Posted on Categories:Nuclear Physics, 核物理, 物理代写

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Quantum effects inside nuclei are fundamental. It is therefore surprising that the volume $\mathcal{V}$ of a nucleus is, to good approximation, proportional to the number of nucleons $A$ with each nucleon occupying a volume of the order of $\mathcal{V}_0=7.2 \mathrm{fm}^3$. In first approximation, stable nuclei are spherical, so a volume $\mathcal{V} \simeq A \mathcal{V}_0$ implies a radius
$$R=r_0 A^{1 / 3} \quad \text { with } \quad r_0=1.2 \mathrm{fm} \quad .$$
We shall see that $r_0$ in (1.9) is the order of magnitude of the range of nuclear forces.

In Chap. 3 we will show how one can determine the spatial distribution of nucleons inside a nucleus by scattering electrons off the nucleus. Electrons can penetrate inside the nucleus so their trajectories are sensitive to the charge distribution. This allows one to reconstruct the proton density, or equivalently the proton probability distribution $\rho_p(r)$. Figure 1.1 shows the charge densities inside various nuclei as functions of the distance to the nuclear center.

We see on this figure that for $A>40$ the charge density, therefore the proton density, is roughly constant inside these nuclei. It is independent of the nucleus under consideration and it is roughly 0.075 protons per $\mathrm{fm}^3$. Assuming the neutron and proton densities are the same, we find a nucleon density inside nuclei of
$$\rho_0 \simeq 0.15 \text { nucleons } \mathrm{fm}^{-3} .$$
If the nucleon density were exactly constant up to a radius $R$ and zero beyond, the radius $R$ would be given by (1.9). Figure 1.1 indicates that the density drops from the above value to zero over a region of thickness $\sim 2 \mathrm{fm}$ about the nominal radius $R$.

## 物理代写|核物理代考Nuclear Physics代写|Binding energies

The saturation phenomenon observed in nuclear radii also appears in nuclear binding energies. The binding energy $B$ of a nucleus is defined as the negative of the difference between the nuclear mass and the sum of the masses of the constituents:
$$B(A, Z)=N m_{\mathrm{n}} c^2+Z m_{\mathrm{p}} c^2-m(A, Z) c^2$$
Note that $B$ is defined as a positive number: $B(A, Z)=-E_B(A, Z)$ where $E_B$ is the usual (negative) binding energy.

The binding energy per nucleon $B / A$ as a function of $A$ is shown in Fig. 1.2. We observe that $B / A$ increases with $A$ in light nuclei, and reaches a broad maximum around $A \simeq 55-60$ in the iron-nickel region. Beyond, it decreases slowly as a function of $A$. This immediately tells us that energy can be released by the “fusion” of light nuclei into heavier ones, or by the “fission” of heavy nuclei into lighter ones.

As for nuclear volumes, it is observed that for stable nuclei which are not too small, say for $A>12$, the binding energy $B$ is in first approximation additive, i.e. proportional to the number of nucleons :
$$B(A, Z) \simeq A \times 8 \mathrm{MeV}$$
or more precisely
$$7.7 \mathrm{MeV}<B(A, Z) / A<8.8 \mathrm{MeV} \quad 12<A<225$$
The numerical value of $\sim 8 \mathrm{MeV}$ per nucleon is worth remembering!
The additivity of binding energies is quite different from what happens in atomic physics where the binding energy of an atom with $Z$ electrons increases as $Z^{7 / 3}$, i.e. $Z^{4 / 3}$ per electron. The nuclear additivity is again a manifestation of the saturation of nuclear forces mentioned above. It is surprising from the quantum mechanical point of view. In fact, since the binding energy arises from the pairwise nucleon-nucleon interactions, one might expect that $B(A, Z) / A$ should increase with the number of nucleon pairs $A(A-1) / 2 .{ }^1$ The additivity confirms that nucleons only interact strongly with their nearest neighbors.

# 核物理代写

$$R=r_0 A^{1 / 3} \quad \text { with } \quad r_0=1.2 \mathrm{fm} \quad .$$

$$\rho_0 \simeq 0.15 \text { nucleons } \mathrm{fm}^{-3} .$$

## 物理代写|核物理代考Nuclear Physics代写|Binding energies

$$B(A, Z)=N m_{\mathrm{n}} c^2+Z m_{\mathrm{p}} c^2-m(A, Z) c^2$$

$$B(A, Z) \simeq A \times 8 \mathrm{MeV}$$

$$7.7 \mathrm{MeV}<B(A, Z) / A<8.8 \mathrm{MeV} \quad 12<A<225$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。