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# 物理代写|核物理代考Nuclear Physics代写|Nucleons and leptons

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## 物理代写|核物理代考Nuclear Physics代写|Nucleons and leptons

Atomic nuclei are quantum bound states of particles called nucleons of which there are two types, the positively charged proton and the uncharged neutron. The two nucleons have similar masses:
$$m_{\mathrm{n}} c^2=939.56 \mathrm{MeV} \quad m_{\mathrm{p}} c^2=938.27 \mathrm{MeV}$$
i.e. a mass difference of order one part per thousand
$$\left(m_{\mathrm{n}}-m_{\mathrm{p}}\right) c^2=1.29 \mathrm{MeV}$$
For nuclear physics, the mass difference is much more important than the masses themselves which in many applications are considered to be “infinite.” Also of great phenomenological importance is the fact that this mass difference is of the same order as the electron mass
$$m_{\mathrm{e}} c^2=0.511 \mathrm{MeV}$$
Nucleons and electrons are spin 1/2 fermions meaning that their intrinsic angular momentum projected on an arbitrary direction can take on only the values of $\pm \hbar / 2$. Having spin $1 / 2$, they must satisfy the Pauli exclusion principle that prevents two identical particles (protons, neutrons or electrons)
from having the same spatial wavefunction unless their spins are oppositely aligned.
Nucleons and electrons generate magnetic fields and interact with magnetic fields with their magnetic moment. Like their spins, their magnetic moments projected in any direction can only take on the values $\pm \mu_{\mathrm{p}}$ or $\pm \mu_{\mathrm{n}}$ :
$$\mu_{\mathrm{p}}=2.792847386(63) \mu_{\mathrm{N}} \quad \mu_{\mathrm{n}}=-1.91304275(45) \mu_{\mathrm{N}} \text {, }$$
where the nuclear magneton is
$$\mu_{\mathrm{N}}=\frac{e \hbar}{2 m_{\mathrm{p}}}=3.15245166(28) \times 10^{-14} \mathrm{MeVT}^{-1} .$$
For the electron, only the mass and the numerical factor changes
$$\mu_{\mathrm{e}}=1.001159652193(40) \mu_{\mathrm{B}},$$
where the Bohr magneton is
$$\mu_{\mathrm{B}}=\frac{q \hbar}{2 m_{\mathrm{e}}}=5.78838263(52) \times 10^{-11} \mathrm{MeVT}^{-1}$$

## 物理代写|核物理代考Nuclear Physics代写|General properties of nuclei

Nuclei, the bound states of nucleons, can be contrasted with atoms, the bound states of nuclei and electrons. The differences are seen in the units used by atomic and nuclear physicists:
$$\begin{array}{lll} \text { length : } & 10^{-10} \mathrm{~m} \text { (atoms) } & \rightarrow 10^{-15} \mathrm{~m}=\mathrm{fm} \text { (nuclei) } \ \text { energy : } \mathrm{eV} \text { (atoms) } & \rightarrow \mathrm{MeV} \text { (nuclei) } \end{array}$$
The typical nuclear sizes are 5 orders of magnitude smaller than atomic sizes and typical nuclear binding energies are 6 orders of magnitude greater than atomic energies. We will see in this chapter that these differences are due to the relative strengths and ranges of the forces that bind atoms and nuclei.
We note that nuclear binding energies are still “small” in the sense that they are only about $1 \%$ of the nucleon rest energies $m c^2$ (1.1). Since nucleon binding energies are of the order of their kinetic energies $m v^2 / 2$, nucleons within the nucleus move at non-relativistic velocities $v^2 / c^2 \sim 10^{-2}$.
A nuclear species, or nuclide, is defined by $N$, the number of neutrons, and by $Z$, the number of protons. The mass number $A$ is the total number of nucleons, i.e. $A=N+Z$. A nucleus can alternatively be denoted as
$$(A, Z) \leftrightarrow{ }^A X \leftrightarrow{ }_Z^A X \leftrightarrow{ }_Z^A X_N$$
where $X$ is the chemical symbol associated with $Z$ (which is also the number of electrons of the corresponding neutral atom). For instance, ${ }^4 \mathrm{He}$ is the helium-4 nucleus, i.e. $N=2$ and $Z=2$. For historical reasons, ${ }^4 \mathrm{He}$ is also called the $\alpha$ particle. The three nuclides with $Z=1$ also have special names
$${ }^1 \mathrm{H}=\mathrm{p}=\text { proton }{ }^2 \mathrm{H}=\mathrm{d}=\text { deuteron }{ }^3 \mathrm{H}=\mathrm{t}=\text { triton }$$
While the numbers $(A, Z)$ or $(N, Z)$ define a nuclear species, they do not determine uniquely the nuclear quantum state. With few exceptions, a nucleus $(A, Z)$ possesses a rich spectrum of excited states which can decay to the ground state of $(A, Z)$ by emitting photons. The emitted photons are often called $\gamma$-rays. The excitation energies are generally in the $\mathrm{MeV}$ range and their lifetimes are generally in the range of $10^{-9}-10^{-15} \mathrm{~s}$. Because of their high energies and short lifetimes, the excited states are very rarely seen on Earth and, when there is no ambiguity, we denote by $(A, Z)$ the ground state of the corresponding nucleus.
Some particular sequences of nuclei have special names:
Isotopes : have same charge $Z$, but different $N$, for instance ${ }{92}^{238} \mathrm{U}$ and ${ }{92}^{235} \mathrm{U}$. The corresponding atoms have practically identical chemical properties, since these arise from the $Z$ electrons. Isotopes have very different nuclear properties, as is well-known for ${ }^{238} \mathrm{U}$ and ${ }^{235} \mathrm{U}$.
Isobars : have the same mass number $A$, such as ${ }^3 \mathrm{He}$ and ${ }^3 \mathrm{H}$. Because of the similarity of the nuclear interactions of protons and neutrons, different isobars frequently have similar nuclear properties.
Less frequently used is the term isotone for nuclei of the same $N$, but different $Z$ ‘s, for instance ${ }^{14} \mathrm{C}_6$ and ${ }^{16} \mathrm{O}_8$.
Nuclei in a given quantum state are characterized, most importantly, by their size and binding energy. In the following two subsections, we will discuss these two quantities for nuclear ground states.

# 核物理代考

## 物理代写|核物理代考Nuclear Physics代写|Nucleons and leptons

$$m_{\mathrm{n}} c^2=939.56 \mathrm{MeV} \quad m_{\mathrm{p}} c^2=938.27 \mathrm{MeV}$$

$$\left(m_{\mathrm{n}}-m_{\mathrm{p}}\right) c^2=1.29 \mathrm{MeV}$$

$$m_{\mathrm{e}} c^2=0.511 \mathrm{MeV}$$

$$\mu_{\mathrm{p}}=2.792847386(63) \mu_{\mathrm{N}} \quad \mu_{\mathrm{n}}=-1.91304275(45) \mu_{\mathrm{N}} \text {, }$$

$$\mu_{\mathrm{N}}=\frac{e \hbar}{2 m_{\mathrm{p}}}=3.15245166(28) \times 10^{-14} \mathrm{MeVT}^{-1} .$$

$$\mu_{\mathrm{e}}=1.001159652193(40) \mu_{\mathrm{B}},$$

$$\mu_{\mathrm{B}}=\frac{q \hbar}{2 m_{\mathrm{e}}}=5.78838263(52) \times 10^{-11} \mathrm{MeVT}^{-1}$$

## 物理代写|核物理代考Nuclear Physics代写|General properties of nuclei

$$\begin{array}{lll} \text { length : } & 10^{-10} \mathrm{~m} \text { (atoms) } & \rightarrow 10^{-15} \mathrm{~m}=\mathrm{fm} \text { (nuclei) } \ \text { energy : } \mathrm{eV} \text { (atoms) } & \rightarrow \mathrm{MeV} \text { (nuclei) } \end{array}$$

$$(A, Z) \leftrightarrow{ }^A X \leftrightarrow{ }_Z^A X \leftrightarrow{ }_Z^A X_N$$

$${ }^1 \mathrm{H}=\mathrm{p}=\text { proton }{ }^2 \mathrm{H}=\mathrm{d}=\text { deuteron }{ }^3 \mathrm{H}=\mathrm{t}=\text { triton }$$

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