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What patterns and insights are gained from an examination of the binding energy of various nuclides? First, we find that BE is approximately proportional to the number of nucleons A in any nucleus. About twice as much energy is needed to pull apart a nucleus like 24 Mg compared with pulling apart 12 C , for example. To help us look at other effects, we divide BE by A size 12{A} {} and consider the binding energy per nucleon    , BE / A size 12{ {"BE"} slash {A} } {} . The graph of BE / A size 12{ {"BE"} slash {A} } {} in [link] reveals some very interesting aspects of nuclei. We see that the binding energy per nucleon averages about 8 MeV, but is lower for both the lightest and heaviest nuclei. This overall trend, in which nuclei with A size 12{A} {} equal to about 60 have the greatest BE / A size 12{ {"BE"} slash {A} } {} and are thus the most tightly bound, is due to the combined characteristics of the attractive nuclear forces and the repulsive Coulomb force. It is especially important to note two things—the strong nuclear force is about 100 times stronger than the Coulomb force, and the nuclear forces are shorter in range compared to the Coulomb force. So, for low-mass nuclei, the nuclear attraction dominates and each added nucleon forms bonds with all others, causing progressively heavier nuclei to have progressively greater values of BE / A size 12{ {"BE"} slash {A} } {} . This continues up to A 60 size 12{A approx "60"} {} , roughly corresponding to the mass number of iron. Beyond that, new nucleons added to a nucleus will be too far from some others to feel their nuclear attraction. Added protons, however, feel the repulsion of all other protons, since the Coulomb force is longer in range. Coulomb repulsion grows for progressively heavier nuclei, but nuclear attraction remains about the same, and so BE / A size 12{ {"BE"} slash {A} } {} becomes smaller. This is why stable nuclei heavier than A 40 size 12{A approx "40"} {} have more neutrons than protons. Coulomb repulsion is reduced by having more neutrons to keep the protons farther apart (see [link] ).

The figure shows a graph of binding energy per nucleon versus atomic mass for different elements. From the graph it can be observed that elements with atomic mass near sixty have greater binding energy per nucleon.
A graph of average binding energy per nucleon, BE / A size 12{ {"BE"} slash {A} } {} , for stable nuclei. The most tightly bound nuclei are those with A size 12{A} {} near 60, where the attractive nuclear force has its greatest effect. At higher A size 12{A} {} s, the Coulomb repulsion progressively reduces the binding energy per nucleon, because the nuclear force is short ranged. The spikes on the curve are very tightly bound nuclides and indicate shell closures.

The image shows a bunch of spherical nucleons inside a nucleus. A circular dashed path is shown which depicts the range of nuclear force and the nucleons inside that range feel nuclear force directly.
The nuclear force is attractive and stronger than the Coulomb force, but it is short ranged. In low-mass nuclei, each nucleon feels the nuclear attraction of all others. In larger nuclei, the range of the nuclear force, shown for a single nucleon, is smaller than the size of the nucleus, but the Coulomb repulsion from all protons reaches all others. If the nucleus is large enough, the Coulomb repulsion can add to overcome the nuclear attraction.

There are some noticeable spikes on the BE / A size 12{ {"BE"} slash {A} } {} graph, which represent particularly tightly bound nuclei. These spikes reveal further details of nuclear forces, such as confirming that closed-shell nuclei (those with magic numbers of protons or neutrons or both) are more tightly bound. The spikes also indicate that some nuclei with even numbers for Z size 12{Z} {} and N size 12{N} {} , and with Z = N size 12{Z=N} {} , are exceptionally tightly bound. This finding can be correlated with some of the cosmic abundances of the elements. The most common elements in the universe, as determined by observations of atomic spectra from outer space, are hydrogen, followed by 4 He , with much smaller amounts of 12 C and other elements. It should be noted that the heavier elements are created in supernova explosions, while the lighter ones are produced by nuclear fusion during the normal life cycles of stars, as will be discussed in subsequent chapters. The most common elements have the most tightly bound nuclei. It is also no accident that one of the most tightly bound light nuclei is 4 He , emitted in α decay.

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Source:  OpenStax, Basic physics for medical imaging. OpenStax CNX. Feb 17, 2014 Download for free at http://legacy.cnx.org/content/col11630/1.1
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