物理代写|核物理代写nuclear physics代考|PHYS585

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Hyperfine Structure

The magnetic moment (vector) of a nucleus is proportional to its spin and is given by
$$\tilde{\boldsymbol{\mu}}_N=g_I \frac{\mu_N}{\hbar} \boldsymbol{I},$$
where $\mu_N$ is the nuclear magneton (4.6), $g_I$ is the nuclear $\mathrm{g}$-factor ${ }^3$ and $\boldsymbol{I}$ is the nuclear spin vector.
The magnetic moment of the atomic electrons is (analogously)
$$\tilde{\boldsymbol{\mu}}_e=g_J \frac{\mu_e}{\hbar} \boldsymbol{J},$$
where $\mu_e$ is the Bohr magneton
$$\mu_e \equiv \frac{e \hbar}{2 m_e} \text {, }$$
where $g_J$ is the atomic $\mathrm{g}$-factor, and $\boldsymbol{J}$ is the total electron angular momentum vector.

These two magnetic moments interact with each other, generating a hyperfine energy shift,

$$\Delta E_{\mathrm{hf}}=\frac{\mu_0}{4 \pi} \tilde{\boldsymbol{\mu}}N \cdot \tilde{\boldsymbol{\mu}}_e\left\langle\frac{1}{r_a^3}\right\rangle=\frac{\mu_0}{4 \pi \hbar^2} g_1 g_J \mu_N \mu_e \boldsymbol{I} \cdot \boldsymbol{J}\left\langle\frac{1}{r_a^3}\right\rangle,$$ where $\mu_0\left(=1 / \epsilon_0 c^2\right)$ is the permeability of the vacuum, and $r_a$ is the radial distance of the electrons from the nucleus. The nuclear and electron angular momenta combine to produce a total angular momentum with quantum number $F$, which takes possible values $$|I-J| \leq F \leq I+J,$$ and using the fact that the entire atomic state is in a simultaneous eigenstate of the operators $F^2, I^2$ and $J^2$ with eigenvalues $F(F+1) \hbar^2, I(I+1) \hbar^2$ and $J(J+1) \hbar^2$, respectively, we may write $$\boldsymbol{I} \cdot \boldsymbol{J}=\frac{\hbar^2}{2}(F(F+1)-I(I+1)-J(J+1)),$$ such that the hyperfine energy shift, $\Delta E{\mathrm{hf}}$, is
\begin{aligned} \Delta E_{\mathrm{hf}} &=\frac{\mu_0}{4 \pi} \frac{1}{2} g_I g_J \mu_N \mu_e\left\langle\frac{1}{r_a^3}\right\rangle(F(F+1)-I(I+1)-J(J+1)) \ &=\frac{\alpha}{2} g_I g_j \frac{\hbar^2}{m_p m_e c}\left\langle\frac{1}{r_a^3}\right\rangle(F(F+1)-I(I+1)-J(J+1)) \end{aligned}

物理代写|核物理代写nuclear physics代考|Isomeric Shift

The wavefunctions for electrons in an $s$-wave $(\ell=0)$ do not vanish at the origin, $\Psi(0) \neq 0$. This means that $s$-wave electrons have a small but non-zero probability of being inside the nucleus. When this is the case, the electrostatic potential between the nucleus and these electrons is smaller than that obtained by treating the nucleus as a point particle. It was pointed out by Richard Weiner [64] that since the effective volume of the nucleus is different for different excited states, this would lead to a

small correction to the energy of the $\gamma$-ray emitted in the transition between two nuclear states.

The shift in energy of a state due to the non-zero volume of a nucleus with charge density $\rho(\mathrm{r})$, interacting with an electron whose wavefunction is $\Psi_e(\boldsymbol{r})$, is given by
$$\Delta E_{\mathrm{vol}}=\frac{e^2}{4 \pi \varepsilon_0} \int d^3 \boldsymbol{r} \int d^3 \boldsymbol{r}^{\prime}\left|\Psi_e(\boldsymbol{r})\right|^2 \rho\left(\boldsymbol{r}^{\prime}\right)\left[\frac{1}{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|}-\frac{1}{|\boldsymbol{r}|}\right]$$
Assuming that the nuclear charge density is spherically symmetric, as well as the $s$-wave electron wavefunctions, the angular integration in (8.14) can be performed to give
$$\Delta E_{\mathrm{vol}}=\frac{4 \pi e^2}{\varepsilon_0} \int r^2 d r\left|\Psi_e(\boldsymbol{r})\right|^2 \int_r^{\infty} d r^{\prime} \rho\left(r^{\prime}\right)\left[r^{\prime}-\frac{r^{\prime 2}}{r} \mid\right.$$
If we treat the nuclear charge density as being uniform inside the nuclear radius, $R$, i.e.
\begin{aligned} \rho(r) &=\frac{3 \angle e}{4 \pi R^3}, \quad(rR), \end{aligned}
the radial integrand is non-zero only for $r<R$. In that region, we can approximate the electron wavefunction by its value at the origin. Radial integration over $r$ and $r^{\prime}$ then gives
$$\Delta E_{\mathrm{vol}}=\frac{4 \pi Z \alpha \hbar c}{10}\left|\Psi_e(0)\right|^2 R^2$$

物理代写|核物理代写核物理学代考|超精细结构

$$\tilde{\boldsymbol{\mu}}_N=g_I \frac{\mu_N}{\hbar} \boldsymbol{I},$$

$$\tilde{\boldsymbol{\mu}}_e=g_J \frac{\mu_e}{\hbar} \boldsymbol{J},$$

$$\mu_e \equiv \frac{e \hbar}{2 m_e} \text {, }$$

$$\Delta E_{\mathrm{hf}}=\frac{\mu_0}{4 \pi} \tilde{\boldsymbol{\mu}}N \cdot \tilde{\boldsymbol{\mu}}_e\left\langle\frac{1}{r_a^3}\right\rangle=\frac{\mu_0}{4 \pi \hbar^2} g_1 g_J \mu_N \mu_e \boldsymbol{I} \cdot \boldsymbol{J}\left\langle\frac{1}{r_a^3}\right\rangle,$$ 哪里 $\mu_0\left(=1 / \epsilon_0 c^2\right)$ 真空的磁导率，和 $r_a$ 是电子到原子核的径向距离。原子核的角动量和电子的角动量结合在一起产生一个具有量子数的总角动量 $F$，它接受可能的值 $$|I-J| \leq F \leq I+J,$$ 利用整个原子状态是同时存在的算子的特征态这一事实 $F^2, I^2$ 和 $J^2$ 带有特征值 $F(F+1) \hbar^2, I(I+1) \hbar^2$ 和 $J(J+1) \hbar^2$，分别，我们可以写 $$\boldsymbol{I} \cdot \boldsymbol{J}=\frac{\hbar^2}{2}(F(F+1)-I(I+1)-J(J+1)),$$ 以至于超精细能量转移， $\Delta E{\mathrm{hf}}$，为
\begin{aligned} \Delta E_{\mathrm{hf}} &=\frac{\mu_0}{4 \pi} \frac{1}{2} g_I g_J \mu_N \mu_e\left\langle\frac{1}{r_a^3}\right\rangle(F(F+1)-I(I+1)-J(J+1)) \ &=\frac{\alpha}{2} g_I g_j \frac{\hbar^2}{m_p m_e c}\left\langle\frac{1}{r_a^3}\right\rangle(F(F+1)-I(I+1)-J(J+1)) \end{aligned}

物理代写|核物理代写核物理代考|同分异构体移位

$s$ -波$(\ell=0)$中的电子波函数在原点$\Psi(0) \neq 0$处不消失。这意味着$s$ -波电子在原子核内部的概率很小，但非零。在这种情况下，原子核和这些电子之间的静电势比把原子核当作点粒子得到的静电势要小。Richard Weiner[64]指出，由于不同激发态下原子核的有效体积是不同的，这将导致

$$\Delta E_{\mathrm{vol}}=\frac{e^2}{4 \pi \varepsilon_0} \int d^3 \boldsymbol{r} \int d^3 \boldsymbol{r}^{\prime}\left|\Psi_e(\boldsymbol{r})\right|^2 \rho\left(\boldsymbol{r}^{\prime}\right)\left[\frac{1}{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|}-\frac{1}{|\boldsymbol{r}|}\right]$$

$$\Delta E_{\mathrm{vol}}=\frac{4 \pi e^2}{\varepsilon_0} \int r^2 d r\left|\Psi_e(\boldsymbol{r})\right|^2 \int_r^{\infty} d r^{\prime} \rho\left(r^{\prime}\right)\left[r^{\prime}-\frac{r^{\prime 2}}{r} \mid\right.$$

\begin{aligned} \rho(r) &=\frac{3 \angle e}{4 \pi R^3}, \quad(rR), \end{aligned}
，只有$r<R$的径向被积函数不为零。在这个区域，我们可以用电子波函数在原点处的值近似它。对$r$和$r^{\prime}$的径向积分得到
$$\Delta E_{\mathrm{vol}}=\frac{4 \pi Z \alpha \hbar c}{10}\left|\Psi_e(0)\right|^2 R^2$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYS161

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Radiation Modes and Selection Rules

As in the case of $\beta$-decay, the emitted photon in $\gamma$-decay can carry off angular momentum $\ell$, which permits a transition between an initial state with spin $I_i$ and a final state with spin $I_f$, provided that angular momentum is conserved, i.e. that the vector sum of $\boldsymbol{I}_f$ and the photon angular momentum, $\boldsymbol{\ell}$, must be equal to $\boldsymbol{I}_i$. The allowed values of $\ell$ are then given by
$$\left|I_i-I_f\right| \leq \ell \leq I_i+I_f .$$
The interactions responsible for $\gamma$-decay are the electromagnetic interactions (as is the case for atomic transitions). There are two types of electromagnetic transitions – electric transitions and magnetic transitions. Electric transitions with angular momentum $\ell=1,2, \ldots$ are denoted by the symbols E1, E2,… They are called “electric $2^l$-pole transitions” – “electric dipole”, “electric quadrupole” etc. Magnetic transitions with angular momentum $\ell=1,2, \ldots$ are denoted by the symbols M1, M2,… They are called “magnetic $2^l$-pole transitions” – “magnetic dipole”, “magnetic quadrupole” etc. The emitted radiation from such transitions is known as “radiation modes”.

Unlike the weak interactions, which mediate $\beta$-decay, the electromagnetic interactions are parity conserving. An electric dipole, $\boldsymbol{d}_E=e \boldsymbol{r}$, is odd under parity transformation so that electric dipole transitions are only permitted between initial and final states of opposite parity. On the other hand, a magnetic dipole is proportional to the spin, $s$, of the nucleon that makes the transition. This is an axial vector and therefore even under parity transformations, implying that magnetic dipole transitions are only permitted between initial and final states of the same parity.

More generally, for an electric transition $\mathrm{E} \ell$, the parities, $\pi$, of the initial and final states are related by
$$\pi_i=(-1)^{\ell} \pi_f,$$
whereas for a magnetic transition $\mathrm{M} \ell$, the parities of the initial and final states are related by
$$\pi_i=(-1)^{(\ell+1)} \pi_f .$$

物理代写|核物理代写nuclear physics代考|Decay Rates

The decay rates for different radiation modes were estimated by Victor Weisskopf [63] in 1951. A rigorous calculation of transition rates effected by electromagnetic interactions requires “Quantum Electrodynamics” (QED), but we can obtain the Weisskopf estimate for electric multipole transitions using Fermi’s golden rule (7.50), with the electric interaction Hamiltonian for the emission of a photon with energy $E_\gamma$ obtained from QED
$$H_{E_\gamma}(\boldsymbol{r})=\sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \Psi_{k_\gamma^*}(\boldsymbol{r}),$$
where $\Psi_{k_y}(\boldsymbol{r})$ is the plane-wave wavefunction for the outgoing photon (in a volume $V)$ with wave number $k_\gamma\left(=E_\gamma / \hbar c\right)$. The decay rate for an electric multipole transition $\mathrm{E} \ell$ of a nuclide with atomic mass number $A$ is then given approximately by 1
$$\lambda_{\mathrm{E} l}\left(A, E_\gamma\right) \approx \frac{2 \alpha c}{r_0} \frac{(\ell+1)}{\ell((2 \ell+1) ! !)^2}\left(\frac{3}{\ell+3}\right)^2\left(\frac{r_0 E_\gamma}{\hbar c}\right)^{(2 \ell+1)} A^{2 \ell / 3},$$
where the nuclear radius, $R$, is given by $R=r_0 A^{1 / 3}$.
The estimate of the decay rates for magnetic transitions involves the nuclear spin. We would expect the magnetic interaction Hamiltonian, $H_M$, to be proportional to the magnetic moment of the nucleon which makes the transition. ${ }^2$ Weisskopf estimated that the magnetic interaction Hamiltonian for a nucleus of radius $R$ can be approximated by $$H_M \approx \sqrt{10} \frac{\hbar}{m_p c R} H_{E_Y}$$
with $H_{E_\gamma}$ given by (8.5). The decay rate for magnetic transitions is therefore
$$\lambda_{\mathrm{M} l}\left(A, E_\gamma\right) \approx 20 \frac{\alpha \hbar^2}{r_0^3 m_p^2 c} \frac{(\ell+1)}{\ell((2 \ell+1) ! !)^2}\left(\frac{3}{\ell+3}\right)^2\left(\frac{r_0 E_\gamma}{\hbar c}\right)^{(2 \ell+1)} A^{(2 l-2) / 3} .$$

物理代写|核物理代写nuclear physics代考|辐射模式和选择规则

$$\left|I_i-I_f\right| \leq \ell \leq I_i+I_f .$$

“electric quadrupole”等。具有角动量的磁跃迁 $\ell=1,2, \ldots$ 用符号 M1、M2、……表示它们被称为“磁性 $2^l$ 极跃 迁”一一“磁偶极子”、“磁四极子”等。从这种跃迁发出的辐射称为“辐射模式”。

$$\pi_i=(-1)^{\ell} \pi_f,$$

$$\pi_i=(-1)^{(\ell+1)} \pi_f$$

物理代写|核物理代写nuclear physics代考|衰减率

$$H_{E_\gamma}(\boldsymbol{r})=\sqrt{\frac{2 \pi \alpha \hbar^3 c^3}{E_\gamma}} \Psi_{k_\gamma^*}(\boldsymbol{r}),$$

$$\lambda_{\mathrm{E} l}\left(A, E_\gamma\right) \approx \frac{2 \alpha c}{r_0} \frac{(\ell+1)}{\ell((2 \ell+1) ! !)^2}\left(\frac{3}{\ell+3}\right)^2\left(\frac{r_0 E_\gamma}{\hbar c}\right)^{(2 \ell+1)} A^{2 \ell / 3},$$

$$H_M \approx \sqrt{10} \frac{\hbar}{m_p c R} H_{E_Y}$$

$$\lambda_{\mathrm{Ml}}\left(A, E_\gamma\right) \approx 20 \frac{\alpha \hbar^2}{r_0^3 m_p^2 c} \frac{(\ell+1)}{\ell((2 \ell+1) ! !)^2}\left(\frac{3}{\ell+3}\right)^2\left(\frac{r_0 E_\gamma}{\hbar c}\right)^{(2 \ell+1)} A^{(2 l-2) / 3} .$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYSICS404

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Fermi’s Golden Rule

The approximate expression for the transition rate for a system due to a perturbing potential is known as Fermi’s golden rule, although it was actually first derived by Paul Dirac [62].

If a time-independent perturbing potential, $H^{\prime}$, is applied to a quantum system in a state $|i\rangle$, energy $E_i$, at time, $t=0$, then the amplitude $a_{f i}(t)$ for the system to have made a transition to the state $|f\rangle$, with energy $E_f$, at time $t$ is given by first order time-dependent perturbation theory to be
$$a_{f i}(t)=2 e^{i \eta}\left\langle f\left|H^{\prime}\right| i\right\rangle \frac{\sin \left(\frac{1}{2}\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)},$$
where $\left\langle f\left|H^{\prime}\right| i\right\rangle$ is the matrix element of the perturbing Hamiltonian between the initial state $|i\rangle$ and final state $|f\rangle$, and $\eta$ is a phase.

The probability, $T_{f i}(t)$, for such a transition to have occurred by time $t$, is then
$$T_{f i}(t)=\left|a_{f i}(t)\right|^2=4\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \frac{\sin ^2\left(\frac{1}{2}\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)^2} .$$
The transition rate, $\lambda_{f i}$, is given by the derivative of $T_{f i}$ with respect to time
$$\lambda_{f i}=\frac{2}{\hbar}\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \frac{\sin \left(\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)} .$$
To determine the total transition rate, $\lambda$, to any final state, we sum over all final states $|f\rangle$. Ilowever, if these final states are in a continuum, this discretè sum is replaced by an integral over final-state energy, $E_f$, with a Jacobian factor equal to the density of states, $\rho\left(E_f\right)$-the number of quantum states per unit energy interval. We then obtain
$$\lambda=\frac{2}{\hbar} \int d E_f\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \rho\left(E_f\right) \frac{\sin \left(\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)} .$$

物理代写|核物理代写nuclear physics代考|Gamma Decay

The emission of $\gamma$-rays from nuclei is the nuclear analogue of the atomic emission of photons, which occur when an electron makes a transition from an excited state either to a lower excited state or to the atomic ground state. Similarly, $\gamma$-rays are emitted when a nucleus in an excited state makes a transition to a lower state. Atomic excitation energies are typically of the order of a few electron volts ( $\mathrm{eV})$, leading to the emission of photons with wavelengths of hundreds of nanometres encompassing the visible spectrum, whereas nuclear excitations are of the order of hundreds of $\mathrm{KeV}$, emitting $\gamma$-rays with wavelengths of the order of a picometre $(1000 \mathrm{fm})$, although some nuclear excitation energies are less than $100 \mathrm{keV}$, so that the emitted photons are strictly classified as $\mathrm{X}$-rays. In contrast to atomic radiation, $\gamma$-rays are usually described in terms of their energies, $E_\gamma$, rather than their wavelengths.
Most excited states have a very short lifetime – of order $10^{-13}-10^{-10} \mathrm{~s}$. However, there are some excited states which are metastable and therefore have a much longer lifetime. An example of this is the nuclide ${ }_{27}^{58} \mathrm{Co}$, which has a metastable excited state with energy $24.9 \mathrm{keV}$ and half-life of about $9 \mathrm{~h}$. Such excited states are called “nuclear isomers” and their decays are called “isomer transitions” – often abbreviated to IT.

An excited state with decay rate $\lambda$ has a mean lifetime $\tau-1 / \lambda$ (see (5.3)). By Heisenberg’s uncertainty principle, this implies that the energy of the excited state has an uncertainty $\frac{1}{2} \hbar / \tau$, so that the spectral line of an emitted $\gamma$-ray has a halfwidth, $\frac{1}{2} \Gamma_\gamma$, which is equal to that uncertainty. The line-width is therefore given by
$$\Gamma_\gamma=\frac{\hbar}{\tau}=\hbar \lambda .$$

物理代写|核物理代写核物理学代考|费米黄金定律

. .

$$a_{f i}(t)=2 e^{i \eta}\left\langle f\left|H^{\prime}\right| i\right\rangle \frac{\sin \left(\frac{1}{2}\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)},$$
，其中$\left\langle f\left|H^{\prime}\right| i\right\rangle$是初始态$|i\rangle$和最终态$|f\rangle$之间的摄动哈密顿量的矩阵元，$\eta$是一个相

$$T_{f i}(t)=\left|a_{f i}(t)\right|^2=4\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \frac{\sin ^2\left(\frac{1}{2}\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)^2} .$$
$\lambda_{f i}$由的导数给出 $T_{f i}$ 关于时间
$$\lambda_{f i}=\frac{2}{\hbar}\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \frac{\sin \left(\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)} .$$

$$\lambda=\frac{2}{\hbar} \int d E_f\left|\left\langle f\left|H^{\prime}\right| i\right\rangle\right|^2 \rho\left(E_f\right) \frac{\sin \left(\left(E_i-E_f\right) t / \hbar\right)}{\left(E_i-E_f\right)} .$$

物理代写|核物理代写核物理学代考|伽马衰变

$$\Gamma_\gamma=\frac{\hbar}{\tau}=\hbar \lambda .$$ 给出

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYS585

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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

One of the useful applications of radioactivity is the “radiometric dating” method for determining the age of rock samples. The concentrations (number of nuclei per unit volume) of different nuclides in a rock sample can be measured using a mass spectrometer. Suppose the concentration of a radioactive nuclide, whose decay constant is $\lambda$, is $X(t)$, where $t$ is the age of the rock sample, and the concentration of its daughter nuclide is $Y(t)$. When the rock was formed these concentrations were $X(0)$ and zero, respectively, where for the moment, we have assumed that the rock contained none of the daughter nuclide when the rock was formed.
From (5.2) we have
$$X(t)=X(0) e^{-\lambda t}$$
and
$$Y(t)=X(0)\left(1-e^{-\lambda t}\right),$$
so that the age of the rock is given by
$$t=\frac{1}{\lambda} \ln \left(1+\frac{Y(t)}{X(t)}\right) .$$
However, in most cases there was a primordial concentration of the daughter nuclide at the time of formation of the rock, i.e. $Y(0) \neq 0$, so that (5.9) is modified to $$Y(t)=Y(0)+X(0)\left(1-e^{-\lambda t}\right)$$
This is known as the “age equation”. One of the most effective ways of determining the primordial concentration of the daughter nuclide is to use the fact that there is usually another stable isotope of the daughter nuclide present in the rock sample, which is not involved in the radioactive process and whose concentration is therefore time-independent. Let us call the concentration of this other isotope $W$. We have the relation
$$Y(0)=\rho W,$$
where $\rho$ is the relative abundance of the two isotopes at the time of rock formation.

A variant of radiometric dating, invented by Willard Libby in 1946 [45], which can be used for determining the age of organic fossils, is “radiocarbon dating”. The carbon isotope, ${ }6^{14} \mathrm{C}$ decays into ${ }_7^{14} \mathrm{~N}$ (nitrogen), via $\beta$-decay with a half-life, $\tau{\frac{1}{2}}$, of 5730 years.

Living organisms absorb the radioactive isotope of carbon ${ }_6^{14} \mathrm{C}$, which is created in the atmosphere by cosmic ray activity. The production of ${ }_6^{14} \mathrm{C}$ from cosmic ray bombardment exactly cancels the rate at which that isotope of carbon decays so that the global concentration of ${ }_6^{14} \mathrm{C}$ remains constant.

A sample of carbon taken from a living organism has a relative abundance, $\rho$, equal to about one part in $10^{12}$. This isotope is being continually circulated by exchanging carbon with the environment (either by photosynthesis or by eating plants which have undergone photosynthesis or by eating other animals that have eaten such plants), so all living organisms – plants or animals – have the same small abundance, $\rho$, of ${ }_6^{14} \mathrm{C}$.

On the other hand, a sample of carbon from a dead object does not exchange its carbon with the environment, and therefore its concentration of ${ }6^{14} \mathrm{C}$ is not replenished as it decays radioactively. Such fossils therefore have a smaller concentration, $\rho^{\prime}(t)$, of ${ }_6^{14} \mathrm{C}$, where $\rho^{\prime}(t)$ depends on the age, $t$, of the fossil. $$t=\frac{\tau{\frac{1}{2}}}{\ln 2} \ln \left(\frac{\rho^{\prime}(t)}{\rho}\right) .$$
This means that from a measurement of the abundance of ${ }_6^{14} \mathrm{C}$ in a fossil sample, one can determine its age. It is not necessary to measure directly the concentration of ${ }_6^{14} \mathrm{C}$ but simply to ascertain the total mass of the carbon in a given sample. Then, provided that any other radioactive nuclides are present in such small quantities that the radioactivity from them is negligible, it is sufficient merely to measure the radioactivity rate from the fossil sample.

核物理代写

$$X(t)=X(0) e^{-\lambda t}$$

$$Y(t)=X(0)\left(1-e^{-\lambda t}\right)$$

$$t=\frac{1}{\lambda} \ln \left(1+\frac{Y(t)}{X(t)}\right)$$

$$Y(t)=Y(0)+X(0)\left(1-e^{-\lambda t}\right)$$

$$Y(0)=\rho W,$$

$$t=\frac{\tau \frac{1}{2}}{\ln 2} \ln \left(\frac{\rho^{\prime}(t)}{\rho}\right)$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYS161

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Decay Rates

The probability of a parent nucleus decaying in $1 \mathrm{~s}$ is called the “decay constant”, (or “decay rate”) $\lambda$. If we have $N(t)$ parent nuclei at time, $t$, then the number of decays expected per second is $\lambda N(t)$. The number of parent nuclei decreases by this amount and so we have
$$\frac{d N(t)}{d t}=-\lambda N(t) .$$
This differential equation has a simple solution – the number of parent nuclei decays exponentially:
$$N(t)=N(0) e^{-\lambda t} \text {. }$$
The time taken for the number of parent nuclei to fall to $1 / e$ of its initial value is called the “mean lifetime” (or simply “lifetime”), $\tau$, of the radioactive nucleus, and we can see from (5.2) that
$$\tau=\frac{1}{\lambda} .$$
More often one talks about the “half-life”, $\tau_{\frac{1}{2}}$, of a radioactive nuclide, which is the time taken for the number of parent nuclei to fall to one-half of its initial value. From (5.2) we can also see that
$$\tau_{\frac{1}{2}}=\frac{\ln 2}{\lambda}=\tau \ln 2 .$$
For example, the uranium isotope ${ }{92}^{238} \mathrm{U}$ has a half-life of $4.47$ billion years, equivalent to $1.41 \times 10^{17} \mathrm{~s}$. From (5.3) and (5.4), this gives a decay constant $$\lambda=\frac{\ln 2}{1.41 \times 10^{17}[\mathrm{~s}]}=4.92 \times 10^{-18} \mathrm{~s}^{-1} .$$ Using the semi-empirical mass formula, the mass of one atom of ${ }{92}^{238} \mathrm{U}$ is $2.22 \times$ $10^5 \mathrm{MeV} / \mathrm{c}^2$, equivalent to $3.97 \times 10^{-25} \mathrm{~kg}$. Thus we would expect a (pure) sample of $1 \mathrm{~g}$ of this isotope to give an average number of radioactive counts of
$$\bar{n}=\frac{10^{-3}[\mathrm{~kg}]}{\left(3.97 \times 10^{-25}[\mathrm{~kg}]\right)} \times\left(4.92 \times 10^{-18} \mathrm{~s}^{-1}\right)=12,380 \mathrm{~Bq}$$

物理代写|核物理代写nuclear physics代考|Random Decay

A nucleus is a sub-microscopic object to which Quantum Physics must be applied. It is therefore not possible to determine exactly when a given radioactive nucleus will decay. The best we can do is determine the probability that it will decay in unit time (the decay constant, $\lambda$ ).

This means that whereas the “expected” number of decays in a sample of $N$ nuclei is $\lambda N$ per second, this does not mean that there will always be precisely this number of decays per second.

The average number of decays over several measurements of duration $1 \mathrm{~s}$ is given by
$$\bar{n}=\lambda N$$
but there will be random fluctuations around this value. A measure of the size of these fluctuations over a set of $\mathcal{N}$ measurements, $n_i$, with average value $\bar{n}$, is given by the “standard deviation”, which is determined as follows:

• For each measurement, determine the deviation of the number of counts, $n_i$ from the average value $\bar{n}$.
• Since this number can be positive or negative with an average value of zero, this quantity is squared – the square of the deviation is always positive.
• Take the average of the square of the deviation.
• The standard deviation, $\sigma$, is the square root of this quantity. For this reason, the standard deviation is also known as the “root-mean-square (r.m.s.) deviation”.
$$\sigma=\sqrt{\frac{1}{\mathcal{N}} \sum_{i=1}^{\mathcal{N}}\left(n_i-\bar{n}\right)^2}$$
More precisely, if the expected number of decays in a particular time period is $\bar{n}$, then the probability, $P(n)$, that there will be $n$ decays in that period is given by the “Poisson distribution” 2

物理代写|核物理代写nuclear physics代考|Decay Rates

$$\frac{d N(t)}{d t}=-\lambda N(t) .$$

$$N(t)=N(0) e^{-\lambda t} .$$

$$\tau=\frac{1}{\lambda} .$$

$$\tau_{\frac{1}{2}}=\frac{\ln 2}{\lambda}=\tau \ln 2 .$$

$$\lambda=\frac{\ln 2}{1.41 \times 10^{17}[\mathrm{~s}]}=4.92 \times 10^{-18} \mathrm{~s}^{-1} .$$

$$\bar{n}=\frac{10^{-3}[\mathrm{~kg}]}{\left(3.97 \times 10^{-25}[\mathrm{~kg}]\right)} \times\left(4.92 \times 10^{-18} \mathrm{~s}^{-1}\right)=12,380 \mathrm{~Bq}$$

物理代写|核物理代写nuclear physics代考|Random Decay

$$\bar{n}=\lambda N$$

• 对于每次测量，确定计数的偏差, $n_i$ 从平均值 $\bar{n}$.
• 由于这个数字可以是正数或负数，平均值为零，所以这个数量是平方的一一偏差的平方总是正的。
• 取偏差平方的平均值。
• 标准差， $\sigma$, 是这个量的平方根。因此，标准偏差也称为”均方根 (rms) 偏差”。
$$\sigma=\sqrt{\frac{1}{\mathcal{N}} \sum_{i=1}^{\mathcal{N}}\left(n_i-\bar{n}\right)^2}$$
更准确地说，如果特定时间段内的预期衰减次数为 $\bar{n}$ ，那么概率， $P(n)$ ，会有 $n$ 该时期的衰减由“泊松分布”2 给出

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYSICS404

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Scintillation Counters

The second major type of detectors utilized in radiation detection are scintillation counters, invented by Sam Curran in 1944 [41] whilst working on the “Manhattan project”. These detectors record light produced when radiation interacts with materials that are luminescent. These materials, called “scintillators”, may be liquid or solid. Gaseous scintillators are also in use, usually for detection of heavy charged particles. Unlike a Geiger counter, a scintillation counter is the most effective device for the detection of $\gamma$-rays, although such a device can also be usefully employed for the detection of $\alpha$ – and $\beta$-particles.

A diagram of a scintillation counter is shown in Fig. 5.3. The radiation particle is incident on a crystal of scintillating material. The interaction causes some of the electrons in the crystal to be promoted to an excited state. They then make a spontaneous transition back to their ground state, emitting photons, usually within the visible spectrum. These photons are incident on a photomultiplier tube. This consists of an electrode of photoelectric material kept at a high negative voltage, called a “photocathode”. The emitted photoelectrons are accelerated down the tube, making collisions with “dynodes” (intermediate electrodes), which emit several electrons for each incident electron. These emitted electrons themselves travel down the tube, making collisions with other dynodes, producing yet more electrons, such that at the end of the tube a macroscopic electric pulse is detected and measured. The magnitude of this pulse carries information about the intensity of the original particle. Therefore, unlike a Geiger counter, this device can also be used to measure the energy of the incident particle. The response time is also very short. The time taken for the photons to travel down the photomultiplier tube is of the order of 1 ns and the recovery time of the photomultiplier tube is a few tens of nanoseconds, so that a scintillation counter can count up to ten million events per second.

The following types of scintillators are in common use: inorganic and organic crystals, organic plastics, and liquids. Solid inorganic crystals are characterized by a high density and high atomic number. They are therefore particularly effective for detecting high energy $\gamma$-rays (above $1 \mathrm{MeV}$ ) due to their greater stopping power compared with lower density materials with lower atomic number. There are many different materials used for the scintillator crystal, each with somewhat different properties, designed to detect specific particles more efficiently, making them more useful for different types of detection. Caesium iodide is often used for counters designed to detect $\alpha$-particles, whereas sodium iodide, doped with small amounts of thallium, is found to be more suitable for $\gamma$-ray detection.

物理代写|核物理代写nuclear physics代考|Semiconductor Detectors

The most efficient modern-day detector is the semiconductor detector. The basic operating principle of semiconductor detector is similar to an ionization chamber, but the medium is now a solid semiconductor material instead of gas. This is a semiconductor diode consisting of adjacent strips of p-type and n-type semiconductors. Near the junction between the two, there is a depletion zone with no free charge carriers – neither electrons nor holes.

This depletion zone (the depth of which determines the sensitive region and hence the performance of the detector) is enhanced by placing the diode between electrodes in “reverse bias mode”, i.e. the n-type semiconductor is connected to the positive electrode and the p-type to the negative, as shown in Fig. 5.4. When a radiation particle enters the depletion zone and its energy is absorbed, a large number of electrons are promoted from the “valence band” to the “conduction band”, leaving positively charged holes in the valence band. These pairs of charge carriers diffuse around the circuit giving rise to an electric impulse whose height is proportional to the energy of the incident particle. The pulse rate is equal to the rate of incidence of radiation particles so that these detectors can measure both the rate of incidence and the energies of incident particles. The energy required to create an electron-hole pair is a few eV, which is an order of magnitude less than a typical ionization energy in a gas. A radiation particle with energy $1 \mathrm{MeV}$ can create hundreds of thousands of pairs of charge carriers. Semiconductor detectors have extremely good energy resolution and can also be used for the detection of relatively low-energy particles.

The first radiation measurement using a semiconductor detector was carried out by Pieter Van Heerden in $1945[42,43]$ using a cooled silver chloride crystal. Nowadays, the semiconductor material is usually silicon or germanium, although cadmium-telluride is also sometimes used. Strips of silicon semiconductor have to be only a few millimetres thick, whereas germanium semiconductor strips can be thicker, with a depletion zone of up to a few centimetres, enabling them to absorb $\gamma$-rays with energies up to a few MeV. Another advantage of the semiconductor detector is that as semiconductors are around 1000 times more dense than gas, they can be made much more compact. Furthermore, semiconductor detectors have better spatial resolution than scintillators by about one order of magnitude and thus have a wide application in high energy physics detectors, in particular at the $\mathrm{LHC}$.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PHYS729

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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

One might ask whether it was correct to assume that classical physics was applicable for the description of Rutherford scattering, which probes sub-atomic scales where we might expect quantum effects to be significant. Of course, at the time of Rutherford’s calculation, Quantum Physics was unknown, but nowadays we know that the incident $\alpha$-particle has an associated de Broglie wave, and that, in general, a wave scattering from a regular configuration of gold atoms will produce a diffraction pattern. The angular scale of such diffraction patterns is of the order of the de Broglie wavelength divided by the mean separation of the gold atoms in the foil.
The de Broglie wavelength, $\lambda$, is given by
$$\lambda=\frac{h}{m_{\alpha} v}=\frac{h}{\sqrt{2 m_{\alpha} T}},$$
( $h$ is Planck’s constant), the mass of an $\alpha$-particle is $6.6 \times 10^{-27} \mathrm{~kg}$, and for $\alpha$ particles with kinetic energy $5 \mathrm{MeV}\left(8 \times 10^{-13} \mathrm{~J}\right)$ this gives a wavelength
$$\lambda \approx 6 \times 10^{-15} \mathrm{~m} .$$
In contrast, the separation of the gold atoms is around $170 \mathrm{~nm}$.
This means that the effect of diffraction from the gold atoms is negligible. On the other hand, the size of the nucleus itself is indeed of the order of the de Broglie wavelength of the incident particles, so that for projectiles with somewhat smaller wavelengths, diffraction patterns can be observed from diffraction off single nuclei and these patterns can yield useful information about the structure of nuclei. This is the subject of Chap. 2 .

物理代写|核物理代写nuclear physics代考|Relation Between Scattering Angle and Impact

The relation between impact parameter, $b$, and scattering angle, $\theta$, is derived using Newton’s second law of motion, Coulomb’s law for the force between the $\alpha$-particle and the nucleus, and conservation of angular momentum.

The initial and final momenta, $\boldsymbol{p}{i}, \boldsymbol{p}{f}$, have equal magnitude, $p$, (elastic scattering with no nuclear recoil is assumed). If we take $p_{i}$ to be along the $z$-axis and the scattering to be in the $x-z$ plane, then in Cartesian coordinates these two vectors are given by
$$\begin{gathered} p_{i}=p(0,0,1) \ p_{f}=p(\sin \theta, 0, \cos \theta) \end{gathered}$$
and the momentum transfer is given by
$$\boldsymbol{q} \equiv \boldsymbol{p}{f}-\boldsymbol{p}{i}=p(\sin \theta, 0,(\cos \theta-1)) .$$
Using Pythagoras’ theorem and some trigonometric manipulation, the momentum transfer, $\boldsymbol{q}$, has a magnitude
$$q=2 p \sin \left(\frac{\theta}{2}\right)$$
The direction of the vector $q$ is along the line joining the nucleus to the point of closest approach of the $\alpha$-particle. It bisects the vectors $\boldsymbol{p}{i}$ and $\boldsymbol{p}{f}$, making an angle $(\pi-\theta) / 2$ with each, as can be seen from Fig. 1.6.

The position vector, $\mathbf{r}$, from the nucleus and the $\alpha$-particle is given in terms of two-dimensional polar coordinates $(r, \phi)$ with the nucleus as the origin. The angle $\phi$ is set such that $\phi=0$ is at the point of closest approach, where $\mathbf{r}$ lies along the vector $\boldsymbol{q}$.

核物理代写

$$\lambda=\frac{h}{m_{\alpha} v}=\frac{h}{\sqrt{2 m_{\alpha} T}},$$
( $h$ 是普朗克常数) ，质量 $\alpha$-粒子是 $6.6 \times 10^{-27} \mathrm{~kg}$ ，并且对于 $\alpha$ 具有动能的粒子 $5 \mathrm{MeV}\left(8 \times 10^{-13} \mathrm{~J}\right)$ 这给 出了一个波长
$$\lambda \approx 6 \times 10^{-15} \mathrm{~m} .$$

物理代写|核物理代写nuclear physics代考|Relation Between Scattering Angle and Impact

$$p_{i}=p(0,0,1) p_{f}=p(\sin \theta, 0, \cos \theta)$$

$$\boldsymbol{q} \equiv \boldsymbol{p} f-\boldsymbol{p} i=p(\sin \theta, 0,(\cos \theta-1))$$

$$q=2 p \sin \left(\frac{\theta}{2}\right)$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|ERICE2022

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|Inconsistency of the “Plum Pudding” Model

Let us consider what would be expected if the “Plum Pudding” model were indeed correct.

We know from Gauss’ law that at a distance $r$ from the centre of the atom, the electric field is determined by the charge enclosed in a sphere of radius $r$ surrounding the centre of the atom.

The volume of a sphere of radius $r$ is proportional to $r^{3}$. Therefore for $r$ smaller than the radius, $R$, of the atom, the electric charge enclosed with a sphere of radius $r$ is a fraction $r^{3} / R^{3}$ of the total electric charge (assuming a uniform distribution of electric charge throughout the “dough”), so that the magnitude of the electric field at a distance $r$ from the centre of the atom is given by
$$\left(\frac{r^{3}}{R^{3}}\right) \frac{Z e}{4 \pi \varepsilon_{0} r^{2}},(r \leq R) .$$
This is a maximum for $r=R$. This means that the scattering angle cannot be larger than the scattering angle corresponding to impact parameter $b=R$. For values of impact parameter $b<R$, the scattering angle decreases as $b$ decreases.

We have seen above that for $\alpha$-particles with typical kinetic energy of $5 \mathrm{MeV}$, this corresponds to a maximum scattering angle of around $3 \times 10^{-4}$ radians $\left(\approx 0.017^{\circ}\right)$. Such an angle would have been far too small to be observed in any of the GeigerMarsden experiments and they certainly would not have observed any scattering exceeding $90^{\circ}$.

物理代写|核物理代写nuclear physics代考|Confirmation of Rutherford Scattering Cross Section

In 1913, Geiger and Marsden [18] performed a far more accurate experiment to check the details of Rutherford’s formula (1.12). They checked the dependence of the rate on the scattering angle and found consistency with the prediction
$$N(\theta) \propto \frac{1}{\sin ^{4}(\theta / 2)}$$
Their results, shown in Fig. 1.5, agree remarkably well.
By using foils of different thickness, they showed that the number of particles scattered through a given angle was proportional to the thickness of the foil, and by using foils made from different metals (tin, silver, copper and aluminium) they were able to show that this number was proportional to the square of the atomic number, $Z$, of the material of the foil.

They were able to slow down the incident $\alpha$-particles, by placing thin sheets of mica immediately in front of the radioactive source. From this they were able to verify that the number of scattered particles was inversely proportional to the fourth power of their velocity, as indicated in (1.12).

物理代写|核物理代写nuclear physics代考|Inconsistency of the “Plum Pudding” Model

$$\left(\frac{r^{3}}{R^{3}}\right) \frac{Z e}{4 \pi \varepsilon_{0} r^{2}},(r \leq R)$$

物理代写|核物理代写nuclear physics代考|Confirmation of Rutherford Scattering Cross Section

1913 年，Geiger 和 Marsden [18] 进行了更准确的实验来检查卢瑟福公式 (1.12) 的细节。他们检查了速率对散 射角的依赖性，并发现与预测的一致性
$$N(\theta) \propto \frac{1}{\sin ^{4}(\theta / 2)}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|核物理代写nuclear physics代考|PX4516

statistics-lab™ 为您的留学生涯保驾护航 在代写核物理nuclear physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写核物理nuclear physics代写方面经验极为丰富，各种代写核物理nuclear physics相关的作业也就用不着说。

• 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|The Geiger-Marsden Experiments

As we shall see later the “Plum Pudding” model predicts that a charged particle which is moving through such a positively charged “dough” will experience a very weak electric force and will only undergo very small angular deflections. In order to verify this, Hans Geiger and Ernest Marsden, at the behest of Ernest Rutherford, carried out three experiments between 1908 and 1910 in which $\alpha$-particles from a radioactive source were incident on a very thin foil of gold (gold was selected because it can be beaten very thin – the foil used by Geiger and Marsden had a thickness of $400 \mathrm{~nm}$ ). The entire apparatus was encased in a tube, which was evacuated in order to minimize energy loss of the $\alpha$-particles before they scattered off the foil. A schematic sketch of the experimental setup is shown in Fig. 1.1.
In the first experiment, [13] a screen was placed behind the gold foil and scintillations caused by the $\alpha$-particles landing on the screen, were observed with a travelling microscope. Although most $(86 \%)$ of the $\alpha$-particles passed through with a deflection of less than $1^{\circ}$, a substantial angular spread of scintillations was observed.

In the second experiment [14], the screen was placed on the incident side of the gold foil in order to observe reflected $\alpha$-particles. The screen was protected from direct $\alpha$-particles by placing an impenetrable lead plate in the direct path of the particles. They nevertheless observed that about one particle in 8000 was reflected by the foil, implying that there had been scattering through an angle of greater than $90^{\circ}$ – way above the limit predicted by the “Plum Pudding” model.

In a third experiment [15], a year later, Geiger and Marsden used several different foils of different thickness and made of different materials. In this experiment, they managed to determine the most probable deflection angle. They showed that the most probable angle of scattering:

1. Increased with increasing thickness of the foil,
2. Increased with the atomic mass of the material the foil,
3. Decreased with increasing velocity of the incident $\alpha$-particles.

物理代写|核物理代写nuclear physics代考|Rutherford’s Scattering Formula

Rutherford’s surprise at the results of the Geiger-Marsden experiment, particularly the fact that some of the $\alpha$-particles were scattered though an angle of more than $90^{\circ}$, led him to state during a lecture at Cambridge University:

In 1911 , he adopted the model postulated 7 years earlier by the Japanese physicist Hantaro Nagaoka [16]. This model comprised of a small positively charged nucleus at the centre of an atom with electrons orbiting around it. Within this model, Rutherford calculated the probability of scattering of the $\alpha$-particles through an angle $\theta$ [17] under the following assumptions:

• The atom contains a nucleus of charge $Z e$, where $Z$ is the atomic number of the atom (i.e. the number of electrons in the neutral atom),
• The nucleus can be treated as a point particle,
• The nucleus is sufficiently massive compared with the mass of the incident $\alpha$ particle that the nuclear recoil may be neglected,
• The laws of classical mechanics and Electromagnetism can be applied and that no other forces are present,
• The collision is elastic.

物理代写|核物理代写nuclear physics代考|The Geiger-Marsden Experiments

1. 随着箔厚度的增加而增加，
2. 随着箔材料的原子质量增加，
3. 随着事件速度的增加而减少一个-粒子。

物理代写|核物理代写nuclear physics代考|Rutherford’s Scattering Formula

1911 年，他采用了日本物理学家长冈半太郎 7 年前提出的模型[16]。该模型由位于原子中心的带正电的小核组成，电子围绕它运行。在这个模型中，卢瑟福计算了散射的概率一个- 粒子通过一个角度一世[17] 在以下假设下：

• 原子含有一个电荷核从和， 在哪里从是原子的原子序数（即中性原子中的电子数），
• 原子核可以看作一个点粒子，
• 与事件的质量相比，原子核的质量足够大一个可以忽略核反冲的粒子，
• 可以应用经典力学和电磁学定律，并且不存在其他力，
• 碰撞是弹性的。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。