物理代写|粒子物理代写particle physics代考|PHYS575

物理代写|粒子物理代写particle physics代考|Stability of Isotopes

Since different isotopes have different atomic mass numbers, they will have different binding energies and some isotopes will be more stable than others. It turns out (and can be seen by looking for the most stable isotopes using the semi-empirical mass formula) that for the lighter nuclei, the stable isotopes have approximately the same number of neutrons as protons. This arises as a result of the asymmetry term.

However, above $A \sim 20$ the number of neutrons required for stability increases up to about one and a half times the number of protons for the heaviest nuclei (Fig. 3.4).
Qualitatively, the reason for this arises from the Coulomb term. Protons bind less tightly than neutrons because they have to overcome the Coulomb repulsion between them. It is therefore energetically favourable to have more neutrons than protons. For nuclides with a large atomic mass number, this Coulomb effect beats the asymmetry effect that favours equal numbers of protons and neutrons. Quantitatively, this phenomenon can be seen from (3.10), where for small $A$ the maximum binding energy occurs for $Z \approx \frac{1}{2} A$, i.e. the same number of protons and neutrons, but as $A$ increases the maximum binding energy occurs for $Z<\frac{1}{2} A$, i.e. nuclides with more neutrons than protons.

物理代写|粒子物理代写particle physics代考|Magic Numbers

It turns out that nuclei have a periodic nature analogous to the periodic nature of atoms described by the Mendeleev periodic table of chemical elements. This feature of nuclei is not described by the semi-empirical mass formula. In particular, it was found that the binding energies predicted by that formula underestimate the actual binding energies of “magic nuclides” for which either the number of neutrons, $N=$ $(A-Z)$, or the number of protons, $Z$, is equal to one of the following “magic numbers”:
$2,8,20,28,50,82,126,184$
This effect is especially pronounced for the case of “doubly magic” nuclides in which both the number of neutrons and the number of protons are equal to magic numbers, for example, for ${ }2^4 \mathrm{He},{ }_8^{16} \mathrm{O},{ }{20}^{40} \mathrm{Ca},{ }{82}^{208} \mathrm{~Pb}$. For ${ }_2^4 \mathrm{He}$ the semi-empirical mass formula predicts a binding energy of $21.69 \mathrm{MeV}$, whilst the measured value is $28.30 \mathrm{MeV}$ (underestimated by about $30 \%$ ), for ${ }_8^{16} \mathrm{O}$ (oxygen) these values are $123.18$ and $127.62 \mathrm{MeV}$, respectively (underestimated by about $3 \%$, for ${ }{20}^{40} \mathrm{Ca}$

(calcium) they are $338.90$ and $342.05 \mathrm{MeV}$ (underestimated by about $1 \%$ ) and for ${ }_{82}^{208} \mathrm{~Pb}$ (lead) they are $1612.2$ and $1636.4 \mathrm{MeV}$ (underestimated by about $1.5 \%$ ).
Magic nuclides have special features related to their binding energy properties, such as: The neutron (or proton) separation energies (the energy required to remove the last neutron (or proton)) peak if $N(Z)$ is equal to a magic number. For example, Fig. $4.1$ shows the neutron separation energy for isotopes of ${ }_{56}^{\mathrm{A}} \mathrm{Ba}$ (barium) as a function of neutron number, $N$. We can see a clear step in the binding energy as the number of neutrons crosses the magic number 82 .

There are more stable isotopes if $Z$ is a magic number and more stable isotones if $N$ is a magic number.

If $N$ is a magic number, then the cross section for neutron capture is much lower (by a factor of 10-100) than for other nuclides as demonstrated in Fig. 4.2.

The energies of the excited states are much higher than the ground state if either $N$ or $Z$ or both are magic numbers. As an example, in Fig. $4.3$ we present the values of the excitation energies for various isotopes of ${ }{82}^{{ }^{\mathrm{A}}} \mathrm{Pb}$, where we can see that the magic nuclide ${ }{82}^{126} \mathrm{~Pb}$ has an excitation energy of about a factor of 3 higher than the other isotopes.

Elements with $Z$ or/and $N$ equal to a magic number have a larger natural abundance than those of nearby elements or isotopes with even values of $Z$ or $N$. Let us take a look, for example, at ${ }{20}^{40} \mathrm{Ca}$. Actually, it makes sense to compare nuclides which differ from magic ones in $N$ or $Z$ by even number, since such nuclides are more stable and therefore have larger natural abundance. One should also note that ${ }{20}^{40} \mathrm{Ca}$ is the heaviest stable isotope with $Z=N$. Its abundance amongst other isotopes of $\mathrm{Ca}$ is about $97 \%$. The previous nuclide with $Z=N$ (not equal to a magic number) is ${ }_{18}^{36} \mathrm{Ar}$ (argon), which has an abundance of only $0.34 \%$ amongst other isotopes of Ar, whilst the $Z=N$ isotope with $Z=22$, ${ }^{44} \mathrm{Ti}$ (titanium) is totally absent.

物理代写|粒子物理代写particle physics代考|Magic Numbers

2,8,20,28,50,82,126,184

（钙）它们是338.90和342.05米和在（低估了大约1%) 并且对于82208 磷b（铅）他们是1612.2和1636.4米和在（低估了大约1.5%）。

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