## 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 ${ }2^4 \mathrm{He},{ }_8^{16} \mathrm{O},{ }{20}^{40} \mathrm{Ca},{ }{82}^{308} \mathrm{~Pb}$. 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 ${ }{8 \lambda}^{\mathrm{A}} \mathrm{Pb}$, where we can see that the magic nuclide ${ }{82}^{126} \mathrm{Ph}$ has an excitation energy of about a factor of 3 higher than the other isotopes.
• Elements with $\mathrm{Z}$ 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 $\mathrm{Ar}$, whilst the $Z=N$ isotope with $Z=22$, ${ }_{22}^{44} \mathrm{Ti}$ (titanium) is totally absent.
• Elements with magic numbers of protons have zero electric quadrupole moment (their charge distribution is spherically symmetric), reflecting their particular stability.

## 物理代写|粒子物理代写Particle Physics代考|Shell Model

The periodicity of the properties of nuclei observed in terms of magic numbers, which is similar to the periodicity of the properties of atoms, gives us a hint about the shell structure of nuclei in analogy to the shell structure of atoms. Indeed magic numbers can be explained in terms of the Shell Model of the nucleus, which considers each nucleon moving in some potential and classifies the energy levels in terms of quantum numbers, $n, \ell$ and $j$, in the same way as the wavefunctions of individual electrons are classified in Atomic Physics. However, contrary to the electromagnetic potential in atoms, the potential in nuclei arises from the strong interactions between nucleons, which have different properties from the electromagnetic interactions that bind electrons in atoms, as we discuss below. The Nuclear Shell Model was proposed by Eugene Gapon and Dmitri Iwanenko in 1932 [32] and was later developed by Eugene Wigner [33], Maria GöppertMayer [34,35] and J. Hans D. Jensen [36].

For a spherically symmetric potential, the wavefunction (neglecting spin for the moment) for a nucleon, whose position, $\mathbf{r}$, from the centre of the nucleus is given by polar coordinates $(r, \theta, \phi)$, has the form
$$\Psi_{n \ell m}=R_{n \ell}(r) Y_{\ell}^m(\theta, \phi),$$
where $Y_{\ell}^m$ are spherical harmonics, which give the angular part of a wavefunction for a particle moving in a spherically symmetric potential.

The energy eigenvalues depend on the principle quantum number, $n$, and the orbital angular momentum quantum number, $\ell$, but are degenerate in the magnetic quantum number, $m$. Unlike the case of a Coulomb potential in Atomic Physics, the quantum number, $\ell$, is not restricted to take values smaller than $n$.

These energy levels come in “bunches” called “shells” with a relatively large energy gap between each shell. In their ground state, the nucleons fill up the available energy levels from the bottom upwards with two protons (and/or two neutrons), with opposite $z$-component of spin, in each available proton (neutron) energy level, as required by the Pauli exclusion principle. Thus a state with a given $n$ and $\ell$ can accommodate up to $2 \times(2 \ell+1)$ protons or neutrons.

Unlike in Atomic Physics, we do not understand, even in principle, the properties of this strong force potential, so we need to make a guess. If we assume a simple harmonic potential (i.e. $V(r) \propto r^2$ ), then we will get equally spaced energy levels and we would not see the shell structure giving rise to magic numbers.

## 物理代写|粒子物理代写粒子物理学代考|魔数

. .物理代写|

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

• 如果$N(Z)$等于一个幻数，则中子(或质子)分离能(去除最后一个中子(或质子)峰值所需的能量)。例如，图$4.1$显示了${ }_{56}^{\mathrm{A}} \mathrm{Ba}$(钡)同位素的中子分离能随中子数$N$的变化。如果$Z$是一个幻数，则会有更稳定的同位素;如果$N$是一个幻数，则会有更稳定的同音。如果$N$是一个神奇的数字，那么中子捕获的截面比图4.2所示的其他核素要低得多(10-100倍)。如果$N$或$Z$或两者都是幻数，激发态的能量远高于基态。例如，在图$4.3$中，我们给出了${ }{8 \lambda}^{\mathrm{A}} \mathrm{Pb}$的各种同位素的激发能值，从中我们可以看到，神奇核素${ }{82}^{126} \mathrm{Ph}$的激发能大约比其他同位素高3倍。
• $\mathrm{Z}$或/和$N$等于一个神奇数字的元素，其自然丰度比附近的偶数值为$Z$或$N$的元素或同位素的自然丰度大。例如，让我们看看${ }{20}^{40} \mathrm{Ca}$。实际上，对$N$或$Z$中与神奇的核素不同的核素进行偶数比较是有意义的，因为这样的核素更稳定，因此具有更大的自然丰度。还应该注意到${ }{20}^{40} \mathrm{Ca}$是与$Z=N$一起最重的稳定同位素。在$\mathrm{Ca}$的其他同位素中，它的丰度大约为$97 \%$。前一个含有$Z=N$(不等于一个神奇数字)的核素是${ }_{18}^{36} \mathrm{Ar}$(氩)，它在$\mathrm{Ar}$的其他同位素中只有$0.34 \%$的丰度，而含有$Z=22$、${ }_{22}^{44} \mathrm{Ti}$(钛)的$Z=N$同位素则完全没有。

## 物理代写|粒子物理代写粒子物理代考|壳层模型

$$\Psi_{n \ell m}=R_{n \ell}(r) Y_{\ell}^m(\theta, \phi),$$
，其中$Y_{\ell}^m$是球谐，给出了在球对称势中运动的粒子波函数的角部分

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