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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

So far, we have assumed that the charge distribution is spherically symmetric. If that were the case we would have

$$\left\langle x^2\right\rangle=\left\langle y^2\right\rangle=\left\langle z^2\right\rangle=\frac{1}{3}\left\langle r^2\right\rangle,$$
where
$$\left.<x^2\right\rangle=\frac{1}{Z e} \int x^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$
$e t c .$
However, for many nuclei this is not the case. They usually still have an axis of symmetry, which we set to be the $z$-axis, so that they are symmetric in the $x-y$ plane but asymmetric in the $x-z$ or $y-z$ planes. In polar coordinates $(r, \vartheta, \phi)$, this means that the charge distribution is a function of the polar angle, $\vartheta$, but for nuclei which still maintain one axis of symmetry (the $z$-axis), the charge distribution is independent of the azimuthal angle, $\phi$.
We can still determine a charge radius
$$R=\int r \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$
even if $\rho(\boldsymbol{r})$ is not a function of the radial component, $r$, alone. However, if we look at the expectation values of the squares of individual component of $\boldsymbol{r}$
$$\left\langle x^2\right\rangle=\int x^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},\left\langle y^2\right\rangle=\int y^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},\left\langle z^2\right\rangle=\int z^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$
we find that these are not equal. Nuclei which nevertheless have an axis of symmetry have the same values of $\left\langle x^2\right\rangle$ and $\left\langle y^2\right\rangle$, but a different value for $\left\langle z^2\right\rangle$.

## 物理代写|粒子物理代写Particle Physics代考|Strong Force Distribution

The protons and neutrons inside a nucleus are held together by a strong nuclear force. This has to be strong enough to overcome the Coulomb repulsion between the protons, but unlike the Coulomb force, it extends only over a short range of a few $\mathrm{fm}$.

Electrons are used to probe the charge distribution of the target nuclei, because they interact with the electric field, but not with the strong forces. Likewise, scattering of neutrons from a nucleus can be used to probe the strong force distribution, but not the electric charge distribution since neutrons are uncharged but interact strongly.

As in the case of electron scattering, the cross section of neutron scattering displays a diffraction pattern if the de Broglie wavelength of the neutrons is of the order of the nuclear size. In such a case the wave from different parts of the nucleus interfere to produce diffraction maxima and minima at different scattering angles. This can be seen in Fig. $2.9$ in which neutrons with kinetic energy $14 \mathrm{MeV}$ are scattered from a $\mathrm{Ni}$ (nickel) nucleus. The de Broglie wavelength of neutrons with kinetic energy $14 \mathrm{MeV}$ is approximately $1.2 \mathrm{fm}$ so that the first minimum of the differential cross section, at a scattering angle of $42^{\circ}$, implies an effective radius of the strong force distribution of a few fm. This is similar to the charge radius of the nucleus. We would expect the total nucleon distribution to have the same range as the proton (charge) distribution. However, whereas the Coulomb potential from the nucleus is long-range, being attenuated as the inverse of the distance from the centre of the nucleus, the strong force is rapidly attenuated and becomes negligible after a few $\mathrm{fm}$ trom the nucleus.

The differential cross section can be expressed in terms of a “scattering amplitude”, $f(\theta, \phi)$ (usually a function of the scattering angle, $\theta$, only, but could in some cases depend on the azimuthal angle, $\phi$, of the outgoing particle), via the relation
$$\frac{d \sigma}{d \Omega}=|f(\theta, \phi)|^2$$

# 粒子物理代考

## 物理代写|粒子物理代写粒子物理代考|电四极矩

$$\left\langle x^2\right\rangle=\left\langle y^2\right\rangle=\left\langle z^2\right\rangle=\frac{1}{3}\left\langle r^2\right\rangle,$$
where
$$\left.<x^2\right\rangle=\frac{1}{Z e} \int x^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$
$e t c .$

$$R=\int r \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$
，即使$\rho(\boldsymbol{r})$不是径向分量$r$的函数。然而，如果我们观察$\boldsymbol{r}$
$$\left\langle x^2\right\rangle=\int x^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},\left\langle y^2\right\rangle=\int y^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},\left\langle z^2\right\rangle=\int z^2 \rho(\boldsymbol{r}) d^3 \boldsymbol{r},$$

## 物理代写|粒子物理代写Particle Physics代考|强力-分布

$$\frac{d \sigma}{d \Omega}=|f(\theta, \phi)|^2$$

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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。

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