21.2 Patterns of Nuclear Stability

The stability of a particular nucleus depends on a variety of factors, and no single rule allows us to predict whether a particular nucleus is radioactive and how it might decay. There are, however, several empirical observations that are helpful in making predictions.

Neutron-to-Proton Ratio

Because like charges repel each other, it may seem surprising that a large number of protons can reside within the small volume of the nucleus. At close distances, however, a strong force of attraction, called the strong nuclear force, exists between nucleons. Neutrons are intimately involved in this attractive force. All nuclei with two or more protons contain neutrons. The more protons packed in the nucleus, the more neutrons are needed to bind the nucleus together. Stable nuclei with low atomic numbers (up to about 20) have approximately equal numbers of neutrons and protons. For nuclei with higher atomic numbers, the number of neutrons exceeds the number of protons. Indeed, the number of neutrons necessary to create a stable nucleus increases more rapidly than the number of protons, as shown in Figure 21.2. Thus, the neutron-to-proton ratios of stable nuclei increase with increasing atomic number.

Figure 21.2 Plot of the number of neutrons versus the number of protons in stable nuclei. As the atomic number increases, the neutron-to-proton ratio of the stable nuclei increases. The stable nuclei are located in the shaded area of the graph known as the belt of stability. The majority of radioactive nuclei occur outside this belt.

The colored band in Figure 21.2 is the area within which all stable nuclei are found and is known as the belt of stability. The belt of stability ends at element 83 (bismuth). All nuclei with 84 or more protons (atomic number 84) are radioactive. For example, all isotopes of uranium, atomic number 92, are radioactive.

The type of radioactive decay that a particular radionuclide undergoes depends to a large extent on its neutron-to-proton ratio compared to those of nearby nuclei within the belt of stability. We can envision three general situations:

  1. Nuclei above the belt of stability (high neutron-to-proton ratios): These neutron-rich nuclei can lower their ratio and move toward the belt of stability by emitting a beta particle. Beta emission decreases the number of neutrons and increases the number of protons in a nucleus, as shown in Equation 21.3.
  2. Nuclei below the belt of stability (low neutron-to-proton ratios): These proton-rich nuclei can increase their ratio by either positron emission or electron capture. Both kinds of decay increase the number of neutrons and decrease the number of protons, as shown in Equations 21.5 and 21.7. Positron emission is more common than electron capture among the lighter nuclei; however, electron capture becomes increasingly common as nuclear charge increases.
  3. Nuclei with atomic numbers 84: These heavy nuclei, which lie beyond the upper right edge of the band of stability, tend to undergo alpha emission. Emission of an alpha particle decreases both the number of neutrons and the number of protons by 2, moving the nucleus diagonally toward the belt of stability.

These three situations are summarized in Figure 21.3.

Figure 21.3 Results of alpha emission PM21031, beta emission PM21032, positron emission PM21033 and electron capture on the number of protons and neutrons in a nucleus. Moving from left to right or from bottom to top, each square represents an additional proton or neutron, respectively. Moving in the reverse direction indicates the loss of a proton or neutron.


Predict the mode of decay of (a) carbon-14; (b) xenon-118.

SOLUTION (a) Carbon has an atomic number of 6. Thus, carbon-14 has 6 protons and 14 - 6 = 8 neutrons, giving it a neutron-to-proton ratio of PM21034 Elements with low atomic numbers normally have stable nuclei with approximately equal numbers of neutrons and protons. Thus, carbon-14 has a high neutron-to-proton ratio, and we expect that it will decay by emitting a beta particle:


This is indeed the mode of decay observed for carbon-14.

(b) Xenon has an atomic number of 54. Thus, xenon-118 has 54 protons and 118 - 54 = 64 neutrons, giving it a neutron-to-proton ratio of PM21036 By examining Figure 21.2, we see that stable nuclei in this region of the belt of stability have higher neutron-to-proton ratios than xenon-118. The nucleus can increase this ratio by either positron emission or electron capture:



In this case both modes of decay are observed.


Predict the mode of decay of (a) plutonium-239; (b) indium-120. Answers: (a) decay; (b) decay

At this point we should note that our guidelines don't always work. For example, thorium-233, PM21039 which we might expect to undergo alpha decay, actually undergoes beta decay. Furthermore, a few radioactive nuclei actually lie within the belt of stability. For example, both PM21040 and PM21041 are stable and lie in the belt of stability; however, PM21042, which lies between them, is radioactive.

Radioactive Series

Some nuclei, like uranium-238, cannot gain stability by a single emission. Consequently, a series of successive emissions occurs. As shown in Figure 21.4, uranium-238 decays to thorium-234, which is radioactive and decays to protactinium-234. This nucleus is also unstable and subsequently decays. Such successive reactions continue until a stable nucleus, lead-206, is formed. A series of nuclear reactions that begins with an unstable nucleus and terminates with a stable one is known as a radioactive series or a nuclear disintegration series. Three such series occur in nature. In addition to the series that begins with uranium-238 and terminates with lead-206, there is one that begins with uranium-235 and ends with lead-207. The third series begins with thorium-232 and ends with lead-208.

Figure 21.4 Nuclear disintegration series for uranium-238. The PM21043 nucleus decays to PM21044. Subsequent decay processes eventually form the stable PM21045 nucleus. Each blue arrow corresponds to the loss of an alpha particle; each red arrow corresponds to the loss of a beta particle.

Further Observations

Two further observations are useful in predicting nuclear stability:

These observations can be understood in terms of the shell model of the nucleus, in which nucleons are described as residing in shells analogous to the shell structure for electrons in atoms. Just as certain numbers of electrons (2, 8, 18, 36, 54, and 86) correspond to stable closed-shell electron configurations, so also the magic numbers of nucleons represent closed shells in nuclei. As an example of the stability of nuclei with magic numbers of nucleons, note that the radioactive series depicted in Figure 21.4 ends with formation of the stable PM21046 nucleus, which has a magic number of protons (82).

Evidence also suggests that pairs of protons and pairs of neutrons have a special stability, analogous to the pairs of electrons in molecules. Thus, stable nuclei with an even number of protons and an even number of neutrons are far more numerous that those with odd numbers (Table 21.3).


Which of the following nuclei are especially stable: PM21047

SOLUTION The PM21048 nucleus (the alpha particle) has a magic number of both protons (2) and neutrons (2) and is very stable. The PM21049 nucleus also has a magic number of both protons (20) and neutrons (20) and is especially stable.

The PM21050 nucleus does not have a magic number of either protons or neutrons. In fact, it has an odd number of both protons (43) and neutrons (55). There are very few stable nuclei with odd numbers of both protons and neutrons. Indeed, technetium-98 is radioactive.


Which of the following nuclei would you expect to exhibit a special stability: PM21051 Answer: PM21052