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## 物理代写|粒子物理代写Particle Physics代考|B-beta-Decay and the Neutrino

The study of beta-decay played a very important role in the development of the entire field of subatomic physics, so we shall briefly review the main steps. We start with a simple kinematical analysis. Consider a particle $A$ which decays into two particles: $A\left(p_A\right) \rightarrow B\left(p_B\right)+C\left(p_C\right)$. The conservation of energy and momentum applied in the rest frame of the decaying particle implies
$$M_A=p_B^0+p_C^0, \quad 0=\boldsymbol{p}_B+\boldsymbol{p}_C$$
and gives
$$p_B^0=\frac{M_A^2+M_B^2-M_C^2}{2 M_A}, \quad p_C^0=\frac{M_A^2+M_C^2-M_B^2}{2 M_A}$$
i.e. the energy of each one of the final particles is fixed. In nuclear decays, both the electron mass $m_e \equiv M_C \simeq 0.5 \mathrm{MeV}$, and the mass difference $M_A-M_B$ between the initial and the final nuclei, typically of order a few $\mathrm{MeV}$, are negligible compared to the nuclear masses which are on the order of several $\mathrm{GeV}$. Therefore equation (6.3) tells us that, if $\beta$-decay is of the form given in (6.1), in the rest frame of the decaying nucleus the electrons are monoenergetic with energy given by
$$E_e \equiv p_C^0 \simeq \Delta M=M_A-M_B$$
The experiments performed up to 1914 did not show any appreciable contradiction with this result. We know today that this was due to lack of sufficient accuracy in the determination of the electron energy. The latter was estimated by the penetration length of the emitted electrons in various materials and the results were only approximate. The best measurements were performed in the chemistry department of the University of Berlin by O. Hahn, a former assistant of Rutherford, and L. Meitner.

## 物理代写|粒子物理代写Particle Physics代考|1932: The First Table of Elementary Particles

Table $6.1$ contains all the particles considered to be elementary in 1932 . We saw previously that the year is not chosen randomly. It is the year of the discovery of the neutron and the formulation of the correct nuclear model. Our basic ideas on the structure of matter date from that year. The table is separated into two parts. The upper part contains two doublets, the proton-neutron and the electron-neutrino which, as we shall explain shortly, can be considered as the constituents of matter. We shall call them matter particles. The lower part contains only one entry, the photon, which is the quantum manifestation of the electromagnetic field. We shall call it the quantum of radiation.
Let us have a first cursory look at each entry separately.

• The protons and the neutrons form the nuclei. We shall call them collectively nucleons. ${ }^5$ We shall see in one of the following sections that this is more than just a common name. They both have spin one-half and obey the Pauli exclusion principle.

Compared to the other particles in Table $6.1$, they appear to be heavy, with masses of order $1 \mathrm{GeV}$ and this fact explains why the mass of the atoms is contained almost entirely in the nuclei.

• The electron is the oldest-known elementary particle. It has a long history but its discovery is attributed to J.J. Thomson who, in 1897 , established that cathode rays consist of corpuscles carrying negative electric charge. Furthermore, he measured the ratio of charge over mass, $e / m$, quite accurately and determined that these properties are independent of the chemical composition of the cathode. In 1900, $\mathrm{H}$. Becquerel proved that the same particles are emitted in $\beta$-decay. The ratio $e / m$ was measured more accurately by R. Millikan in 1909 . The electrons also have spin one-half and obey the Pauli exclusion principle.
• We have already presented the story of the neutrino. ${ }^6$ In Table $6.1$ we see a question mark in the entry of its mass. It shows our ignorance concerning its precise value. We shall come back in more detail to the problem of the neutrino mass in Chapter 20.
• At the end of the matter particles in the table we see a line under the title “anti-particles”. When Dirac proposed his equation in 1928 , it was meant to be a relativistic wave equation for the electron. It was, however, soon realised by Dirac that the same equation admits solutions describing a positively charged particle with the same mass and spin as the electron. We shall introduce and study the Dirac equation in Chapter 7. Dirac thought for a while to identify this new solution with the proton and he published a paper along these lines in 1930, but he soon understood that it is in fact a new particle. In 1931, he predicted the existence of such an anti-electron, which we call the positron ${ }^7$ and the following year C.D. Anderson detected this particle ${ }^8$ in the cosmic rays using a cloud chamber, which was invented by C.T. Wilson two decades earlier. ${ }^9$ This successful prediction was a triumph of the Dirac theory. Concerning the antiparticle of the neutrino, we saw previously that, by convention whose origin will be clear later, we called anti-neutrino the particle conjectured by Pauli. So, in 1932 this was the only one that people knew. The existence of what we call today the “neutrino” was established in the same indirect way in 1934 when F. and I. Joliot-Curie discovered a reaction that looked like $\beta$-decay but the emitted particle was a positron rather than an electron. In the notation of nuclear physics $(A, Z)$ changes into $(A, Z-1)$ according to
$$(A, Z) \rightarrow(A, Z-1)+e^{+}+\nu$$

## 物理代写|粒子物理代写Particle Physics代考|B-beta\$-Decay and the Neutrino 学分析开始。考虑一个粒子$A$衰变为两个粒子:$A\left(p_A\right) \rightarrow B\left(p_B\right)+C\left(p_C\right)$. 在亯变粒子的静止框架中应用的能量和动量守恒 意味着 $$M_A=p_B^0+p_C^0, \quad 0=\boldsymbol{p}_B+\boldsymbol{p}_C$$ 并给出 $$p_B^0=\frac{M_A^2+M_B^2-M_C^2}{2 M_A}, \quad p_C^0=\frac{M_A^2+M_C^2-M_B^2}{2 M_A}$$ 即每个最終粒子的能量是固定的。在核言变中，电子质量$m_e \equiv M_C \simeq 0.5 \mathrm{MeV}$, 和质量差$M_A-M_B$在初始核和最终核之 间，通常为几个$\mathrm{MeV}$，与核质量相比可以忽略不计$\mathrm{GeV}$. 因此等式 (6.3) 告诉我们，如果$\beta$-哏变具有 (6.1) 中给出的形式，在 㚆变核的静止框架中，电子是单能的，能量由下式给出 $$E_e \equiv p_C^0 \simeq \Delta M=M_A-M_B$$ 直到 1914 年进行的实验并没有显示出与这个结果有任何明显的矛盾。我们今天知道这是由于在确定电子能量方面缺乏足够的淮确 性。后者是通过发射电子在各种材料中的穿透长度来估计的，结果只是近似值。最好的测量是由卢瑟福的前助理 O. Hahn 和 L. Meitner 在柏林大学的化学系进行的。 ## 物理代写|粒子物理代写Particle Physics代考|1932: The First Table of Elementary Particles 桌子6.1包含所有在 1932 年被认为是基本的粒子。我们之前看到，年份不是随机选择的。这是发现中子和制定正确核模型的一 年。我们关于物质结构的基本思想是从那一年开始的。该表分为两部分。上半部分包含两个双峰，质子-中子和电子-中微子，正 如我们稍后将解释的，它们可以被认为是物质的组成部分。我们称它们为物质粒子。下半部分只包含一个条目，光子，它是电磁场 的量子表现。我们称之为辐射量子。 让我们先粗略地分别看一下每个条目。 • 质子和中子形成原子核。我们将它们统称为核子。我们将在以下部分中看到，这不仅仅是一个通用名称。他们都自旋了一 半并且邅守泡利不相容原理。 与表中其他粒子相比6.1，它们看起来很重, 有大量的秩序$1 \mathrm{GeV}$这个事实解释了为什么原子的质量几乎完全包含在原子核中。 • 电子是已知最古老的基本粒子。它有着悠久的历史，但它的发现归功于 JJ Thomson，他在 1897 年确立了阴极射线由带有 负电荷的微粒组成。此外，他测量了电荷与质量的比率，$e / m$，非常准确地确定这些特性与阴极的化学成分无关。1900 年，$\mathrm{H}$. 贝古勒尔证明了相同的粒子在$\beta$-亳变。比例$e / m R$. Millikan 在 1909 年更准确地测量了。电子也有二分之一的自旋 并邅守泡利不相容原理。 • 我们已经介绍了中微子的故事。${ }^6$在表中6.1我们在其质量的条目中看到了一个问号。它显示了我们对其精确价值的无知。 我们将在第 20 章更详细地讨论中微子质量问题。 • 在表中物质粒子的末尾，我们看到标题为“反粒子”的一行。当狄拉克在 1928 年提出他的方程时，它本来是一个电子的相对 论波动方程。然而，狄拉克很快意识到，同样的方程允许描述一个与电子具有相同质量和自旋的带正电粒子的解。我们将在 第 7 章介绍和研究狄拉克方程。狄拉克想了一阵子想用质子来确定这个新解，并在 1930 年发表了一篇沿此思路的论文，但 他很快就明白它实际上是一个新粒子。1931年，他预言了这种反电子的存在，我们称之为正电子${ }^7$第二年 CD Anderson 检 测到了这个粒子${ }^8$在宇宙射线中使用云室，这是由 CT Wilson 二十年前发明的。${ }^9$这一成功的预则是狄拉克理论的胜利。关 于中微子的反粒子，我们之前已经看到，按照贯例，其起源将在后面清楚，我们称反中微子为泡利猜想的粒子。所以，在 1932 年，这是人们唯一知道的。我们今天所说的“中微子”的存在是在 1934 年以同样的间接方式确立的，当时$\mathrm{F}$. 和$\mathrm{I}$. Joliot-Curie 发现了一种看起来像$\beta$-毫变，但发射的粒子是正电子而不是电子。在核物理符号中$(A, Z)$变成$(A, Z-1)\$ Joliot-C
$$(A, Z) \rightarrow(A, Z-1)+e^{+}+\nu$$

## 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。