31.6 Binding energy  (Page 3/6)

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What is $\text{BE}/A$ For an alpha particle?

Calculate the binding energy per nucleon of ${}^{4}\text{He}$ , the $\alpha$ particle.

Strategy

To find $\text{BE}/A$ , we first find BE using the Equation $\text{BE}=\left\{\left[\text{Zm}\left({}^{1}\text{H}\right)+{\text{Nm}}_{n}\right]-m\left({}^{A}\text{X}\right)\right\}{c}^{2}$ and then divide by $A$ . This is straightforward once we have looked up the appropriate atomic masses in Appendix A .

Solution

The binding energy for a nucleus is given by the equation

$\text{BE}=\left\{\left[\text{Zm}\left({}^{1}\text{H}\right)+{\text{Nm}}_{n}\right]-m\left({}^{A}\text{X}\right)\right\}{c}^{2}\text{.}$

For ${}^{4}\text{He}$ , we have $Z=N=2$ ; thus,

$\text{BE}=\left\{\left[2m\left({}^{1}\text{H}\right)+{2m}_{n}\right]-m\left({}^{4}\text{He}\right)\right\}{c}^{2}\text{.}$

Appendix A gives these masses as $m\left({}^{4}\text{He}\right)=\text{4.002602 u}$ , $m\left({}^{1}\text{H}\right)=\text{1.007825 u}$ , and ${m}_{n}=\text{1.008665 u}$ . Thus,

$\text{BE}=\left(0\text{.}\text{030378 u}\right){c}^{2}\text{.}$

Noting that $\text{1 u}=\text{931}\text{.}\text{5 MeV/}{c}^{2}$ , we find

$\text{BE}=\left(\text{0.030378}\right)\left(\text{931}\text{.}\text{5 MeV/}{c}^{2}\right){c}^{2}=28.3 MeV\text{.}$

Since $A=4$ , we see that $\text{BE}/A$ is this number divided by 4, or

$\text{BE}/A=\text{7.07 MeV/nucleon}\text{.}$

Discussion

This is a large binding energy per nucleon compared with those for other low-mass nuclei, which have $\text{BE}/A\approx \text{3 MeV/nucleon}$ . This indicates that ${}^{4}\text{He}$ is tightly bound compared with its neighbors on the chart of the nuclides. You can see the spike representing this value of $\text{BE}/A$ for ${}^{4}\text{He}$ on the graph in [link] . This is why ${}^{4}\text{He}$ is stable. Since ${}^{4}\text{He}$ is tightly bound, it has less mass than other $A=4$ nuclei and, therefore, cannot spontaneously decay into them. The large binding energy also helps to explain why some nuclei undergo $\alpha$ decay. Smaller mass in the decay products can mean energy release, and such decays can be spontaneous. Further, it can happen that two protons and two neutrons in a nucleus can randomly find themselves together, experience the exceptionally large nuclear force that binds this combination, and act as a ${}^{4}\text{He}$ unit within the nucleus, at least for a while. In some cases, the ${}^{4}\text{He}$ escapes, and $\alpha$ decay has then taken place.

There is more to be learned from nuclear binding energies. The general trend in $\text{BE}/A$ is fundamental to energy production in stars, and to fusion and fission energy sources on Earth, for example. This is one of the applications of nuclear physics covered in Medical Applications of Nuclear Physics . The abundance of elements on Earth, in stars, and in the universe as a whole is related to the binding energy of nuclei and has implications for the continued expansion of the universe.

For reaction and binding energies and activity calculations in nuclear physics

1. Identify exactly what needs to be determined in the problem (identify the unknowns) . This will allow you to decide whether the energy of a decay or nuclear reaction is involved, for example, or whether the problem is primarily concerned with activity (rate of decay).
2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
3. For reaction and binding-energy problems, we use atomic rather than nuclear masses. Since the masses of neutral atoms are used, you must count the number of electrons involved. If these do not balance (such as in ${\beta }^{+}$ decay), then an energy adjustment of 0.511 MeV per electron must be made. Also note that atomic masses may not be given in a problem; they can be found in tables.
4. For problems involving activity, the relationship of activity to half-life, and the number of nuclei given in the equation $R=\frac{\text{0.693}N}{{t}_{1/2}}$ can be very useful. Owing to the fact that number of nuclei is involved, you will also need to be familiar with moles and Avogadro’s number.
5. Perform the desired calculation; keep careful track of plus and minus signs as well as powers of 10.
6. Check the answer to see if it is reasonable: Does it make sense? Compare your results with worked examples and other information in the text. (Heeding the advice in Step 5 will also help you to be certain of your result.) You must understand the problem conceptually to be able to determine whether the numerical result is reasonable.

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