Nuclear states have definite parity.

Here “parity” is what happens to the wave function when the nucleus is rotated 180$\POW9,{\circ}$ and then seen in the mirror, chapter 7.3. The wave function can either stay the same, (called parity 1 or even parity), or it can change sign, (called parity $\vphantom0\raisebox{1.5pt}{$-$}$1 or odd parity). Parity too does not depend on the details of the nuclear force. It is a consequence of the fact that the forces of nature behave the same way when seen in the mirror.

To be sure, it has been discovered that the so-called “weak force” does not behave the same when seen in the mirror. But the weak force is, like it says, weak. The chances of finding a nucleus in a given energy state with the “wrong” parity can be ballparked at 10$\POW9,{-14}$, [29, pp. 313ff]. That is almost always negligible. Only if, say, a nuclear process is strictly impossible solely because of parity, then the uncertainty in parity might give it a very slight possi­bility of occurring anyway.

Parity is commonly indicated by $\pi$. And physicists usually list the spin and parity of a nucleus together in the form $J^\pi$. If you have two quantities like spin and parity that have nothing to do with one another, what is better than show one as a super­script of the other? But do not start raising $J$ to the power $\pi$! You should be translating this into common sense as follows:

$$J^\pi \quad\Rightarrow\quad j_{\rm {N}}^{\pm} \quad\Rightarrow\quad j_{\rm {N}}\mbox{ and }\pm$$

As a numerical example, 3$\POW9,{-}$ means a nucleus with spin 3 and odd parity. It does not mean a nucleus with spin 1/3, (which is not even possible; spins can only be integer or half-integer.)

Key Points

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Nuclei form the centers of atoms.

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Nuclei consist of protons and neutrons. Therefore protons and neutrons are called nucleons.

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Protons and neutrons themselves consist of quarks. But for practical purposes, you may as well forget about that.

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Neutrons are electri­cally neutral. Protons are positively charged.

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Nuclei are held together by the so-called nuclear force.

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The nuclear force is approximately independent of whether the nucleons are protons or neutrons. That is called charge independence. Charge symmetry is a more accurate, but also more limited version of charge independence.

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Nuclear states, including the ground state, have definite nuclear energy $E$. The differen­ces in energy between nuclear states are so large that they produce small but measurable differen­ces in the nuclear mass $m_{\rm {N}}$.

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Nuclear states also have definite nuclear spin $j_{\rm {N}}$. Nuclear spin is the azimuthal quantum number of the net angular momentum of the nucleus. Many references indicate it by $J$ or $I$.

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Nuclear states have definite parity $\pi$. At least they do if the so-called weak force is ignored.

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Never use an SI unit when talking to a nuclear physicist.