## 物理代写|粒子物理代写Particle Physics代考|PHYS522

statistics-lab™ 为您的留学生涯保驾护航 在代写粒子物理Particle Physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写粒子物理Particle Physics代写方面经验极为丰富，各种代写粒子物理Particle Physics相关的作业也就用不着说。

• 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

## 物理代写|粒子物理代写Particle Physics代考|Confirmation of Rutherford Scattering Cross Section

In 1913 , Geiger and Marsden [18] performed a far more accurate experiment to check the details of Rutherford’s formula (1.12). They checked the dependence of the rate on the scattering angle and found consistency with the prediction
$$N(\theta) \propto \frac{1}{\sin ^4(\theta / 2)} .$$
Their results, shown in Fig. 1.5, agree remarkably well.
By using foils of different thickness, they showed that the number of particles scattered through a given angle was proportional to the thickness of the foil, and by using foils made from different metals (tin, silver, copper and aluminium) they were able to show that this number was proportional to the square of the atomic number, $Z$, of the material of the foil.

They were able to slow down the incident $\alpha$-particles, by placing thin sheets of mica immediately in front of the radioactive source. From this they were able to verify that the number of scattered particles was inversely proportional to the fourth power of their velocity, as indicated in (1.12).

## 物理代写|粒子物理代写Particle Physics代考|What About Quantum Effects

One might ask whether it was correct to assume that classical physics was applicable for the description of Rutherford scattering, which probes sub-atomic scales where we might expect quantum effects to be significant. Of course, at the time of Rutherford’s calculation, Quantum Physics was unknown, but nowadays we know that the incident $\alpha$-particle has an associated de Broglie wave, and that, in general, a wave scattering from a regular configuration of gold atoms will produce a diffraction pattern. The angular scale of such diffraction patterns is of the order of the de Broglie wavelength divided by the mean separation of the gold atoms in the foil.
The de Broglie wavelength, $\lambda$, is given by
$$\lambda=\frac{h}{m_\alpha v}=\frac{h}{\sqrt{2 m_\alpha T}},$$
( $h$ is Planck’s constant), the mass of an $\alpha$-particle is $6.6 \times 10^{-27} \mathrm{~kg}$, and for $\alpha$ particles with kinetic energy $5 \mathrm{MeV}\left(8 \times 10^{-13} \mathrm{~J}\right)$ this gives a wavelength
$$\lambda \approx 6 \times 10^{-15} \mathrm{~m}$$
In contrast, the separation of the gold atoms is around $170 \mathrm{~nm}$.
This means that the effect of diffraction from the gold atoms is negligible. On the other hand, the size of the nucleus itself is indeed of the order of the de Broglie wavelength of the incident particles, so that for projectiles with somewhat smaller wavelengths, diffraction patterns can be observed from diffraction off single nuclei and these patterns can yield useful information about the structure of nuclei. This is the subject of Chap. $2 .$

## 物理代写|粒子物理代写粒子物理学代考|卢瑟福散射截面的确认

1913年，Geiger和Marsden进行了一个精确得多的实验来检验Rutherford公式(1.12)的细节。他们检查了速率对散射角的依赖性，发现与预测一致
$$N(\theta) \propto \frac{1}{\sin ^4(\theta / 2)} .$$

## 物理代写|粒子物理代写粒子物理学代考|关于量子效应

. . .

( $h$ 是普朗克常数)，an的质量 $\alpha$-particle是 $6.6 \times 10^{-27} \mathrm{~kg}$，和 $\alpha$ 具有动能的粒子 $5 \mathrm{MeV}\left(8 \times 10^{-13} \mathrm{~J}\right)$ 它给出的波长
$$\lambda \approx 6 \times 10^{-15} \mathrm{~m}$$相比之下，金原子的分离是在周围 $170 \mathrm{~nm}$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|粒子物理代写Particle Physics代考|PHY357H1

statistics-lab™ 为您的留学生涯保驾护航 在代写粒子物理Particle Physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写粒子物理Particle Physics代写方面经验极为丰富，各种代写粒子物理Particle Physics相关的作业也就用不着说。

• 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

## 物理代写|粒子物理代写Particle Physics代考|Flux and Cross Section

The “flux”, $F$, of incident particles is defined as the number of incident particles arriving per unit area per second at the target.

The number of particles, $d N(b)$, with impact parameter between $b$ and $b+d b$ is the flux multiplied by the area between two concentric circles of radius $b$ and $b+d b$ (see Fig. 1.4)
$$d N(b)=F 2 \pi b d b .$$

Differentiating (1.3) gives us
$$d b=-\frac{D}{4 \sin ^2(\theta / 2)} d \theta$$
which allows us to write an expression for the number of $\alpha$-particles scattered through an angle between $\theta$ and $\theta+d \theta$ after substituting (1.5) and (1.3) into (1.4):
$$d N(\theta)=F \pi \frac{D^2}{4} \frac{\cos (\theta / 2)}{\sin ^3(\theta / 2)} d \theta$$
(the minus sign has been dropped as it merely indicates that as $b$ increases, the scattering angle $\theta$ decreases $-d N(\theta)$ must be positive).

The “differential cross section”, $d \sigma / d \theta$, with respect to the scattering angle is the number of scatterings between $\theta$ and $\theta+d \theta$ per unit flux, per unit range of angle, i.e.
$$\frac{d \sigma}{d \theta}=\frac{d N(\theta)}{F d \theta}=\pi \frac{D^2}{4} \frac{\cos (\theta / 2)}{\sin ^3(\theta / 2)}$$
It is more usual to quote the differential cross section with respect to a given interval of solid angle, $d \Omega$. Solid angle is defined such that an area element, $d S$, of a sphere of radius $r$ subtends a solid angle (at the centre of the sphere)
$$d \Omega=\frac{d S}{r^2} .$$
The unit of solid angle is the “steradian” (sr). Solid angle is related to the scattering angle $\theta$ and the “azimuthal angle”, $\phi$, by
$$d \Omega=\sin \theta d \theta d \phi$$

## 物理代写|粒子物理代写Particle Physics代考|Inconsistency of the “Plum Pudding” Model

Let us consider what would be expected if the “Plum Pudding” model were indeed correct.

We know from Gauss’ law that at a distance $r$ from the centre of the atom, the electric field is determined by the charge enclosed in a sphere of radius $r$ surrounding the centre of the atom.

The volume of a sphere of radius $r$ is proportional to $r^3$. Therefore for $r$ smaller than the radius, $R$, of the atom, the electric charge enclosed with a sphere of radius $r$ is a fraction $r^3 / R^3$ of the total electric charge (assuming a uniform distribution of electric charge throughout the “dough”), so that the magnitude of the electric field at a distance $r$ from the centre of the atom is given by
$$\left(\frac{r^3}{R^3}\right) \frac{Z e}{4 \pi \varepsilon_0 r^2},(r \leq R)$$
This is a maximum for $r=R$. This means that the scattering angle cannot be larger than the scattering angle corresponding to impact parameter $b=R$. For values of impact parameter $b<R$, the scattering angle decreases as $b$ decreases.

We have seen above that for $\alpha$-particles with typical kinetic energy of $5 \mathrm{MeV}$, this corresponds to a maximum scattering angle of around $3 \times 10^{-4}$ radians $\left(\approx 0.017^{\circ}\right)$. Such an angle would have heen far ton small to he observed in any of the GeigerMarsden experiments and they certainly would not have observed any scattering exceeding $90^{\circ}$.In fact, the scattering from the “Plum Pudding” model is expected to be even smaller as the above estimate neglects any attractive force between the $\alpha$-particle and the electrons (the “plums”) embedded in the “dough”.

## 物理代写|粒子物理代写粒子物理代考|通量和截面

.

$$d N(b)=F 2 \pi b d b .$$

$$d b=-\frac{D}{4 \sin ^2(\theta / 2)} d \theta$$
，这允许我们在将(1.5)和(1.3)代入(1.4)之后，写出$\alpha$ -粒子通过$\theta$和$\theta+d \theta$之间的角度散射的数量的表达式:
$$d N(\theta)=F \pi \frac{D^2}{4} \frac{\cos (\theta / 2)}{\sin ^3(\theta / 2)} d \theta$$
(负号被删除了，因为它仅仅表示随着$b$的增加，散射角度$\theta$减小$-d N(\theta)$一定是正的) “微分截面”，$d \sigma / d \theta$，相对于散射角是$\theta$到$\theta+d \theta$之间的每单位通量，每单位角度范围的散射数，即
$$\frac{d \sigma}{d \theta}=\frac{d N(\theta)}{F d \theta}=\pi \frac{D^2}{4} \frac{\cos (\theta / 2)}{\sin ^3(\theta / 2)}$$

$$d \Omega=\frac{d S}{r^2} .$$

$$d \Omega=\sin \theta d \theta d \phi$$

## 物理代写|粒子物理代写粒子物理学代考|“Plum Pudding”模型的不一致性

.

$$\left(\frac{r^3}{R^3}\right) \frac{Z e}{4 \pi \varepsilon_0 r^2},(r \leq R)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|粒子物理代写Particle Physics代考|PHYS3717

statistics-lab™ 为您的留学生涯保驾护航 在代写粒子物理Particle Physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写粒子物理Particle Physics代写方面经验极为丰富，各种代写粒子物理Particle Physics相关的作业也就用不着说。

• 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

## 物理代写|粒子物理代写Particle Physics代考|The Geiger-Marsden Experiments

As we shall see later the “Plum Pudding” model predicts that a charged particle which is moving through such a positively charged “dough” will experience a very weak electric force and will only undergo very small angular deflections. In order to verify this, Hans Geiger and Ernest Marsden, at the behest of Ernest Rutherford, carried out three experiments between 1908 and 1910 in which $\alpha$-particles from a radioactive source were incident on a very thin foil of gold (gold was selected because it can be beaten very thin – the foil used by Geiger and Marsden had a thickness of $400 \mathrm{~nm}$ ). The entire apparatus was encased in a tube, which was evacuated in order to minimize energy loss of the $\alpha$-particles before they scattered off the foil. A schematic sketch of the experimental setup is shown in Fig. 1.1.
In the first experiment, [13] a screen was placed behind the gold foil and scintillations caused by the $\alpha$-particles landing on the screen, were observed with a travelling microscope. Although most $(86 \%)$ of the $\alpha$-particles passed through with a deflection of less than $1^{\circ}$, a substantial angular spread of scintillations was observed.

In the second experiment [14], the screen was placed on the incident side of the gold foil in order to observe reflected $\alpha$-particles. The screen was protected from direct $\alpha$-particles by placing an impenetrable lead plate in the direct path of the particles. They nevertheless observed that about one particle in 8000 was reflected by the foil, implying that there had been scattering through an angle of greater than $90^{\circ}$ – way above the limit predicted by the “Plum Pudding” model.

In a third experiment [15], a year later, Geiger and Marsden used several different foils of different thickness and made of different materials. In this experiment, they managed to determine the most probable deflection angle. They showed that the most probable angle of scattering:

1. Increased with increasing thickness of the foil,
2. Increased with the atomic mass of the material the foil,
3. Decreased with increasing velocity of the incident $\alpha$-particles.

## 物理代写|粒子物理代写Particle Physics代考|Rutherford’s Scattering Formula

Rutherford’s surprise at the results of the Geiger-Marsden experiment, particularly the fact that some of the $\alpha$-particles were scattered though an angle of more than $90^{\circ}$, led him to state during a lecture at Cambridge University:

It was almost as incredible as if you fired a 15 -inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision. ..
In 1911, he adopted the model postulated 7 years earlier by the Japanese physicist Hantaro Nagaoka [16]. This model comprised of a small positively charged nucleus at the centre of an atom with electrons orbiting around it. Within this model, Rutherford calculated the probability of scattering of the $\alpha$-particles through an angle $\theta[17]$ under the following assumptions:

• The atom contains a nucleus of charge $Z e$, where $Z$ is the atomic number of the atom (i.e. the number of electrons in the neutral atom),
• The nucleus can be treated as a point particle,
• The nucleus is sufficiently massive compared with the mass of the incident $\alpha$ particle that the nuclear recoil may be neglected,
• The laws of classical mechanics and Electromagnetism can be applied and that no other forces are present,
• The collision is elastic.
If the collision between the nucleus and incident particle, with kinetic energy $T$ and electric charge $z e^1$ were head-on, as shown in Fig. 1.2, the distance of closest approach $D$ is obtained by equating the initial kinetic energy to the Coulomb energy at closest approach, i.e.
$$T=\frac{z Z e^2}{4 \pi \varepsilon_0 D}$$
so that the distance of closest approach is given by
$$D=\frac{z Z e^2}{4 \pi \varepsilon_0 T}$$
at which point the $\alpha$-particle reverses direction.
In general, the collision is not head-on, but is described by a quantity, $b$, called the “impact parameter”. This is the perpendicular distance between the nucleus and the initial line of the incident projectile, as shown in Fig. 1.3.

## 物理代写|粒子物理代写粒子物理学代考|盖格-马斯登实验

1. 随箔片厚度的增加而增加，
2. 随箔片材料原子质量的增加而增加，
3. 随入射$\alpha$ -粒子速度的增加而减少
物理代写|粒子物理代写粒子物理学代考|卢瑟福散射公式
卢瑟福对盖格-马斯登实验的结果感到惊讶，特别是有些$\alpha$粒子的散射角度超过了$90^{\circ}$，这使他在剑桥大学的一次演讲中说这简直不可思议，就像你向一张薄纸发射一枚15英寸的炮弹，它却打回来打在你身上。经过思考，我意识到这种向后散射一定是一次碰撞的结果。1911年，他采用了7年前由日本物理学家长冈汉太郎提出的模型。这个模型由一个位于原子中心的带正电的小原子核组成，原子核周围环绕着电子。在这个模型中，Rutherford计算了$\alpha$ -粒子通过$\theta[17]$角度散射的概率，在以下假设条件下:原子含有一个带电荷的原子核 $Z e$，其中 $Z$ 为原子的原子序数(即中性原子中的电子数)，
4. 核可视为点粒子，
5. 与入射质量相比，核的质量足够大 $\alpha$ 粒子的核后坐力可以忽略，
6. 经典力学和电磁学定律可以应用，并且没有其他力存在，
7. 碰撞是弹性的。如果原子核与入射粒子之间发生碰撞，则具有动能 $T$ 电荷 $z e^1$ 如图1.2所示，最接近的距离 $D$ 将初始动能与最接近处的库仑能相等，即
$$T=\frac{z Z e^2}{4 \pi \varepsilon_0 D}$$
使最接近的距离由
给出$$D=\frac{z Z e^2}{4 \pi \varepsilon_0 T}$$
$\alpha$粒子反转方向。一般来说，碰撞不是正面的，而是用一个量来描述的， $b$，称为“冲击参数”。这是核与入射弹的初始线之间的垂直距离，如图1.3所示。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。