14.3 Draft: 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”

. Since each proton has an electric charge

, equal to 1.602 18 10 $\POW9,{-19}$ C, the total nuclear charge is

. 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

. 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

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

, 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. Nuclei with the same number of neutrons are called “isotones.” How clever, to replace the p in isotopes with an n.

The name of a nucleus indicates its number of protons

; for example, hydrogen” means

, “helium

. To also indicate the number of neutrons, the convention is to follow the name by the “mass number, or “nucleon number”

$\vphantom0\raisebox{1.5pt}{$=$}$

. 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 contains both a proton and 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, however; 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 it is 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 both protons and neutrons have spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ , as do electrons, so a difference of one neutron switches the net atomic spin between half-integer (helium-3) and integer (helium-4). That makes the helium-3 atom a fermion but the helium-4 one a boson. At extremely low temperatures it makes a big difference in behavior, chapter 11.

In terms of symbols, 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 ${}\fourIdx{4}{2}{}{}{\rm {He}}$ .

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.

Sometimes the element symbol is also followed by the number of neutrons as a subscript. However, that then raises the question whether H

stands for hydrogen-3 or a hydrogen molecule. The neutron number can readily by found by subtracting the atomic number from the mass number,

$\begin{displaymath} \fbox{$\displaystyle N = A - Z $} \end{displaymath}$

(14.1)

It may also be noted that the atomic number is technically 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 in the periodic table. To create further confusion, 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:** Chart of the nuclides.
$\begin{figure}\centering \setlength{\unitlength}{1pt} \resizebox{405pt}{!}{ ... ...{X: given data less than 50\%}}} \end{picture}}}} \end{picture}} \end{figure}$

**Figure 14.2:** Nuclear decay modes. [pdf][con]
$\begin{figure}\centering \setlength{\unitlength}{1pt} \begin{picture}(405,56... ...{\makebox(0,0)[l]{$\fourIdx{4}{2}{}{}{\rm He}$}} } \end{picture} \end{figure}$

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 observed but unstable. It is conventional to represent both the stable and unstable nuclei in a Chart Of the Nuclides, (CON), like the one shown in figure 14.1. In the CON, the tiny green squares are the stable nuclei. Squares of colors other than green represent unstable nuclei. The horizontal position of each square gives the number of neutrons

(selected magic values are listed along the horizontal axis). The vertical position gives the number of protons

, (the crucial number that determines what chemical properties an atom with that nucleus has). Note that

is lead, the last element with at least one stable nucleus. In fact, lead has four stable isotopes and naturally occuring lead atoms have a fair chance of having any one of the four as nucleus. The fact that lead is so unusually stable has a lot to do with the fact that lead is right on top of the

magic line, and close to the

magic line. But more on that later.

Before continuing the discussion of the CON, a graphical problem must be addressed. While you cannot argue about taste, 99.9% of readers would surely agree that 14.1 is (a) ugly as hell, and (b) requires a magnifying glass to read. The Chart Of the Nuclides is well suited for printing out on two yards of paper and hanging on the wall of your office in its full glory. But in a book, it really does not work. It could be made slightly bigger if printed out sideways, but rotating a monitor with coffee cups on it and cables attached is a bit awkward. And a crick in your neck is not that great either.

Based on these considerations, from now on, this book will no longer plot the neutron number

along the horizontal axis, but the “neutron excess”

. The neutron excess is how many more neutrons there are than protons. That is an important number, maybe even more important than the absolute number of neutrons. The corresponding modified chart of the nuclides is in figure 14.2. It will be called the RECON (Revised Chart of the Nuclei). It gives you something you can view in comfort.

But admittedly there are some disadvantages. In the RECON the isotones (the lines connecting nuclei with the same number of neutrons) are no longer vertical; now they slope down by 45 $\POW9,{\circ}$ . That cannot be helped. Similarly the isobars, the lines connecting nuclei with the same total number of nucleons, no longer slope down by 45 $\POW9,{\circ}$ like in the CON. In the RECON they slope down with the smaller slope 1/2: going down one square to the previous chemical element now requires that you go two squares to the right to stay on the same isobar.

So be it. The good news is that the neutron excess is a lot more relevant to nuclear stability than the absolute number of nucleons. For example, at low values of

, the green band in RECON figure 14.2 is vertical, demonstrating very clearly that indeed for light nuclei, the number of neutrons must be about equal to the number of protons. Also for heavier nuclei, the RECON shows much more clearly exactly how much the relative number of neutrons goes up to mitigate the effect of the Coulomb repulsions between protons.

There is an other advantage to the RECON. It has to do with the fact that those nuclei in which both the number of protons and the number of neutrons is even, the even-even nuclei, are found to have enhanced stability. On the other hand those nuclei in which both the number of protons and the number of neutrons is odd, the odd-odd nuclei, are found to have reduced stability. Simply put, protons like to pair up, and so do neutrons.

It works out that in the RECON, the even-even and odd-odd nuclei end up on the same vertical lines. (These vertical lines alternate with vertical lines of even-odd and odd-even nuclei.) So wherever in figure 14.2 you see a vertical line with alternating green and non-green squares, well, the stable green squares are the even-even nuclei and the non-green ones in between the odd-odd ones. The pattern very convincingly demonstrates that indeed even-even nuclei are a lot more stable than odd-odd ones. (In the CON, the equivalent lines slant by 45 $\POW9,{\circ}$ and are not by far as striking.)

In the intermediate vertical lines in the RECON, where you do not see such a periodic variation of stability, you find the even-odd and odd-even nuclei. Note that on these lines, the vertical extent of green squares is much less than on the adjacent lines with even-even nuclei. This demonstrates graphically that even-even nuclei are not just a lot more stable than odd-odd ones; they are also a lot more stable than even-odd and odd-even ones.

All this also makes it easy to figure out whether a given nucleus is is even-even or odd-odd in the RECON. Look whether the vertical line it is on has a series of alternating green and non-green squares; if so, then that is a line of even-even and odd-odd nuclei. The adjacent two lines then contain even-odd and odd-even nuclei. (In the region of heaviest nuclei, you can typically look at the yellow alpha-decay nuclei as a substitute for the green nuclei.) Alternatively, if you see two green squares immediately above each other in the RECON above

, then that vertical line consists of even-odd and odd-even nuclei; there are no stable odd-odd nuclei above

. (Conversely, the RECON illustrates that quite clearly too.)

Note also that the mass number

is odd on the even-odd, odd-even vertical lines. And

is even on the even-even, odd-odd vertical lines. While the mass number by iself does not have that much physical meaning, nuclear physicists often use “odd mass number nuclei” as a shorthand for “even-odd and odd-even nuclei.”

If you are really a CON man or woman, there is nothing wrong with that. You can always click on the [con] link provided in the legend of the figure to load the figure in CON format as a separate pdf file. Conversely, if you really like the RECON format and you want to print it and hang it on your wall, click on the [pdf] link instead for a printable version. And either type of pdf can be readily magnified to see details more clearly.

Let's look at some of the details of RECON figure 14.2. 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.

Like in the CON, isotopes are found on the same horizontal line in the RECON. As mentioned, the horizontal position of each square in RECON figure 14.2 indicates the neutron excess. For example, hydrogen-2 and helium-4 both have equal numbers of protons and neutrons. So they are at the same horizontal position, zero, 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 closely packed 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 14.2. That is ${}\fourIdx{3}{1}{}{}{\rm {H}}$ , hydrogen-3 or tritium. It is unstable. After on average about twelve years, it will turn into helium-3. In particular, one of the two neutrons turns into a positively charged proton. So there are still three nucleons, the mass number has stayed the same, but the atomic number has increased one unit. In terms of RECON figure 14.2, the nucleus has changed into one that is one place up and two places to the left.

Since charge is conserved, the creation of the positive charge can only happen if the neutron emits a compensating negative charge; the neutron does so by emitting an electron. 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 observed radiation was merely high energy 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 14.2 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 rather than too many neutrons, they can turn their excess protons 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 neutral neutron.

However, a nucleus has a much easier way to get rid of one unit of net positive charge: it can swipe a negatively charged electron from the atom it is in. This is called electron capture (EC). An electron neutrino is emitted in this process.

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. These stars then emit enormous amounts of high-energy neutrinos, taking along a large amount of the available energy of the star.

Of course, inverse beta decay is not really inverse beta decay, because in beta decay the emitted electron does not go into an empty atomic orbit, and in beta decay no neutrino is absorbed; instead an antineutrino is emitted.

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 fullest. 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 14.2. 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.

There are a variety of other ways in which nuclei may decay. If the number of protons or neutrons is really excessive, the nucleus may just kick out one of the bums instead of convert it. Nuclei that do that are marked with P,” respectively “N in figure 14.2,

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

by 4 and the atomic number

by 2. The nucleus moves two places straight down in RECON figure 14.2.

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

Another process, “gamma decay,” is not shown in figure 14.2. In gamma decay, an excited nucleus transitions to a lower energy state and emits the released energy as very energetic electromagnetic radiation. This is much like the spontaneous decay of excited electron levels in atoms, which too releases electromagnetic radiation. However, the electromagnetic radiation emitted in gamma decay is much more powerful than that emitted by atomic electrons, as nuclear energies are so much higher than those of atomic electrons. Unlike the decays shown in figure 14.2, in gamma decay the type of nucleus does not change; there is no change in the number of protons nor neutrons.

Unlike gamma decay, the nuclear decays shown in figure 14.2 are from their their non-excited “ground state.” But the shown decays are commonly associated with additional gamma radiation, since the decay tends to leave the changed nucleus in an excited state.

Gamma decay as a separate process, not directly caused by another process, is often referred to as an “isomeric transition” (IT) or “internal transition.” In nuclear physics, an isomer is a long-lived excited state of a nucleus.

Besides gamma decay, a second way that an excited 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 already outside the nucleus to start with, completely out of the atom than internal conversion (IC)? It can produce some of that hilarious confusion with the similar sounding term internal transition. Internal conversion is usually included in the term isomeric transition.

Figure 14.2 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 2003 lists that contribution as unknown and only mentions that beta-minus decay is negligible.

**Table 14.2:** Alternate names for nuclei.
$\begin{table}\begin{displaymath} \begin{array}{ccccccccc} \hline\hline \mbox{... ...0}{84}{}{}{\rm Po} \\ \hline\hline \end{array} \end{displaymath} \end{table}$

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.)

14.3 Draft: Overview of Nu­clei

14.3 Draft: Overview of Nuclei