# 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|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, ${ }{22}^{47}$ 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|Magnetic Dipole Moments

Since nuclei with an odd number of protons and/or neutrons have intrinsic spin, they also generally possess a magnetic dipole moment.

The unit of the magnetic dipole moment for a nucleus, the “nuclear magneton”, is defined as
$$\mu_N=\frac{e \hbar}{2 m_p},$$
which is analogous to the Bohr magneton but with the electron mass replaced by the proton mass. It is defined such that the magnetic moment of a proton due to its orbital angular momentum $\ell$ is $\mu_N \ell$. Experimentally it is found that the magnetic moment of the proton due to its spin is
$$\mu_p=2.79 \mu_N=5.58 \mu_N s, \quad\left(s=\frac{1}{2}\right)$$
and that of the neutron is
$$\mu_n=-1.91 \mu_N=-3.82 \mu_N s, \quad\left(s=\frac{1}{2}\right) .$$
If we apply a magnetic field in the $z$-direction to a nucleus, then the unpaired proton with orbital angular momentum $\boldsymbol{\ell}, \operatorname{spin} s$ and total angular momentum $j$ will give a contribution to the $z$-component of the nuclear magnetic moment
$$\mu_z=\left(5.58 s_z+\ell_z\right) \mu_N .$$

## 物理代写|粒子物理代写粒子物理代考|核基态的自旋和宇称性

$$\Psi\left(-r_1,-r_2 \cdots-r_A\right)=\eta \Psi\left(r_1, r_2 \cdots r_A\right)$$核基态的自旋和宇称态通常可以由壳层模型确定。质子和中子倾向于成对，所以每对的总角动量为零，而且每对都有相等的宇称 $(\eta=1)$。因此，未配对的中子和/或质子定义了核自旋和宇称。因此，我们有以下内容:

• 奇奇核:在这种情况下，存在一个总角动量为$j_1$的未配对质子和一个总角动量为$j_2$的未配对中子。原子核的总自旋是这些角动量的(矢量)和，可以取$\left|j_1-j_2\right|$和$\left|j_1+j_2\right|$之间的值(单位阶跃)。宇称由$(-1)^{\left(\ell_1+\ell_2\right)}$给出，其中$\ell_1$和$\ell_2$分别为未配对质子和中子的轨道角动量

## 物理代写|粒子物理代写粒子物理学代考|磁偶极矩

$$\mu_N=\frac{e \hbar}{2 m_p},$$
，它类似于玻尔磁子，但用质子质量取代了电子质量。它的定义是这样的，质子的磁矩由于其轨道角动量$\ell$是$\mu_N \ell$。实验发现，质子的自旋磁矩为
$$\mu_p=2.79 \mu_N=5.58 \mu_N s, \quad\left(s=\frac{1}{2}\right)$$
，中子的磁矩为
$$\mu_n=-1.91 \mu_N=-3.82 \mu_N s, \quad\left(s=\frac{1}{2}\right) .$$

$$\mu_z=\left(5.58 s_z+\ell_z\right) \mu_N .$$ 作出贡献

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