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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

## 物理代写|核物理代写nuclear physics代考|Measurement of Line-Widths

The Mössbauer effect provides an extremely accurate method for measuring $\gamma$-ray line-widths.

Mössbauer [66] used this method to measure the line-width of the decay of the excited state of ${ }{77}^{191} \mathrm{Ir}$ (iridium) with excitation energy $129 \mathrm{keV}$. This state is produced in the $\beta$-decay of ${ }{76}^{191} \mathrm{Os}$ (osmium). The source, consisting of osmium embedded in a thin crystal, was moved with a constant velocity $\pm v$ relative to the absorber, which consisted of a crystal of iridium. The motion of the absorber introduces a Doppler shift of the spectral line (in the frame of the absorber)
$$\Delta E_{\text {Doppler }}=E_\gamma \frac{v}{c},$$
so that the absorber is once again slightly off resonance, but as the velocity of the absorber was only a few millimetres per second, this energy shift was only a few parts in $10^{11}$.

The results are shown in Fig. 8.2. The intensity, $\mathcal{I}^{\text {Ir }}$, of $\gamma$-radiation penetrating the iridium absorber was compared with the intensity, $\mathcal{I}^{\mathrm{Pt}}$, of $\gamma$-radiation penetrating an absorber consisting of platinum, which was used as a control. The platinum absorber does not have a resonance around $129 \mathrm{keV}$, and for sufficiently large source velocities, the penetrating intensities through the two absorbers are the same. The half-width is obtained from the velocity at which the absorption was one-half of the peak absorption. This was found to be $$v_{1 / 2}=0.52 \pm 0.06 \mathrm{~cm} \mathrm{~s}^{-1}$$
corresponding to a line-width ${ }^5$
$$\Gamma=2 \frac{v_{1 / 2}}{c} E_\gamma=4.6 \pm 0.6 \times 10^{-6} \mathrm{eV}$$

## 物理代写|核物理代写nuclear physics代考|Measurement of Isomeric Shift and Quadrupole Splitting

Very small changes in $\gamma$-ray energies can be measured using the Mössbauer technique (known as “Mössbauer spectroscopy”). This can be used to observe the tiny energy changes discussed in the previous section. The velocity, $v$, of the absorber relative to the source, which is required to bring a $\gamma$-ray of energy $E_\gamma$ back into resonance, is related to an energy shift, $\Delta E_\gamma$, between the resonant energy of the absorber and that of the source given by
$$\Delta E_\gamma=E_\gamma \frac{v}{c} .$$
If we have two closely spaced lines, which require velocities $v_1$ and $v_2$ to bring them back into resonance, then the energy splitting between the lines is given by
$$\left(\Delta E_\gamma\right){\text {split }}=\left(\Delta E\gamma\right)1-\left(\Delta E\gamma\right)2=E\gamma \frac{\left(v_1-v_2\right)}{c} .$$
The absorption of the $14.4 \mathrm{keV}$ line of ${ }_{26}^{57} \mathrm{Fe}$ (iron) is a good example of a line which is split due to the quadrupole interaction. This has been widely studied in many experiments. The excited state has spin $I=\frac{3}{2}$, whereas the ground state has $I=\frac{1}{2}$. This leads to quadrupole splitting. There can also be an isomeric shift if the source and absorber are in different chemical environments.

In one such experiment [65], the source was stainless steel, whereas in the absorber the iron was doubly ionized $\mathrm{Fe}^{++}$embedded in a crystal of $\mathrm{FeSO}4$. The different chemical environments of source and absorber led to an isomeric shift. The results of this experiment are displayed in Fig. 8.3, which shows two absorption lines – one with velocity $+3.4 \mathrm{~mm} \mathrm{~s}^{-1}$ and the other with velocity $-0.2 \mathrm{~mm} \mathrm{~s}^{-1}$. corresponding to energy shifts $\left(\Delta E\gamma\right)1=1.6 \times 10^{-7} \mathrm{eV}$ and $\left(\Delta E\gamma\right)_2=-9.6 \times$ $10^{-9} \mathrm{eV}$, respectively. These shifts are caused by the sum of the electric quadrupole shift and an isomer shift. Using (8.17) with $I=\frac{3}{2}$, and $m_I=\frac{3}{2}$ and $\frac{1}{2}$, respectively, we have $$\begin{gathered} \left(\Delta E_\gamma\right)1=\frac{\mathcal{Q} V{z z}}{4}+\left(\Delta E_\gamma\right){\text {isomer }} \ \left(\Delta E\gamma\right)2=-\frac{\mathcal{Q} V{z z}}{4}+\left(\Delta E_\gamma\right){\text {isomer }} \end{gathered}$$ This gives us an isomer shift $$\left(\Delta E\gamma\right){\text {isomer }}=7.5 \times 10^{-8} \mathrm{eV} .$$ For the electric quadrupole shifts, $\Delta E{\text {quad }}$, of the two lines, we find
$$\Delta E_{\text {quad }}=\pm \frac{\mathcal{Q} V_{z z}}{4}=\pm 8.5 \times 10^{-8} \mathrm{eV} .$$
The electric quadrupole moment of ${ }{26}^{57} \mathrm{Fe}$ is $0.08 \mathrm{eb}$ (electron-barns). This allows us to estimate the electric field gradient $$V{z z}=4.25 \times 10^{22} \mathrm{~V} \mathrm{~m}^{-2}$$

# 核物理代写

## 物理代写|核物理代写核物理代考|测量线宽

Mössbauer效应为测量$\gamma$射线线宽提供了一种极为精确的方法

Mössbauer[66]用这种方法测量了${ }{77}^{191} \mathrm{Ir}$(铱)激发态衰变的线宽，激发能为$129 \mathrm{keV}$。这种状态是在${ }{76}^{191} \mathrm{Os}$(锇)的$\beta$衰变中产生的。源由镶嵌在薄晶体中的锇组成，相对于由铱晶体组成的吸收器以恒定速度$\pm v$移动。吸收器的运动引入谱线的多普勒频移(在吸收器的框架中)
$$\Delta E_{\text {Doppler }}=E_\gamma \frac{v}{c},$$
，因此吸收器再次轻微地失去共振，但由于吸收器的速度只有几毫米每秒，这种能量移只是$10^{11}$ .

$$\Gamma=2 \frac{v_{1 / 2}}{c} E_\gamma=4.6 \pm 0.6 \times 10^{-6} \mathrm{eV}$$

## 物理代写|核物理代写核物理代考|测量同分异构体移位和四极子分裂

$$\Delta E_{\text {quad }}=\pm \frac{\mathcal{Q} V_{z z}}{4}=\pm 8.5 \times 10^{-8} \mathrm{eV} .$$的电四极矩 ${ }{26}^{57} \mathrm{Fe}$ 是 $0.08 \mathrm{eb}$ (电子仓)。这使我们可以估计电场梯度 $$V{z z}=4.25 \times 10^{22} \mathrm{~V} \mathrm{~m}^{-2}$$

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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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