# 物理代写|粒子物理代写particle physics代考|PHYS763

## 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particle physics代考|Spin and Parity of Nuclear Ground States

Nuclear states have an intrinsic spin and a well-defined parity, $\eta=\pm 1$, determined by the behaviour of the wavefunction of all the nucleons under mirror reversal $(r \rightarrow-r)$, with the centre of the nucleus at the origin.
$$\Psi\left(-r_1,-r_2 \cdots-r_A\right)=\eta \Psi\left(r_1, r_2 \cdots r_A\right) .$$
The spin and parity of nuclear ground states can usually be determined from the Shell Model. Protons and neutrons tend to pair up so that the total angular momentum of each pair is zero and each pair has even parity ( $\eta=1)$. Therefore, the unpaired neutron and/or proton define nuclear spin and parity. Thus, we have the following:

• Even-even nuclides (both $Z$ and $A$ even) have zero intrinsic spin and even parity.
• Odd- $A$ nuclei have one unpaired nucleon. The spin of the nucleus is equal to the $j$-value of that unpaired nucleon and the parity is $(-1)^{\ell}$, where $\ell$ is the orbital angular momentum of the unpaired nucleon. For example, ${ }{22}^{47} \mathrm{Ti}$ has an even number of protons and 25 neutrons. Twenty of the neutrons fill the shells up to magic number 20 and there are 5 in the $1 f{\frac{7}{2}}$ state $\left(\ell=3, j=\frac{7}{2}\right)$. Four of these form pairs, and the remaining one leads to a nuclear spin of $\frac{7}{2}$ and parity $(-1)^3=-1$
• Odd-odd nuclei: In this case, there is an unpaired proton whose total angular momentum is $j_1$ and an unpaired neutron whose total angular momentum is $j_2$. The total spin of the nucleus is the (vector) sum of these angular momenta and can take values between $\left|j_1-j_2\right|$ and $\left|j_1+j_2\right|$ (in unit steps). The parity is given by $(-1)^{\left(\ell_1+\ell_2\right)}$, where $\ell_1$ and $\ell_2$ are the orbital angular momenta of the unpaired proton and neutron, respectively.

For example, ${ }3^6 \mathrm{Li}$ (lithium) has 3 neutrons and 3 protons. The first two of each fill the $1 s$ level and the third is in the $1 p{\frac{3}{2}}$ level. The orbital angular momentum of each is $\ell=1$, so the parity is $(-1) \times(-1)=+1$ (even), but the spin can take any value between 0 and 3 .

## 物理代写|粒子物理代写particle physics代考|Shell Model

$$\Psi_{n \ell m}=R_{n \ell}(r) Y_{\ell}^m(\theta, \phi),$$

## 物理代写|粒子物理代写particle physics代考|Spin and Parity of Nuclear Ground States

$$\Psi\left(-r_1,-r_2 \cdots-r_A\right)=\eta \Psi\left(r_1, r_2 \cdots r_A\right) .$$

• 偶偶核素 (两者 $Z$ 和 $A$ even) 具有零固有自旋和偶校验。
• 奇怪的- $A$ 核有一个不成对的核子。原子核的自旋等于 $j$-那个不成对的核子的值和宇称是 $(-1)^{\ell}$ ，在哪 里 $\ell$ 是末配对核子的轨道角动量。例如， $22^{47} \mathrm{Ti}$ 有偶数个质子和 25 个中子。二十个中子填满壳层，达 到幻数 20 ，其中有 5 个 $1 f \frac{7}{2}$ 状态 $\left(\ell=3, j=\frac{7}{2}\right)$. 其中四个形成对，剩下的一个导致核自旋 $\frac{7}{2}$ 和平价 $(-1)^3=-1$
• 奇奇核: 在这种情况下，有一个末配对的质子，其总角动量为 $j_1$ 和一个末配对的中子，其总角动量为 $j_2$ . 原子核的总自旋是这些角动量的 (矢量) 和，可以取值之间 $\left|j_1-j_2\right|$ 和 $\left|j_1+j_2\right|$ (以单位步骤) 。奇 偶性由下式给出 $(-1)^{\left(\ell_1+\ell_2\right)}$ ， 在哪里 $\ell_1$ 和 $\ell_2$ 分别是末配对质子和中子的轨道角动量。
例如， $3^6 \mathrm{Li}$ (锂) 有 3 个中子和 3 个质子。每个的前两个填充 $1 s$ 水平和第三是在 $1 p \frac{3}{2}$ 等级。每个的轨道角 动量是 $\ell=1$ ，所以奇偶性是 $(-1) \times(-1)=+1$ (偶数)，但自旋可以取 0 到 3 之间的任何值。

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