# 物理代写|核物理代写nuclear physics代考|PHYS529

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## 物理代写|核物理代写nuclear physics代考|Weisskopf’s Decay Rate Estimate

The matrix element of the electric transition Hamiltonian (see (8.5)) between in initial nuclear state with wavefunction $\Psi_i(r)$ and a final state with wavefunction $\Psi_f(r)$ is
$$\left\langle f\left|H_{E_\gamma}\right| i\right\rangle=\int d^3 \boldsymbol{r} \sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \Psi_f^(\boldsymbol{r}) \Psi_{k_\gamma}^(\boldsymbol{r}) \Psi_i(\boldsymbol{r}),$$
where $\Psi_{k_y}^*(\boldsymbol{r})$ is the photon wavefunction.
In order to extract the $2^{\ell}$-multipole $(\mathrm{E} \ell)$ part of this matrix element, we need to expand the plane-wave wavefunction of the photon in terms of spherical harmonics, $Y_{\ell m}(\theta, \phi)$. Using the fact that the integrand in (8.24) only has support for $r$ less than or of the order of the nuclear radius, $R$, (the nuclear wavefunctions are negligible outside this region) and that the wavelengths of the emitted $\gamma$-ray is always much larger than the nuclear radius, we can assume that
$$k_\gamma r \ll 1,$$
so that we can keep only the leading power of $k_\gamma r$ in the coefficient of each spherical harmonic in the expansion of the plane-wave photon wavefunction. This expansion then gives (in spherical polar coordinates)
$$\Psi_{k_\gamma}(r, \theta, \phi)=\frac{1}{\sqrt{V}} \sum_{\ell=0}^{\infty} \sum_{m=-l}^{\ell} \frac{\sqrt{(\ell+1)}}{\sqrt{\ell}(2 \ell+1) ! !}\left(k_\gamma r\right)^{\ell} Y_{\ell m}(\theta, \phi)$$
The matrix element of the interaction Hamiltonian for the transition of an initial nucleon state with wavefunction $\Psi_i(\boldsymbol{r})$ and a final nucleon state with wavefunction $\Psi_f(\boldsymbol{r})$, emitting a photon with energy $E_\gamma$ and orbital angular momentum $\ell$ is then
$$\left\langle f\left|H_{E_\gamma}\right| i\right\rangle_l=\frac{1}{\sqrt{V}} \sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \sum_{m=-l}^{\ell} \frac{\sqrt{(\ell+1)}}{\sqrt{\ell}(2 \ell+1) ! !}$$

## 物理代写|核物理代写nuclear physics代考|Nuclear Transmutation

In the same way that molecules can interact with each other, exchanging atoms or ions, to produce different molecules, nuclei can interact with each other, exchanging protons and/or neutrons. If the total binding energies of the final-state nuclides is larger than that of the initial nuclides, then energy is liberated in the reaction (the reaction has a positive $Q$-factor); otherwise, the initial-state nuclei must be accelerated to a minimum kinetic energy before the reaction can take place.

Such a nuclear reaction is called nuclear “transmutation”. This term is applied to all nuclear reactions including radioactive decay.

The first demonstration of this transmutation was carried out by John Cockroft and Ernest Walton in 1932 [69]. They built the first particle accelerator, which accelerated protons up to a kinetic energy of $0.7 \mathrm{MeV}$, using pulsed or $\mathrm{AC}$ voltages. The accelerated protons were used to bombard a target of ${ }_3^7 \mathrm{Li}$ and set at an angle of $45^{\circ}$ to the proton beam. This gave rise to the reaction
$$p+{ }_3^7 \mathrm{Li} \rightarrow{ }_2^4 \mathrm{He}+{ }_2^4 \mathrm{He} .$$
The final-state $\alpha$-particles were observed perpendicular to the direction of the proton beam, using zinc sulphide screens. They were observed to be moving in opposite directions with the same energy and at right angles to the direction of the proton beam. By conservation of momentum, this meant that the final-state particles had equal mass and therefore were different from the initial-state particles. This experiment was popularly described as “splitting the atom”.

## 物理代写|核物理代写核物理代考|韦斯科普夫的衰变率估计

$$\left\langle f\left|H_{E_\gamma}\right| i\right\rangle=\int d^3 \boldsymbol{r} \sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \Psi_f^(\boldsymbol{r}) \Psi_{k_\gamma}^(\boldsymbol{r}) \Psi_i(\boldsymbol{r}),$$
，其中$\Psi_{k_y}^*(\boldsymbol{r})$是光子波函数。为了提取这个矩阵元素的$2^{\ell}$ -多极$(\mathrm{E} \ell)$部分，我们需要将光子的平面波波函数展开为球形谐波，$Y_{\ell m}(\theta, \phi)$。利用(8.24)中的被积函数只支持$r$小于或与核半径$R$同级(核波函数在此区域外可以忽略不计)，以及发射出的$\gamma$射线的波长总是远远大于核半径的事实，我们可以假设
$$k_\gamma r \ll 1,$$
，这样我们就可以在平面波光子波函数展开的各个球面谐波系数中只保留$k_\gamma r$的前导幂。这个展开得到(在球极坐标下)
$$\Psi_{k_\gamma}(r, \theta, \phi)=\frac{1}{\sqrt{V}} \sum_{\ell=0}^{\infty} \sum_{m=-l}^{\ell} \frac{\sqrt{(\ell+1)}}{\sqrt{\ell}(2 \ell+1) ! !}\left(k_\gamma r\right)^{\ell} Y_{\ell m}(\theta, \phi)$$

$$\left\langle f\left|H_{E_\gamma}\right| i\right\rangle_l=\frac{1}{\sqrt{V}} \sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \sum_{m=-l}^{\ell} \frac{\sqrt{(\ell+1)}}{\sqrt{\ell}(2 \ell+1) ! !}$$

## 物理代写|核物理代写核物理代考|核嬗变

1932年，约翰·科克罗夫特和欧内斯特·沃尔顿对这种转变进行了首次论证[69]。他们建造了第一个粒子加速器，利用脉冲或$\mathrm{AC}$电压将质子加速到动能$0.7 \mathrm{MeV}$。加速的质子被用来轰击一个${ }_3^7 \mathrm{Li}$的目标，并与质子束成$45^{\circ}$的角度。这导致了反应
$$p+{ }_3^7 \mathrm{Li} \rightarrow{ }_2^4 \mathrm{He}+{ }_2^4 \mathrm{He} .$$
，最终状态$\alpha$ -粒子垂直于质子束的方向，使用硫化锌屏蔽。他们被观察到以相同的能量向相反的方向运动，并且与质子束的方向成直角。根据动量守恒，这意味着最终态粒子的质量相等，因此与初始态粒子不同。这个实验通常被描述为“原子分裂”

## 有限元方法代写

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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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