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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|The Gravitational Redshift

One of the most spectacular applications of the Mössbauer effect was the measurement of the gravitational shift of a photon as it falls through a gravitational field. Einstein’s General Theory of Relativity predicts that a photon emitted with energy $E$, travelling downwards from a height $h$ in a gravitational field with acceleration $g$, will increase its energy by
$$\Delta E_{\text {grav }}=\frac{E g h}{c^2} .$$
In 1959, Robert Pound and Glen Rebka [67] detected this gravitational shift using the Mössbauer effect. They placed a source sample of ${ }_{26}^{57} \mathrm{Fe}$ on the top of the tower of the Jefferson Laboratory at Harvard University, $22.6$ metres above ground level, and an absorber sample at the bottom of the tower, with a scintillation counter below it. From (8.23), the resonant velocity at which the source must move relative to the absorber is
$$v=\frac{g h}{c}=7.4 \times 10^{-4} \mathrm{~mm} \mathrm{~s}^{-1} .$$
This very small velocity, at which absorption was observed, was measured by placing the sample at the top of the tower in the cone of a loudspeaker to which they applied a pure sinusoidal signal with frequency that ranged between 10 and $50 \mathrm{~Hz}$. Absorption occurs once every cycle when the velocity of the loudspeaker is exactly equal to the resonant velocity. The determination of the precise phase of the oscillation at which this absorption is observed allows one to calculate the velocity of the membrane of the loudspeaker at which absorption occurs. In order to improve the accuracy, the experiment was conducted both with the upper sample as the source ${ }^6$ and the lower sample as the absorber and the other way around.

They obtained a result which was within $10 \%$ of the theoretical result. This accuracy was later improved to $1 \%$ [68].

## 物理代写|核物理代写nuclear physics代考|Internal Conversion

As well as decaying by the emission of a $\gamma$-ray, a nucleus in an excited state can interact directly with an inner atomic electron and free it from its atomic binding. The electron is then emitted with a fixed energy equal to the energy difference of the excited state and the ground state (or lower excited state) minus the atomic binding energy of the electron.

The emitted electron is not created inside the nucleus as in $\beta$-decay, nor is it accompanied by the emission of an antineutrino. This type of electron emission is not classified as $\beta$-decay and is known as “internal conversion”.

An example is shown in Fig. 8.4. An excited state of ${ }{81}^{203} \mathrm{Tl}$ (thallium), with energy $279 \mathrm{keV}$ above the ground state, is created from the $\beta$-decay of ${ }{80}^{203} \mathrm{Hg}$ (mercury). The $\beta$-decay has a $Q$-value of $214 \mathrm{keV}$, so that we find the usual spectrum of $\beta$-decay electrons with kinetic energy up to $214 \mathrm{keV}$. Since the daughter nuclide is created in the excited state, these events are followed almost immediately by the emission of a $\gamma$-ray with energy $279 \mathrm{keV}$. In addition, there are two peaks of observed electrons at 195 and $264 \mathrm{keV}$. These are internal conversion electrons, whose binding energies are $85 \mathrm{keV}$ (K-shell) and $15 \mathrm{keV}$ (L-shell), so that these electrons are emitted with energies $(279-85)=194 \mathrm{keV}$ and $(279-15)=264 \mathrm{keV}$, respectively.

These electrons are emitted as a result of the Coulomb interaction between the nucleus and the inner atomic electrons. This interaction is largest if the probability of finding the electron near the nucleus is relatively large. The electrons emitted in internal conversion are therefore nearly always in an $s$-wave state, whose wavefunction does not vanish at the origin.

The ejected electron leaves a vacancy in an inner atomic level, which is then filled by an electron from an outer level. This electron transition is accompanied by the emission of a photon, usually in the X-ray range.

Internal conversion is always possible as an alternative to $\gamma$-ray emission. The ratio of the rate of internal electron emissions to the rate of $\gamma$-ray emissions is called the “internal conversion coefficient” and is denoted by $\alpha_{{n}}$ for the emission of an electron from the shell ${n}(n=K, L, \ldots)$. If $\lambda_\gamma$ is the decay rate for $\gamma$-decay of an excited state, then the $K$-shell internal conversion rate for that state is $\alpha_K \lambda_\gamma$ etc., so that the total decay rate for the state is $\left(1+\sum_n \alpha_{{n}}\right) \lambda_\gamma$. In the example above, the internal conversion coefficient for the $\mathrm{K}$-shell electron is $\alpha_K=0.163$, meaning that for every $1000 \gamma$-ray emissions, there are 163 internal conversion $\mathrm{K}$ shell electron emissions. As can be seen from rig. $8.4$, the internal conversion factor for the emission of an electron from the L-shell is smaller by about a factor of 3 . Internal conversion coefficients increase with atomic number, $Z$, and decrease with increasing $\gamma$-ray energy.

# 核物理代写

## 物理代写|核物理代写核物理学代考|引力红移

.

Mössbauer效应最引人注目的应用之一是测量光子在穿过引力场时的引力位移。爱因斯坦的广义相对论预言，一个能量为$E$的光子，在一个加速度为$g$的引力场中从高度$h$向下运动，其能量将增加
$$\Delta E_{\text {grav }}=\frac{E g h}{c^2} .$$
。1959年，Robert Pound和Glen Rebka[67]利用Mössbauer效应探测到这种引力转移。他们把${ }_{26}^{57} \mathrm{Fe}$的源样本放在哈佛大学杰斐逊实验室的塔顶，离地面$22.6$米，把吸收体样本放在塔底，下面有一个闪烁计数器。从(8.23)，共振速度时，源必须移动相对于吸收器是
$$v=\frac{g h}{c}=7.4 \times 10^{-4} \mathrm{~mm} \mathrm{~s}^{-1} .$$

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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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