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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|核物理代写nuclear physics代考|Decay Mechanism

The mean lifetime of $\alpha$-decaying nuclides varies from the order of $10^{-7} \mathrm{~s}$ to $10^{10}$ years. We can understand this by investigating the mechanism for $\alpha$-decay.

What happens is that two protons from the highest proton energy levels and two neutrons from the highest neutron energy levels combine to form an $\alpha$-particle inside the nucleus – this is known as a “quasi-bound state”. It acquires an energy that is approximately equal to $Q_\alpha$ (neglecting the small correction due to the recoil of the nucleus).

The $\alpha$-particle is bound to the nucleus in a potential well created by the strong, short-range nuclear forces. There is also a Coulomb repulsion between this “quasi-” $\alpha$-particle and the rest of the nucleus. Since the strong-interaction force is rapidly attenuated at distances from the centre exceeding the nuclear radius, $R$, we can make the approximation that the nuclear potential well can be taken to be a spherical well, of radius $R$.

The Coulomb repulsion between the $\alpha$-particle, (charge $2 e$ ), and the daughter nucleus is described by the potential (written in terms of the fine structure constant, $\alpha$ )
$$V_c=\frac{2 Z_D \alpha \hbar c}{r},$$
where $Z_D \equiv(Z-2)$ is the atomic number of the daughter nuclide. It is convenient to rewrite this as
$$V_c=\frac{Q_\alpha r_b}{r},$$
where $r_b$ is the radial distance at which the kinetic energy of the $\alpha$-particle inside the nucleus is equal to the Coulomb energy i.e.
$$r_b=\frac{2 Z_D \alpha \hbar c}{Q_\alpha} .$$
The nuclear potential well combined with the Coulomb potential forms a potential barrier as shown in Fig. $6.1$.

## 物理代写|核物理代写nuclear physics代考|Empirical Formulae

In 1911 Hans Geiger and John Nuttall [47] observed that the logarithm of the halflives of different $\alpha$-decays was a linear function of the inverse square root of the $Q$-value:
$$\log {10}\left(t{\frac{1}{2}}\right) \approx \frac{f}{\sqrt{Q_\alpha}}+g .$$
Comparing this with the theoretical expression, (6.20), the coefficient $f$ has a $Z$ dependence given by
$$f=1.72 Z_D$$
From (6.20), the parameter $g$ is given by
$$g=-1.42 \sqrt{Z_D} A_D^{1 / 6}-21 .$$
This is not a constant but varies over the range of $\alpha$-decaying nuclides from around $-46$ to $-58$. It turns out, however, that using the Geiger-Nuttall formula with $g$ fixed at the value of $-54$ gives a better fit to data. This fit is shown in Fig. 6.2. One can see that this fit is very approximate and sometimes has large errors of up to 3 in the logarithm of the half-life (i.e. 3 orders of magnitude in the half-life). The purely theoretical prediction $(6.20)$ can produce even larger errors for the absolute values of the half-lives. On the other hand it reproduces the dependence on the $Q$-values reasonably well, although it is clear from Fig. 6.2, comparing the best fit to data (green line) with the theoretical slope (blue line), that the theoretical slope is around $10 \%$ too large.

# 核物理代写

## 物理代写|核物理代写nuclear physics代考|Decay Mechanism

$$V_c=\frac{2 Z_D \alpha \hbar c}{r},$$

$$V_c=\frac{Q_\alpha r_b}{r},$$

$$r_b=\frac{2 Z_D \alpha \hbar c}{Q_\alpha} .$$

## 物理代写|核物理代写nuclear physics代考|Empirical Formulae

1911 年 Hans Geiger 和 John Nuttall [47] 观察到不同的半衰期的对数 $\alpha$-decays 是反平方根的线性函数 $Q$-价值:
$$\log 10\left(t \frac{1}{2}\right) \approx \frac{f}{\sqrt{Q_\alpha}}+g .$$

$$f=1.72 Z_D$$

$$g=-1.42 \sqrt{Z_D} A_D^{1 / 6}-21$$

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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。

assignmentutor™您的专属作业导师
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