“This parabola and the general behavior of the binding energy withThe semiempirical mass formula consists of five terms, which together predict the binding energy of a nucleus having atomic numberZandAcan be explained remarkably well by the semiempirical mass formula [Evans (1955, Chapter 11); Eisberg and Resnick, (1985, p. 528)].”

*Z*and mass number

*A*.

- The first term is negative, and arises from the binding caused by the short range nuclear force. It is proportional to
*A*, which implies that it increases with the volume of the nucleus (this term assumes that the nuclear density is constant; the “liquid drop model”). - The second term represents a positive correction caused by surface tension, arising because nucleons at the surface of the nucleus feel an attractive force from only one side (the nuclear interior). It is proportional to surface area, or
*A*^{2/3}. - All the positively charged protons repel each other, and this effect is accounted for by a positive term for the Coulomb energy, proportional to
*Z*^{2}/*A*^{1/3}. - Everything else being equal, nuclei tend to be more stable if they have the same number of protons and neutrons. This behavior is reflected in an asymmetry term containing (
*Z*-*A*/2)^{2}/*A*. It is zero if*A*= 2*Z*(an equal number of protons and neutrons) and is positive otherwise. - Finally, a pairing term is negative if both the number of protons and neutrons is even, positive if both are odd, and zero if one is even and the other odd.

What can this formula explain? One example is the plot of average binding energy per nucleon as a function of

*A*given in Fig. 17.3. At low

*A*, this function predicts a very low binding energy because of the surface term (very small nuclei have a large surface-to-volume ratio). As

*A*increases, the surface term becomes less important, but the Coulomb term increases as the nucleus is packed with more and more positive charge. For nuclei above about

*A*= 60, the Coulomb term causes the binding energy to decrease as

*A*increases. Therefore, the binding energy per nucleon reaches a peak for isotopes of elements such as iron and nickel, the most stable of nuclei, because of a competition between the surface and Coulomb terms. Although Russ and I did not mention it in our book, the smooth curve that most of the data cluster about in Fig. 17.3 is the prediction of the semiempirical mass formula.

If you hold

*A*constant, you can examine the binding energy as a function of

*Z*. This case is important for beta decay (in which a neutron is converted to a proton and an electron) and positron decay (in which a proton is converted to a neutron and a positron). The two terms in the semiempirical mass formula containing

*Z*—the Coulomb term and the asymmetry term—combine to give a quadratic shape for the binding energy, as shown in Fig. 17.6. For odd

*A*, the resulting parabola predicts the stable isotope (

*Z*) for that

*A*. For even

*A*, the pairing term results in two parabolas, one for even

*Z*and one for odd

*Z*(Fig. 17.7).

In their textbook, Eisberg and Resnick conclude that

“The liquid drop model is the oldest, and most classical nuclear model. At the time the semiempirical mass formula was first developed, mass data was available, but not much else was known about nuclei. The parameters were purely empirical, and there was not even a qualitative understanding of the asymmetry and pairing terms. Nevertheless, the formula was significant because it described fairly accurately the masses of hundreds of nuclei in terms of only five parameters.”

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