14.4 Overview of Nuclei

This section introduces basic terminology and concepts of nuclei. It also gives an overview of the ways that they can decay.

The number of protons in a nucleus is called its “atomic number” $Z$. Since each proton has an electric charge $e$, equal to 1.602,18 10$\POW9,{-19}$ C, the total nuclear charge is $Ze$. While protons attract nearby protons and neutrons in the nucleus with the short-range nuclear force, they also repel other protons by the long-range Coulomb force. This force too is very strong at nuclear distances. It makes nuclei with more than 82 protons unstable, because for such large nuclei the longer range of the Coulomb forces becomes a major factor.

The number of neutrons in a nucleus is its neutron number $N$. Neutrons have no charge, so they do not produce Coulomb repulsions. Therefore, the right amount of neutrons has a stabilizing effect on nuclei. However, too many neutrons is not stable either, because neutrons by themselves are unstable particles that fall apart in about 10 minutes. Combined with protons in a nucleus, neutrons can be stable.

Since neutrons have no charge, they also do not attract the electrons in the atom or molecule that the nucleus is in. Therefore only the atomic number $Z$ is of much relevance for the chemical properties of an atom. It determines the position in the periodic table of chemistry, chapter 5.9. Nuclei with the same atomic number $Z$, so with the same place in the periodic table, are called “isotopes.” (In Greek, “iso” means equal and “topos” place.)

However, the number of neutrons does have a secondary effect on the chemical properties, because it changes the mass of the nucleus. And the number of neutrons is of critical importance for the nuclear properties. To indicate the number of neutrons in a nucleus, the convention is to follow the element name by the “mass number, or “nucleon number” $A$ $\vphantom0\raisebox{1.5pt}{$=$}$ $N+Z$. It gives the total number of nucleons in the nucleus.

For example, the normal hydrogen nucleus, which consists of a lone proton, is hydrogen-1. The deuterium nucleus, which also contains a neutron, is hydrogen-2, indicating that it contains two nucleons total. Because it has the same charge as the normal hydrogen nucleus, a deuterium atom behaves chemi­cally almost the same as a normal hydrogen atom. For example, you can create water with deuterium and oxygen just like you can with normal hydrogen and oxygen. Such water is called “heavy water.” Don’t drink it, the difference in chemical properties is still sufficient to upset biological systems. Trace amounts are harmless, as can be appreciated from the fact that deuterium occurs naturally. About 1 in 6,500 hydrogen nuclei in water on earth are deuterium ones.

The normal helium nucleus contains two protons plus two neutrons, so is it called helium-4. There is a stable isotope, helium-3, that has only one neutron. In the atmosphere, one in a million helium atoms has a helium-3 nucleus. While normally, there is no big difference between the two isotopes, at very low cryogenic temperatures they do behave very differen­tly. The reason is that the helium-3 atom is a fermion while the helium-4 atom is a boson. Protons and neutrons have spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$, as do electrons, so the difference of one neutron switches the net atom spin between integer and half integer. That turns a bosonic atom into a fermionic one or vice-versa. At extremely low temperatures it makes a difference, chapter 11.

It is conven­tional to precede the element symbol by the mass number as a super­script and the atomic number as a subscript. So normal hydrogen-1 is indicated by $\fourIdx{1}{1}{}{}{\rm {H}}$, hydrogen-2 by $\fourIdx{2}{1}{}{}{\rm {H}}$, helium-3 by $\fourIdx{3}{2}{}{}{\rm {He}}$, and helium-4 by $\fourIdx{4}{2}{}{}{\rm {He}}$.

Sometimes the element symbol is also followed by the number of neutrons as a subscript. However, that then raises the question whether H$_2$ stands for hydrogen-3 or a hydrogen molecule. The neutron number can readily by found by subtracting the atomic number from the mass number,

$$\fbox{$\displaystyle N = A - Z $}$$ (14.1)

so this book will leave it out. It may also be noted that the atomic number is redundant, since the chemical symbol already implies the number of protons. It is often left away, because that confuses people who do not remember the atomic number of every chemical symbol by head.

Isotopes have the same chemical symbol and atomic number, just a different mass number. However, deuterium is often indicated by chemical symbol ${\rm {D}}$ instead of ${\rm {H}}$. It is hilarious to see people who have forgotten this search through a periodic table for element “${\rm {D}}$.” For additional fun, the unstable hydrogen-3 nucleus, with one proton and two neutrons, is also called the “tritium” nucleus, or “triton,” and indicated by ${\rm {T}}$ instead of $\fourIdx{3}{1}{}{}{\rm {H}}$. The helium-3 nucleus is also called the “helion.” Fortunately for us all, helion starts with an h.

Figure 14.1: Nuclear decay modes. [pdf]

The nuclei mentioned above are just a tiny sample of the total of 256 nuclei that are stable and a much greater number still that are not. Figure 14.1 shows the stable nuclei as green squares. The leftmost green square in the bottom row is the hydrogen-1 nucleus, and the green square immediately to the right of it is hydrogen-2, deuterium. The green squares on the second-lowest row are helium-3 and helium-4 respectively.

More generally, isotopes are found on the same horizontal line in the figure. Selected values of the atomic number $Z$ are shown as labeled horizontal lines. Similar, selected values of the neutron number $N$ are shown as labeled lines sloping down at 45 degrees. Nuclei with the same number of neutrons are called “isotones.” How clever, to replace the p in isotopes with an n.

The horizontal position of each square in figure 14.1 indicates the “neutron excess” $N-Z$. For example, hydrogen-2 and helium-4 both have neutron excess zero, they have equal numbers of protons and neutrons. So they are at the same horizontal position in the figure. Similarly, hydrogen-1 and helium-3 both have a neutron excess of minus one. The figure shows that stable light nuclei have about the same number of neutrons as protons. However, for the heaviest nuclei, there are about 50% more neutrons than protons. For heavy nuclei, too many protons would mean too much Coulomb repulsion.

Many isotopes are unstable and decay sponta­neously, liberating energy. For example, consider the blue square to the right of $\fourIdx{2}{1}{}{}{\rm {H}}$ in figure 14.1. That is $\fourIdx{3}{1}{}{}{\rm {H}}$, hydrogen-3 or tritium. It is unstable. After on average about twelve years, it will emit an electron. The electron carries away one unit of negative charge; that turns a neutron with zero net charge into a positively charged proton. So hydrogen-3, with one proton and two neutrons, changes into helium-3, with two protons and one neutron. The mass number has stayed the same but the atomic number has increased one unit. In terms of figure 14.1, the nucleus has changed into one that is one place up and two places to the left.

For historical reasons, a decay process of this type is called “beta decay” ($\beta$-decay) instead of “electron emission;” initially it was not recognized that the emitted radiation was simply electrons. And the name could not be changed later, because that would add clarity. (An anti­neutrino is also emitted, but it is almost impossible to detect: solar neutrinos will readily travel all the way through the earth with only a miniscule chance of being captured.)

Nuclei with too many neutrons tend to use beta decay to turn the excess into protons in order to become stable. Figure 14.1 shows nuclei that suffer beta decay in blue. Since in the decay process they move towards the left, they move towards the stable green area. Although not shown in the figure, a lone neutron also suffers beta decay after about 10 minutes and so turns into a proton.

If nuclei have too many protons instead, they can turn them into neutrons by emitting a positron. The positron, the anti-particle of the electron, carries away one unit of positive charge, turning a positively charged proton into a neutron.

However, a nucleus has a much easier way to get rid of one unit of net positive charge: it can swipe an electron from the atom. This is called “electron capture” (EC). Electron capture is also called K-capture of L-capture, depending on the electron shell from which the electron is swiped. It is also referred to as “inverse beta decay,” especially within the context of “neutron stars.” These stars are so massive that their atoms collapse under gravity and the electrons and protons combine into neutrons.

Of course, “inverse beta decay” is not inverse beta decay, because in beta decay the emitted electron does not go into an empty atomic orbit, and an anti­neutrino is emitted instead of a neutrino absorbed.

Positron emission is also often called “beta-plus decay” ($\beta^+$-decay). After all, if you do have obsolete terminology, it is fun to use it to the most. Note that NUBASE 2003 uses the term beta-plus decay to indicate either positron emission or electron capture. In analogy with the beta-plus terminology, electron emission is also commonly called beta-minus decay or negatron emission. Some physicists leave the “r” away to save trees and talk about positons and negatons.

The nuclei that suffer beta-plus decay or electron capture are shown as red squares in figure 14.1. In the decay, a proton turns into a neutron, so the nucleus moves one place down and two places towards the right. That means that these nuclei too move towards the stable green area.

In either beta-minus or beta-plus decay, the mass number $A$ does not change. Nuclei with the same mass number are called “isobars.” Yes, this conflicts with the established usage of the word isobar for lines of constant pressure in meteorology, but in this case physicists have blown it. There is not likely to be any resulting confusion unless there is a nuclear winter.

There are a variety of other ways in which nuclei may decay. As shown in figure 14.1, if the number of protons or neutrons is really excessive, the nucleus may just kick the bums out instead of convert them.

Similarly, heavy nuclei that are weakened by Coulomb repulsions tend to just throw some nucleons out. Commonly, a $\fourIdx{4}{2}{}{}{\rm {He}}$ helium-4 nucleus is emitted, as this is a very stable nucleus that does not require much energy to create. Such an emission is called “alpha decay” ($\alpha$-decay) because helium-4 emission would be easily under­standable. Alpha decay reduces the mass number $A$ by 4 and the atomic number $Z$ by 2. The nucleus moves two places straight down in figure 14.1.

If nuclei are really oversized, they may just disinte­grate completely; that is called sponta­neous fission.

Another process, “gamma decay,” is much like sponta­neous decay of excited electron levels in atoms. In gamma decay an excited nucleus transitions to a lower energy state and emits the released energy as very energetic electro­magnetic radiation. The nucleus remains of the same type: there is no change in the number of protons or neutrons. Nuclear emissions are commonly associated with additional gamma radiation, since the emission tends to leave the nucleus in an excited state. Gamma decay as a separate process is often referred to as an “isomeric transition” (IT) or “inter­nal transition.”

A second way that a nucleus can get rid of excess energy is by throwing an electron from the atomic electron cloud surrounding the nucleus out of the atom. You or I would probably call that something like electron ejection. But what better name for throwing an electron that is not part of the nucleus completely out of the atom than “inter­nal conversion” (IC)? Inter­nal conversion is usually included in the term isomeric transition.

Figure 14.1 mixes colors if more than one decay mode occurs for a nucleus. The dominant decay is often immediately followed by another decay process. The subsequent decay is not shown. Data are from NUBASE 2003, without any later updates. The blank square right at the stable region is silver 106, and has a half-life of 24 minutes. Other sources list it as decaying through the expected electron capture or positron emission. But NUBASE lists that contribution as unknown and only mentions that beta-minus decay is negligible.

Table 14.2: Alternate names for nuclei.

Since so many outsiders know what nuclear symbols mean, physicists prefer to use obsolete names to confuse them. Table 14.2 has a list of names used. The abbrevi­ations refer to historical names for decay products of radium (radium emanation, radium A, etc.)

If you look closer at which nuclei are stable or not, it is seen that stability tends to be enhanced if the number of protons and/or neutrons is even and reduced if it is odd. Physicists therefore speak of even-even nuclei, even-odd nuclei, etcetera. Note that if the mass number $A$ is odd, the nucleus is either even-odd or odd-even. If the mass number is even, the nucleus is either even-even or odd-odd. Odd mass number nuclei tend to be easier to analyze.

Key Points

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Nuclei consist of protons and neutrons held together by the nuclear force.

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Protons and neutrons are collectively referred to as nucleons.

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Protons also repel each other by the Coulomb force.

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The number of protons in a nucleus is the atomic number $Z$. The number of neutrons is the neutron number $N$. The total number of nucleons $Z+N$ is the mass number or nucleon number $A$.

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Nuclei with the same number of protons corre­spond to atoms with the same place in the periodic table of chemistry. Therefore nuclei with the same atomic number are called isotopes.

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To promote confusion, nuclei with the same number of neutrons are called isotones, and nuclei with the same total number of nucleons are called isobars.

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For an example notation, consider $\fourIdx{4}{2}{}{}{\rm {He}}$. It indicates a helium atom nucleus consisting of 4 nucleons, the left super­script, of which 2 are protons, the left subscript. Since it would not be helium if it did not have 2 protons, that subscript is often left away. This nucleus is called helium-4, where the 4 is again the number of nucleons.

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Nuclei can decay by various mechanisms. To promote confusion, emission of helium-4 nuclei is called alpha decay. Emission of electrons is called beta decay, or $\beta$ decay, or beta-minus decay, or $\beta^-$ decay, or negatron emission, or negaton emission, but never electron emission. Emission of positrons (positons) may be called beta-plus decay, or $\beta^+$ decay, but beta-plus decay might be used to also indicate electron capture, depending on who used the term. Electron capture may also be called K-capture or L-capture or even inverse beta decay, though it is not. Emission of electro­magnetic radiation is called gamma decay or $\gamma$ decay. More extreme decay mechanisms are proton or neutron emission, and sponta­neous fission. Kicking an electron in the electron cloud outside the nucleus completely free of the atom is called inter­nal conversion.

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No, this is not a story made up by this book to put physicists in a bad light.

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Odd mass numbers corre­spond to even-odd or odd-even nuclei. Even mass numbers corre­spond to either even-even nuclei, which tend to have relatively high stability, or to odd-odd ones, which tend to have relatively low stability.