## 物理代写|固体物理代写Solid-state physics代考|KYA322

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Electrical conductivity

The first application of the Drude theory is to predict the direct-current electrical conductivity of a metal. Let $\mathbf{v}{\mathrm{d}}$ be the electron drift velocity under the action of an externally-applied uniform and constant electric field $\mathbf{E}$. The overall dynamical effect of the collisions experienced by the accelerated electrons is described as a frictional term in their Newton equation of motion $$-e \mathbf{E}=m{\mathrm{e}} \dot{\mathbf{v}}{\mathrm{d}}+\beta \mathbf{v}{\mathrm{d}},$$
where $\beta$ is a coefficient to be determined. Basically, the added frictional term forces the electron distribution to relax towards the equilibrium Fermi-Dirac one when the external electric field is removed. In a steady-state condition we have $d \mathbf{v}{\mathrm{d}} / d t=0$ and therefore $$-\frac{e}{m{\mathrm{e}}} \mathbf{E}=\frac{\beta}{m_{\mathrm{e}}} \mathbf{v}{\mathrm{d}},$$ which naturally ${ }^4$ leads to defining $\beta=m{\mathrm{e}} / \tau_{\mathrm{e}}$. This allows us to calculate the electron drift velocity as $$\mathbf{v}{\mathrm{d}}=-\frac{e \tau{\mathrm{e}}}{m_{\mathrm{e}}} \mathbf{E},$$
from which we obtain the steady-state charge current density $\mathbf{J}{\mathrm{q}}$ $$\mathbf{J}_q=-n_e e \mathbf{v}_d=\frac{n_e e^2 \tau_e}{m_e} \mathbf{E},$$ and the Drude expression for the direct-current conductivity $\sigma{\mathrm{e}}$
$$\sigma_{\mathrm{e}}=\frac{n_e e^2 \tau_e}{m_e},$$
which links this quantity to few microscopic physical parameters associated either with the charge carriers $\left(e\right.$ and $\left.m_{\mathrm{e}}\right)$ or to the specific material $\left(n_{\mathrm{e}}\right.$ and $\left.\tau_{\mathrm{e}}\right)$. The conductivity is the inverse of the electrical resistivity $\rho_{\mathrm{e}}=1 / \sigma_{\mathrm{e}}$, a physical property which is easily measured: therefore, the Drude theory allows for a direct estimation of the order of magnitude of the relaxation time related to the charge current ${ }^5$ which turns out to be as small as $\tau_{\mathrm{e}} \sim 10^{-14} \mathrm{~s}$; its predicted value is reported in table $7.1$ for some selected metallic elements. By applying the kinetic theory to the (classical) electron gas, we can estimate the electron thermal velocity $v_{\mathrm{e}}^{\text {th }}$ by means of the equipartition theorem ${ }^6$ and accordingly define the electron mean free path $\lambda_e \sim 1-10 \AA$ which represents the average distance covered by an electron between two successive collisions. It is reassuring to get a number which is comparable with the typical interatomic distance in a crystalline solid: this supports the robustness of the Drude model.

## 物理代写|固体物理代写Solid-state physics代考|Optical properties

Another success of the classical free electron gas theory is that it correctly predicts the optical properties of metals, which are found to strongly reflect any electromagnetic radiation in the visible spectrum, while at higher frequency they are able to absorb [5], as shown in figure $7.1$ in the paradigmatic case of aluminium.

In order to estimate the optical reflectivity of a free electron gas, we need to evaluate its frequency-dependent refractive index $\sqrt{\epsilon_{\mathrm{r}}}$, where $\epsilon_{\mathrm{r}}$ is the relative permittivity of the metal [5]. Let $\mathbf{E}(t)=\mathbf{E}_0 \exp (-i \omega t)$ be a time-varying and uniform electric field applied to a metallic sample, where $\mathbf{E}_0$ and $\omega$ are its amplitude and frequency, respectively. Following the same path which led to equation (7.3), we write the electron equation of motion as
$$-e \mathbf{E}_0 \exp (-i \omega t)=m_e \mathbf{V}_d(t)+\frac{m_e}{\tau_e} \mathbf{v}_d(t),$$

where we have introduced the time dependence in $\mathbf{v}{\mathrm{d}}(t)$ since we understand that, under the action of an oscillating electric field, the drift velocity of a free electron also follows a periodic variation with the same frequency. More specifically, we write $\mathbf{v}{\mathrm{d}}(t)=\mathbf{v}{\mathrm{d}, 0} \exp (-i \omega t)$. From equation (7.8) we easily get the drift velocity ${ }^7$ $$\mathbf{v}{\mathrm{d}, 0}=-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \mathbf{E}0,$$ and by integration we obtain the time-dependent displacement $\mathbf{s}(t)$ of the electron $$\mathbf{s}(t)=\int_0^t \mathbf{v}{\mathrm{d}}\left(t^{\prime}\right) d t^{\prime}=\frac{e \tau_e}{i \omega m_{\mathrm{e}}} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \mathbf{E}0 \exp (-i \omega t),$$ where with no loss of generality we have set $\mathbf{s}(0)=0$ for convenience. We can now calculate the polarisation (that is the induced electric dipole moment per unit volume) $\mathbf{P}=-n{\mathrm{e}} e \mathbf{s}(t)$ and, through the standard relation $\epsilon_0 \epsilon_{\mathrm{r}} \mathbf{E}=\epsilon_0 \mathbf{E}+\mathbf{P}$, eventually obtain the relative permittivity of the metal as
$$\epsilon_{\mathrm{r}}=1-\frac{n_e e^2}{\epsilon_0 m_e} \frac{\tau_e}{i \omega} \frac{1}{1-i \omega \tau_{\mathrm{e}}}=1-\frac{\tau_e}{i \omega} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \omega_{\mathrm{p}}^2,$$

## 物理代写|固体物理代写固态物理代考|电导率

Drude理论的第一个应用是预测金属的直流电导率。让 $\mathbf{v}{\mathrm{d}}$ 是在外加均匀恒定电场作用下的电子漂移速度 $\mathbf{E}$。被加速的电子所经历的碰撞的整体动力效应被描述为牛顿运动方程中的摩擦项 $$-e \mathbf{E}=m{\mathrm{e}} \dot{\mathbf{v}}{\mathrm{d}}+\beta \mathbf{v}{\mathrm{d}},$$
where $\beta$ 是一个待确定的系数。基本上，当外部电场被去除时，增加的摩擦项迫使电子分布向平衡费米-狄拉克分布放松。在稳态条件下 $d \mathbf{v}{\mathrm{d}} / d t=0$ 因此 $$-\frac{e}{m{\mathrm{e}}} \mathbf{E}=\frac{\beta}{m_{\mathrm{e}}} \mathbf{v}{\mathrm{d}},$$ 自然地 ${ }^4$ 导致了定义 $\beta=m{\mathrm{e}} / \tau_{\mathrm{e}}$。这允许我们计算电子漂移速度为 $$\mathbf{v}{\mathrm{d}}=-\frac{e \tau{\mathrm{e}}}{m_{\mathrm{e}}} \mathbf{E},$$

$$\sigma{\mathrm{e}}=\frac{n_e e^2 \tau_e}{m_e},$$
，它将这个量与几个与载流子相关的微观物理参数联系起来 $\left(e\right.$ 和 $\left.m_{\mathrm{e}}\right)$ 或者是特定的材料 $\left(n_{\mathrm{e}}\right.$ 和 $\left.\tau_{\mathrm{e}}\right)$。电导率是电阻率的倒数 $\rho_{\mathrm{e}}=1 / \sigma_{\mathrm{e}}$，这是一种很容易测量的物理性质:因此，德鲁德理论允许直接估计与电荷电流相关的弛豫时间的数量级 ${ }^5$ 它的大小是 $\tau_{\mathrm{e}} \sim 10^{-14} \mathrm{~s}$;其预测值见表 $7.1$ 对于一些选定的金属元素。将动力学理论应用于(经典)电子气，我们可以估计电子热速度 $v_{\mathrm{e}}^{\text {th }}$ 通过均分定理 ${ }^6$ 并相应地定义了电子的平均自由程 $\lambda_e \sim 1-10 \AA$ 它表示电子在两次连续碰撞之间经过的平均距离。得到一个与晶体固体中典型原子间距离相当的数字是令人放心的:这支持了Drude模型的鲁棒性

## 物理代写|固体物理代写固态物理代考|光学性质

$$-e \mathbf{E}_0 \exp (-i \omega t)=m_e \mathbf{V}_d(t)+\frac{m_e}{\tau_e} \mathbf{v}_d(t),$$

，其中我们在$\mathbf{v}{\mathrm{d}}(t)$中引入了时间依赖性，因为我们知道，在振荡电场的作用下，自由电子的漂移速度也遵循相同频率的周期变化。更具体地说，我们写$\mathbf{v}{\mathrm{d}}(t)=\mathbf{v}{\mathrm{d}, 0} \exp (-i \omega t)$。由式(7.8)我们可以很容易地得到漂移速度${ }^7$$\mathbf{v}{\mathrm{d}, 0}=-\frac{e \tau_{\mathrm{e}}}{m_{\mathrm{e}}} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \mathbf{E}0,$$，通过积分我们可以得到电子的时变位移$\mathbf{s}(t)$$\mathbf{s}(t)=\int_0^t \mathbf{v}{\mathrm{d}}\left(t^{\prime}\right) d t^{\prime}=\frac{e \tau_e}{i \omega m_{\mathrm{e}}} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \mathbf{E}0 \exp (-i \omega t),$$，其中为了方便起见，在不失一般性的情况下，我们设为$\mathbf{s}(0)=0$。我们现在可以计算出极化率(即单位体积的感应电偶极矩)$\mathbf{P}=-n{\mathrm{e}} e \mathbf{s}(t)$，并通过标准关系$\epsilon_0 \epsilon_{\mathrm{r}} \mathbf{E}=\epsilon_0 \mathbf{E}+\mathbf{P}$，最终得到金属的相对介电常数为
$$\epsilon_{\mathrm{r}}=1-\frac{n_e e^2}{\epsilon_0 m_e} \frac{\tau_e}{i \omega} \frac{1}{1-i \omega \tau_{\mathrm{e}}}=1-\frac{\tau_e}{i \omega} \frac{1}{1-i \omega \tau_{\mathrm{e}}} \omega_{\mathrm{p}}^2,$$

## 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|固体物理代写Solid-state physics代考|PHYSICS7544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|General features of the metallic state

Metals are characterised at the macroscopic level by the ability to conduct electricity. Phenomenologically, the charge transport properties are defined by their resistivity which typically ranges in between $10^{-8}$ and $10^{-6} \Omega \mathrm{m}$ at $T=300 \mathrm{~K}$. The presence of impurities detrimentally affects the charge transport in these materials and, therefore, their conductivity is typically lowered by increasing the concentration of defects. Finally, the resistivity is found to decrease monotonically with decreasing temperature ${ }^1$.

The metallic state is very common in Nature, since more than two thirds of the elements are in fact good conductors. They are preferentially found on the left-hand side of the periodic table; accordingly, their atomic ground-state configuration typically consists in a large majority of electrons hosted by core states and just a few others found in valence states, as shown in appendix A. The number $n_{\mathrm{e}}$ of valence electrons per $\mathrm{cm}^3$ is given by the product (number of atoms per mole) $\times$ (number of moles per $\left.\mathrm{cm}^3\right) \times($ number of valence electrons per atom) or equivalently
$$n_{\mathrm{e}}=\mathcal{N}{\mathrm{A}} \frac{d{\mathrm{m}}}{A} Z_{\mathrm{v}},$$
where $d_{\mathrm{m}}$ is the mass density of the metal, while the symbols $\mathcal{N}{\mathrm{A}}, Z{\mathrm{v}}$, and $A$ are the Avogadro number, the number of valence electrons per atom (chemical valence), and the atomic mass number, respectively, previously defined in sections 1.2.1 and 1.3.2. As reported in table $7.1$ this corresponds to a typical number density of the order of $10^{22}$ electrons $\mathrm{cm}^{-3}$, which is much larger than found in any ordinary atomic or molecular gas in normal conditions of temperature and pressure ${ }^2$. We can also assign a volume per electron, which corresponds to a sphere of radius $r_{\mathrm{e}}$ defined so that
$$\frac{4}{3} \pi r_{\mathrm{e}}^3=\frac{1}{n_{\mathrm{e}}} .$$
If we compare the calculated values of $r_{\mathrm{e}}$ with the typical interatomic distances in crystals (which are of the order of few $\AA$ ), we come to the conclusions that in metals there is plenty of room available to valence electrons. Finally, we take into consideration that they are only weakly bound to their ion core: therefore, it is quite reasonable to assume that, upon collecting many atoms to form the crystal, they homogeneously delocalise throughout the interstitial regions, thus giving rise to unidirectional metal bonds, as anticipated in figure $2.22$ and related discussion.
This body of phenomenological evidence supports the idea of modelling the conduction gas of a metal as a homogeneous gas of delocalised, free, independent, and charged particles. Although based on very drastic approximations, this picture is nevertheless promising to describe at least the main features of metals.

## 物理代写|固体物理代写Solid-state physics代考|The classical (Drude) theory of the conduction gas

A first simple approach to the physics of the free electron gas is purely classical, mostly based on the kinetic theory of gases [1]. In the Drude theory of the metallic state [2-4] electrons are described as point-like charged particles, confined within the volume of a solid specimen. The very drastic approximations of free and independent particles outlined in the previous section are slightly corrected by assuming that electrons occasionally undergo collisions with ion vibrations, with other electrons and with lattice defects possibly hosted by the sample; the key simplifying assumption is that we define a unique relaxation time $\tau_e$ (thus averaging among all possible scattering mechanisms) defined such that $1 / \tau_e$ is the probability per unit time for an electron to experience a collision of whatever kind ${ }^3$. This approach is usually referred to as the relaxation time approximation. The free-like and independent-like characteristics of the particles of the Drude gas are instead exploited by assuming that between two collisions electrons move according to the Newtons equations of motion, that is uniformly and in straight lines. Collisions are further considered as instantaneous events which abruptly change the electron velocities; also, they are assumed to be the only mechanism by which the Drude gas is able to reach the thermal equilibrium. In other words, the velocity of any electron emerging from a scattering event is randomly distributed in space, while its magnitude is related to the local value of the temperature in the microscopic region of the sample close to the scattering place (local equilibrium).

## 物理代写|固体物理代写固态物理代考|金属态的一般特征

$$n_{\mathrm{e}}=\mathcal{N}{\mathrm{A}} \frac{d{\mathrm{m}}}{A} Z_{\mathrm{v}},$$
where $d_{\mathrm{m}}$ 是金属的质量密度，而符号呢 $\mathcal{N}{\mathrm{A}}, Z{\mathrm{v}}$，以及 $A$ 分别为阿伏伽德罗数、每个原子的价电子数(化学价)和原子质量数，定义见1.2.1节和1.3.2节。如表所示 $7.1$ 这对应一个典型的数量级的数字密度 $10^{22}$ 电子 $\mathrm{cm}^{-3}$，比在正常温度和压力下的任何普通原子或分子气体都要大得多 ${ }^2$。我们也可以给每个电子指定一个体积，它对应一个半径为球面的体积 $r_{\mathrm{e}}$ 定义使
$$\frac{4}{3} \pi r_{\mathrm{e}}^3=\frac{1}{n_{\mathrm{e}}} .$$的计算值 $r_{\mathrm{e}}$ 与晶体中典型的原子间距离(这是数量级的 $\AA$ )，我们得出结论:在金属中，价电子有很大的空间。最后，我们考虑到它们只与离子核弱结合:因此，我们可以很合理地假设，在聚集许多原子形成晶体时，它们在整个间隙区均匀地离域，从而产生单向金属键，如图所示 $2.22$ 及相关讨论。这一系列现象学证据支持将金属的传导气体建模为离域的、自由的、独立的和带电粒子的均匀气体的想法。尽管是基于非常极端的近似，但这幅图至少有希望描述金属的主要特征。

## 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|固体物理代写Solid-state physics代考|PHYSICS3544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|The Bloch theorem

We now derive a formal result due to the translational invariance of any crystal lattice discussed in chapter $2 .$

Let us start from the single electron approximation developed in section 1.4.1, where we proved that the Schrödinger problem given by equation (1.22) must be solved for each crystalline electron. The electron Hamiltonian operator $\hat{H}(\mathbf{r})$
$$\hat{H}(\mathbf{r})=-\frac{\hbar^2}{2 m_e} \nabla^2+\hat{V}{\mathrm{cfp}}(\mathbf{r}),$$ is obviously depending upon the position $\mathbf{r}$ of the particle within the crystal and, because of the property of translational invariance of the lattice, we have $$\hat{H}(\mathbf{r})=\hat{H}\left(\mathbf{r}+\mathbf{R}{\mathbf{l}}\right),$$
as shown in figure 2.3. In order to formally treat such an invariance, it is useful to introduce the translation operator $\hat{\mathrm{R}}{\mathrm{R}_1}$ whose action on a generic space function $f(\mathbf{r})$ is defined as $$\hat{T}{\mathrm{R}1} f(\mathbf{r})=f\left(\mathbf{r}+\mathbf{R}{\mathrm{l}}\right) \text {. }$$
The translational invariance is revealed by stating that the one-electron Hamiltonian operator commutes with the translation operator, that is: $\left[\hat{H}(\mathbf{r}), \hat{T}{\mathrm{R}{]}}\right]=0$. Therefore, the solutions $\psi(\mathbf{r})$ of equation (1.22) are also eigenfunctions of the translation operator
$$\hat{T}{\mathbf{R}} \psi(\mathbf{r})=t\left(\mathbf{R}{\mathbf{l}}\right) \psi(\mathbf{r}),$$
where the number $t\left(\mathbf{R}1\right)$ is the eigenvalue of $\hat{T}{\mathrm{R}{\mathrm{R}}}$; it is intuitive to figure out that, according to equation (2.1), $t\left(\mathbf{R}_1\right)$ depends on the set $\left{n_1, n_2, n_3\right}$. Furthermore, by composing two translations $$\hat{T}{\mathbf{R}1} \hat{T}{\mathbf{R}1 \psi} \psi(\mathbf{r})=\hat{T}{\mathbf{R}_1+\mathbf{R}_1} \psi(\mathbf{r})=t\left(\mathbf{R}_1+\mathbf{R}_1\right) \psi(\mathbf{r}),$$
we understand that $t\left(\mathbf{R}_1+\mathbf{R}_1\right)=t\left(\mathbf{R}_1\right) t\left(\mathbf{R}_1\right)$.
We assume that the electron wavefunctions have been properly normalised
$$\int_V|\psi(\mathbf{r})|^2 d \mathbf{r}=1,$$
over the finite volume $V$ of the crystalline sample we are studying.

## 物理代写|固体物理代写Solid-state physics代考|Electrons in a periodic potential

Just as it has been possible to obtain the general form of the wavefunction of crystalline electrons without taking into consideration any materials-specific property, so we are about to derive the general structure of the energy spectrum for valence electrons by only considering the periodicity of the single-particle potential $\hat{V}{\mathrm{efp}}(\mathbf{r})=\hat{V}{\mathrm{efp}}\left(\mathbf{r}+\mathbf{R}_{\mathrm{l}}\right)$.

To this aim we discuss the simple case of a one-dimensional monoatomic crystal under the construction usually referred as Kronig-Penney model. The situation sketched in figure $6.1$ is further simplified by approximating the crystal field potential with a function $V(x)$ which consists in a periodic sequence of potential wells spanning the core regions, separated by finite barriers occupying the interstitial ones. This idealised situation is represented in figure 6.3. We remark that we have for convenience set the zero of the potential at the bottom of the wells, while $a$ and $b$, respectively, indicate the width of the wells and barriers. Therefore, the lattice periodicity is $a+b$ or, equivalently, in this case the lattice vectors are written as $R_1=n(a+b)$ with $n=0, \pm 1, \pm 2, \pm 3, \ldots$ (see equation (2.1)). We understand that $a<b$ by guessing that interstitial regions are larger than core ones ${ }^{12}$.

Thanks to the crystal periodicity, it is sufficient to solve the quantum problem of a valence electron under the action of the Kronig-Penney potential $V(x)$ only for a single pair of adjacent core and interstitial regions. With reference to figure $6.3$ we write
$$V(x)=\left{\begin{array}{ccl} 0 & 0<x<a & \text { core region } \ V_0 & -b \leqslant x \leqslant 0 & \text { interstitial region, } \end{array}\right.$$
so that where we have indicated by $\psi(x)$ and $E$ the electron wavefunction and energy, respectively.

## 物理代写|固体物理代写固态物理学代考|布洛赫定理

$$\hat{H}(\mathbf{r})=-\frac{\hbar^2}{2 m_e} \nabla^2+\hat{V}{\mathrm{cfp}}(\mathbf{r}),$$显然取决于晶体内粒子的位置$\mathbf{r}$，并且由于晶格的平移不变性的性质，我们有$$\hat{H}(\mathbf{r})=\hat{H}\left(\mathbf{r}+\mathbf{R}{\mathbf{l}}\right),$$

。平移不变性是通过说明单电子哈密顿算符与平移算符交换，即$\left[\hat{H}(\mathbf{r}), \hat{T}{\mathrm{R}{]}}\right]=0$来揭示的。因此，式(1.22)的解$\psi(\mathbf{r})$也是平移算子
$$\hat{T}{\mathbf{R}} \psi(\mathbf{r})=t\left(\mathbf{R}{\mathbf{l}}\right) \psi(\mathbf{r}),$$

，我们理解$t\left(\mathbf{R}_1+\mathbf{R}_1\right)=t\left(\mathbf{R}_1\right) t\left(\mathbf{R}_1\right)$ .

$$\int_V|\psi(\mathbf{r})|^2 d \mathbf{r}=1,$$

## 物理代写|固体物理代写固态物理学代考|周期电势中的电子

$$V(x)=\left{\begin{array}{ccl} 0 & 0<x<a & \text { core region } \ V_0 & -b \leqslant x \leqslant 0 & \text { interstitial region, } \end{array}\right.$$
，这样我们分别用$\psi(x)$和$E$表示电子波函数和能量。

## 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|固体物理代写Solid-state physics代考|KYA322

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|The physical origin of the LO–TO splitting

The derivation of the LST relation anticipated in equation (3.21) is rigorously framed only within the theory of the dielectric properties of crystalline solids $[9,10]$, indeed an advanced topic of solid state physics. Good for us, it is possible to claborate a phenomenological model which, although derived under some important simplifying assumptions, nevertheless leads to a result of general validity. More specifically, we are going to consider a dielectric ionic crystal containing just two atoms in its unit cell.
Let the dielectric crystal be subject to the action of an external macroscopic electric field $\mathbf{E}$. Because of polarisation phenomena, the local electric field $\mathbf{E}{\mathrm{loc}}$ found at any position $\mathbf{r}$ within the crystal differs from the applied one: the theory of the dielectric properties of crystalline solids displays exactly here. We are not developing this calculation; rather, we assume that the local field is known. The electrostatic action on the two ions within the unit cell causes their displacements, but since such a perturbation occurs on a length scale much longer that the typical interatomic distances, we can assume that equally charged ions move as a whole. Accordingly, in harmonic approximation we can guess the ionic equations of motion in the form $$\left{\begin{array}{l} m{+} \ddot{\mathbf{u}}{+}=-K\left(\mathbf{u}{+}-\mathbf{u}{-}\right)+e \mathbf{E}{\mathrm{loc}} \ m_{-} \ddot{\mathbf{u}}{-}=+K\left(\mathbf{u}{+}-\mathbf{u}{-}\right)-e \mathbf{E}{\mathrm{loc}} \end{array}\right.$$
where for sake of simplicity we have assumed just one force constant $K$ for any interaction and indicated by $m_{\pm}$and $\mathbf{u}{\pm}$respectively the mass and the displacement of the positive $(+)$ and negative $(-)$ ion. By further setting $\mathbf{w}=\left(\mathbf{u}{+}-\mathbf{u}{-}\right)$and $1 / m=1 / m{+}+1 / m_{-}$we derive a forced oscillator equation
$$\ddot{\mathbf{w}}=\frac{e}{m} \mathbf{E}_{\mathrm{loc}}-\frac{K}{m} \mathbf{w}$$

## 物理代写|固体物理代写Solid-state physics代考|Quantum theory of harmonic crystals

Moving to a quantum description, as simple as it may appear, represents a major conceptual step forward in our search for a truly fundamental description of lattice dynamics. To appreciate its relevance, we anticipate a result more extensively discussed in the next chapter. The specific heat of a crystal described as an assembly of classical harmonic oscillators is calculated to be independent of temperature (Dulong-Petit law). Contrary to this prediction, experimental measurements provide evidence that the specific heat becomes vanishingly small as $T \rightarrow 0$, thus proving that it is in fact temperature-dependent. Only a full quantum treatment is able to reconcile theoretical predictions to measurements.

Based on the theory developed in the previous section, we will agree to describe each classical (sq) vibrational mode as a quantum one-dimensional harmonic oscillator [1-3] whose energy is restricted to the values $\left(n_{s q}+1 / 2\right) \hbar \omega_s(\mathbf{q})$ where $n_{s q}=0,1,2, \ldots$ is the vibrational quantum number and $\omega_s(\mathbf{q})$ is obtained by diagonalising the dynamical matrix. Since the vibrational energy levels are equally spaced, we can look at the state with energy $\left(n_{s q}+1 / 2\right) \hbar \omega_s(\mathbf{q})$ as a single $n_{s q}$ th excited state or, equivalently, as the state obtained by adding $n_{s q}$ identical energy quanta $\hbar \omega_s(\mathbf{q})$. We will adopt this second approach since it is especially effective in describing the dynamical and thermal characteristics of a crystal lattice through the properties of $a$ gas of pseudo-particles, hereafter named phonons. This choice introduces a corpuscular description of lattice dynamics, where phonons are the energy quanta of the ionic displacement field ${ }^{15}$.
Let us now consider in some detail the physics of phonons. First of all, we clarify that phonons, similarly to photons, are named pseudo-particles since they do not have a mass. Furthermore, in addition to an energy $\hbar \omega_s(\mathbf{q})$, they also carry a momentum $\hbar$ q. Since such a momentum is exact, the uncertainty principle imposes that the phonon position is totally undetermined and, therefore, they must be understood as delocalised pseudo-particles. This is consistent with the fact that their corresponding non-interacting classical vibrational modes extend throughout the system ${ }^{16}$.

## 物理代写|固体物理代写Solid-state physics代考|The physical origin of the LO–TO splitting

$$m+\ddot{\mathbf{u}}+=-K(\mathbf{u}+-\mathbf{u}-)+e \mathbf{E l o c} m_{-} \ddot{\mathbf{u}}-=+K(\mathbf{u}+-\mathbf{u}-)-e \mathbf{E l o c}$$
【正确的。
where forsakeofsimplicitywehaveassumedjustone forceconstant $\$ K \$$foranyinteractionand Iddot {\backslash \operatorname{mathbf}{\mathrm{w}}}=\backslash \operatorname{frac}{\mathrm{e}} \mathrm{m}} \backslash \operatorname{mathbf}{\mathrm{E}}_{-}{\operatorname{mathrm}{\operatorname{loc}}}-\mid \mathrm{frac}{\mathrm{K}}{\mathrm{m}} \backslash \mathrm{mathbf}{\mathrm{w}} \ \$$

## 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|固体物理代写Solid-state physics代考|PHYSICS7544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Diatomic linear chain

Let us now turn to consider the one-dimensional model of minimal complexity for a lattice with a basis, namely a diatomic linear chain. We need to define two ion masses $M_1$ and $M_2$ and two effective springs $\gamma^{(L)}$ and $\xi^{(L)}$, respectively, coupling ions within the same unit cell or belonging to nearest neighbouring unit cells. Ion positions are now indicated as $R_{l, 1}=R_l+R_1$ and $R_{l, 2}=R_l+R_2$, where $R_l$ labels the lth unit cell, while $R_1$ and $R_2$ specify the ion within the basis. The situation is sketched in figure $3.4$ and once again we start by considering longitudinal oscillations.

The equations of motion for the two ions in the lth unit cell form a system of two differential equations
$$\left{\begin{array}{l} M_1 \ddot{u}{l, 1}=\gamma^{(L)}\left(u{l, 2}-u_{l, 1}\right)+\xi^{(L)}\left(u_{l-1,2}-u_{l, 1}\right) \ M_2 \ddot{u}{l, 2}=\xi^{(L)}\left(u{l+1,1}-u_{l, 2}\right)+\gamma^{(L)}\left(u_{l, 1}-u_{l, 2}\right), \end{array}\right.$$
We seek solutions for this system of the same form given in equation (3.10). However, for further convenience, it is useful to rewrite the amplitude as $\left|\mathcal{A}q\right| \rightarrow\left|\mathcal{A}_q\right|\left|\mathrm{a}_i(q)\right|$ and the phase as $\varphi(q) \rightarrow \varphi(q)+\phi_i(q)$ (where $i=1,2$ labels the ion within the basis) since, as we will prove soon, the terms $\left|a_i(q)\right|$ and $\phi_i(q)$ are determined by the very equations of motion, rather than by the boundary conditions as instead $\left|\mathcal{A}_q\right|$ and $\varphi(q)$. By this choice, we write $$u{l, i}=\frac{1}{\sqrt{N M_i}} \sum_q\left|\mathcal{A}q\right|\left|a_i(q)\right| \cos \left[q R{l, i}-\omega(q) t+\varphi(q)+\phi_i(q)\right],$$
which, if inserted in equations (3.11), leads to the following matrix equation
$$\left(\begin{array}{ll} \mathcal{D}{11} & \mathcal{D}{12} \ \mathcal{D}{21} & \mathcal{D}{22} \end{array}\right)\left(\begin{array}{l} a_1 \ a_2 \end{array}\right)=\omega^2\left(\begin{array}{l} a_1 \ a_2 \end{array}\right),$$

## 物理代写|固体物理代写Solid-state physics代考|Dynamics of three-dimensional crystals

The dynamical properties of a three-dimensional solid with arbitrary crystalline structure or basis configuration can only be described by means of a heavy formalism $[5,6,8]$, which somewhat hides the underlying physics. This is the pedagogical reason why we have preliminarily treated the model system corresponding to a linear chain: we will extensively make use of concepts developed in that framework. As for the formal procedures, we will instead follow the same line of action adopted in section 3.1. In particular, we will assume that a suitable force field describing the interactions among ions is available (see appendix D); once for all, therefore, the force constants defined in equation (3.2) are given as known. More important, however, is the fact that we will rely on the harmonic approximation.
Before starting to develop our theory, we preliminarily remark that in a threedimensional crystal containing $N_{\text {atom }}$ atoms in its unit cell, we have $3 N_{\text {atom }}$ ionic degrees of freedom (per unit cell) and, therefore, an equal number of branches in the vibrational dispersion relations; among them we will find 3 acoustic and $3\left(N_{\text {atom }}-1\right)$ optical branches.
In the harmonic approximation, the equations of motion are written as ${ }^9$
$$M_b \ddot{u}i(l b)=-\sum{j l^{\prime} b^{\prime}} U_{i j}\left(l b, l^{\prime} b^{\prime}\right) u_j\left(l^{\prime} b^{\prime}\right),$$
for which we guess solutions in the form
$$u_i(l b)=\frac{\mathrm{a}_i(b \mid \mathbf{q})}{\sqrt{M_b}} e^{i q \cdot \mathbf{R}_i} e^{-i \omega t},$$
where $M_b$ is the mass of the $b$ th ion in the basis ${ }^{10}, \mathbf{q}$ is the wavevector of the vibrational wave and $a_i(b \mid q)$ describes the amplitude of the corresponding ionic oscillations. By inserting this guessed displacement into the equations of motion we get
$$\sum_{j b^{\prime}} \mathcal{D}{i j}\left(b b^{\prime} \mid \mathbf{q}\right) a_j\left(b^{\prime} \mid \mathbf{q}\right)=\omega^2 a_i(b \mid \mathbf{q})$$ where the quantities $$\mathcal{D}{i j}\left(b b^{\prime} \mid \mathbf{q}\right)=\frac{1}{\sqrt{M_b M_{b^{\prime}}}} \sum_{l^{\prime}} U_{i j}\left(l b, l^{\prime} b^{\prime}\right) e^{-i \mathbf{q} \cdot\left(\mathbf{R}l-\mathbf{R}_r\right)}$$ define the dynamical matrix of the crystal. It is important to remark that in this equation just a single summation of the cell index $l^{\prime}$ appears since the force constants $U{i j}\left(l b, l^{\prime} b^{\prime}\right)$ of an ideal crystal depend on the pair $\left(l, l^{\prime}\right)$ just through their difference ${ }^{11}$. This also reflects the choice of the guessed solution for the ionic displacements in the form of a Bloch wave.

## 物理代写|固体物理代写Solid-state physics代考|Diatomic linear chain

$\$ \$$Veft { M_1 \ddot{u} l, 1=\gamma^{(L)}\left(u l, 2-u_{l, 1}\right)+\xi^{(L)}\left(u_{l-1,2}-u_{l, 1}\right) M_2 \ddot{u} l, 2=\xi^{(L)}\left(u l+1,1-u_{l, 2}\right)+\gamma^{(L)}\left(u_{l, 1}-u_{l, 2}\right) 【正确的。 Weseeksolutions forthissystemofthesame formgiveninequation (3.10). However, for furthercon \mathrm{t}+ Ivarphi(q)+\phi_i(q)\right], which, ifinsertedinequations(3.11), leadstothe followingmatrixequation 剩下（ \begin{array}{llll}\mathcal{D} 11 & \mathcal{D} 12 & \mathcal{D} 21 & \mathcal{D} 22\end{array} 右左 ( a_1 a_2 Iright)=lomega^2lleft( a_1 a_2 右）， \ \$$

## 物理代写|固体物理代写Solid-state physics代考|Dynamics of three-dimensional crystals

$\$ \$$forwhichweguesssolutionsinthe form where \ M_b \$$ isthemassofthe $\$ b \$$thioninthebasis \^{10}, \mathbf{q} isthewavevectorofthevibrationalwaveand mathbf {q} \backslash right )=\backslash o m e g a^{\wedge} 2 a_{-} i(b \backslash m i d \backslash mathbf {q}) wherethequantities Imathbf{R}_rıright )} \ \$$ 定义了晶体的动力学矩阵。重要的是要注意，在这个等式中只有一个细胞指数的总和 $l^{\prime}$ 由 于力常数出现 $U i j\left(l b, l^{\prime} b^{\prime}\right)$ 理想晶体的大小取决于对 $\left(l, l^{\prime}\right)$ 只是通过他们的不同 ${ }^{11}$. 这也反映了对布洛赫波形式的 离子位移的猜测解的选择。

## 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|固体物理代写Solid-state physics代考|PHYSICS3544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Conceptual layout

The description of crystal structures developed in chapter 2 relies on an implicit (but really very strong) assumption, namely! ions are clamped at their lattice positions. This is, also, the situation assumed to define the eigenvalue problem for the total electron wavefunction given in equation (1.15) within the framework developed under the adiabatic approximation. While the assumption of static lattice is useful in the above contexts, it is either conceptually wrong and inadequate to describe many important solid state phenomena.

First of all, we recall that ions, although comparatively much more massive than electrons, have in fact a finite mass: therefore, according to fundamental quantum mechanics [1-4], they always (even at zero temperature) have a non-vanishing mean square momentum. We can reconcile crystallography with the quantum uncertainty principle by assuming that the mean position of an ion (obtained by averaging over its zero-point motion) corresponds to $\mathbf{R}1$ in Bravais lattices or to $\mathbf{R}_1+\mathbf{R}{\mathrm{b}}$ for lattices with a basis (see equations (2.1) and (2.3), respectively).

This is not enough. Still considering ions at rest (although in some ‘average’ meaning) is inconsistent with a number of experimental evidences, including (but not limited to): thermal expansion, melting, thermal conductivity, sound propagation, inelastic scattering of electromagnetic waves or particles (electrons as well as neutrons). All together these phenomena provide a robust body of experimental evidence that lattice ions do undergo some kind of motion. The aim of this chapter is to fully characterise the corresponding lattice dynamics.

We will accomplish this task at first under the leading adiabatic and classical approximations (see sections 1.3.4 and 1.4.2, respectively). Non-classical dynamical features will appear later, by a suitable quantisation procedure operated on the ionic classical displacement field. In developing our classical phenomenological theory of lattice dynamics, we will assume that there exists a many-body potential energy $U=U(\mathbf{R})$ governing the motion of the ions ${ }^1$. Basically, $U$ contains the ion-ion Coulomb interaction energy as well as their kinetic energy, as conceptualised in section 1.3.4.

## 物理代写|固体物理代写Solid-state physics代考|Monoatomic linear chain

Let us consider a linear chain where $N$ identical ions of mass $M$ are placed at distance $a$ when they are at rest in equilibrium positions. This corresponds to a onedimensional Bravais crystal with lattice spacing $a$; the primitive unit cell is obtained by the Wigner-Seitz construction as a segment of length $a$ with the ion placed at its midpoint. By adopting Born-von Karman boundary conditions, the ionic positions are indicated as $R_l=l a$ with $l=0,1,2, \ldots, N-1$. Finally, following the force constant approach discussed in the previous section, we represent the interactions between nearest neighbouring ions as harmonic springs. The situation is sketched in figure $3.1$.

Let us consider a longitudinal vibration of the chain, that is a displacement pattern in which the ions move along the chain direction. The classical equation of motion for the lth ion is
$$M \ddot{u}l=\gamma^{(L)}\left(u{l+1}+u_{l-1}-2 u_l\right),$$

where $\gamma^{(L)}$ is the force constant of the effective spring. Suggested by the elementary mechanics of a vibrating wire, we seek a solution in the form
$$u_l=\frac{1}{\sqrt{N M}}\left|\mathcal{A}_q\right| \cos \left[q R_l-\omega(q) t+\varphi(q)\right],$$
where the normalising factor $(N M)^{-1 / 2}$ has been introduced for further convenience, while $\left|\mathcal{A}_q\right|$ and $\varphi(q)$ are the amplitude and the initial phase of the wave ${ }^6$. Of course, $q$ and $\omega(q)$ are the wavenumber and the angular frequency of the travelling wave, respectively. Replacing equation (3.7) into equation (3.6) leads to
$$M \omega^2(q)=2 \gamma^{(L)}[1-\cos (q a)]=4 \gamma^{(L)} \sin ^2\left(\frac{1}{2} q a\right),$$
which is known as the dispersion relation and it is shown in figure 3.2(top). This representation is redundant since it ignores translational periodicity: it makes no difference in the displacement $u_l$ by increasing $q \rightarrow q+G$ with $G=2 m \pi / a$ a reciprocal lattice vector of the linear chain crystal ( $m$ is any positive or negative integer number). It is therefore customary to adopt the reduced zone scheme: the dispersion relation is represented only for $q \in 1 \mathrm{BZ}$ or, equivalently, for $q \in[-\pi / a,+\pi / a]$ as shown in figure 3.2(bottom). The actual number of allowed $q$ is determined by the imposed boundary conditions: since it must be $u_0=u_N$ then
$$q=\frac{2 \pi}{a} \frac{\xi}{N} \quad \text { with } \quad \xi=0,1,2,3, \ldots, N-1$$

## 物理代写|固体物理代写Solid-state physics代考|Monoatomic linear chain

$$M \ddot{u} l=\gamma^{(L)}\left(u l+1+u_{l-1}-2 u_l\right),$$

$$u_l=\frac{1}{\sqrt{N M}}\left|\mathcal{A}_q\right| \cos \left[q R_l-\omega(q) t+\varphi(q)\right],$$

$$M \omega^2(q)=2 \gamma^{(L)}[1-\cos (q a)]=4 \gamma^{(L)} \sin ^2\left(\frac{1}{2} q a\right)$$

$$q=\frac{2 \pi}{a} \frac{\xi}{N} \quad \text { with } \quad \xi=0,1,2,3, \ldots, N-1$$

## 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|固体物理代写Solid-state physics代考|KYA322

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Setting up the atomistic model for a solid state system

Trying to plug the full atomistic picture into condensed matter physics is a hopeless enterprise and an unreasonable choice as well: the resulting mathematical problem would be too complicated to be solved by any analytical or numerical tool and, furthermore, several details specific to the single atomic system are actually marginal when matter is organised in condensates. In order to proceed, we need approximations. Far from being a fallback choice, this way of proceeding will allow us to bring out the most salient physical aspects of the solid state, avoiding an excess of detail that, in reality, would not translate into new meaningful knowledge. We are therefore going to develop a hierarchy of approximations that will actually constitute the backbone of our working model for crystalline solids. In the following chapters these approximations will be critically readdressed whenever some phenomenology questions their validity.

In general, we will treat electric, charge current, and magnetic effects according to the classical Maxwell electromagnetism; on the other hand, ion and electron physics will be described according to quantum mechanics. However, there will be some exceptions to this general choice.

First of all, we remark that the process of emission or absorption of electromagnetic energy by any material system will he descrihed through the concept of photon. This approach represents the most basic way to include the quantum nature of electromagnetic radiation into our elementary theory. We will not go any further because any improvement of this picture, admittedly simplified, would fall beyond the scope of this tutorial introduction to solid state physics.

Finally, the dynamics of crystal lattices will be firstly treated by classical mechanics in order to easily catch the phenomenology of ionic vibrations. Next, a fully quantum picture will be developed through the concept of phonon.

## 物理代写|固体物理代写Solid-state physics代考|Frozen-core approximation

To a large extent, the chemical properties of an atom are dictated by its valence electrons [2]. In particular, valence electrons rule over the formation mechanism of interatomic bonds, so ultimately affecting most of the physical properties in a condensed matter system. This suggests that core electrons are expected to play a minor role in determining most of solid state properties. We can exploit this observation by introducing the frozen-core approximation which will greatly simplify the picture. This approximation can be cast in a very simple form according to the scheme
\begin{aligned} \text { atom } &=\underbrace{\text { nucleus }+\text { core electrons }}{\text {ion }}+\text { valence electrons } \ &=\text { ion }+\text { valence electrons, } \end{aligned} which suggests the following: we will implement the atomistic description of a crystalline solid assuming that it consists of a collection of ions and valence electrons. The former will be described as point-like objects with a nuclear mass specific to their chemical species $^{16}$ and carrying a positive charge. If there are in total $Z=Z{\mathrm{c}}+Z_{\mathrm{v}}$ electrons (where $Z_{\mathrm{c}}$ and $Z_{\mathrm{v}}$ are the number of core and valence electrons, respectively), then the ionic charge $Q$ will be assigned the value $Q=+Z_{\mathrm{v}}$, in units of the elementary charge $\rho$.

The main advantage of the frozen-core approximation is a dramatic reduction of the number of electronic degrees of freedom to deal with: for instance, the main features of the electronic structure in a silicon crystal will be studied by considering just four valence electrons for each ion, instead of the full 14 electron set found in a silicon atom. In conclusion, hereafter when referring to ‘electrons’ we will actually mean ‘valence electrons’, while core ones will never be addressed since they are attached to nuclei forming ions.

## 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|固体物理代写Solid-state physics代考|PHYSICS 7544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Angular and magnetic momenta

In addition to their charge and mass, electrons are further characterised by their spin $[2,10]$ : an intrinsic angular momentum $\mathbf{S}$, whose square modulus $S^{2}$ and $z$-component obey the following quantisation rules
\begin{aligned} &S^{2}=s(s+1) \hbar^{2} \ &S_{z}=m_{s} \hbar \end{aligned}
with $s=1 / 2$ and $m_{s}=\pm 1 / 2$ (spin ‘up’ or ‘down’) known as the spin quantum numbers and $\hbar=h / 2 \pi=1.05446 \times 10^{-34} \mathrm{~J}$ s is the reduced Planck constant. An intrinsic spin magnetic moment $\mathbf{M}{S}$, similarly quantised, is attributed to each electron according to $$\mathbf{M}{S}=-g_{S} \frac{\mu_{B}}{\hbar} \mathbf{S},$$
where $\mu_{B}=e \hbar / 2 m_{\mathrm{e}}=9.2732 \times 10^{-24} \mathrm{~J} \mathrm{~T}^{-1}$ is the Bohr magneton, and $g_{S} \sim 2$ is the spin g-factor.

Similarly, each nucleus, in addition to being charged, also carries a magnetic moment $\mathbf{M}{N}$ [11] which for our purposes is conveniently defined as $$\mathbf{M}{N}=g_{N} \frac{\mu_{\mathrm{N}}}{\hbar} \mathbf{N},$$
where $g_{N}$ is the nuclear g-factor (a dimensionless constant), $\mu_{\mathrm{N}}$ is the nuclear magneton
$$\mu_{\mathrm{N}}=\frac{m_{\mathrm{e}}}{m_{\mathrm{p}}} \mu_{B}=5.05082 \times 10^{-27} \mathrm{~J} \mathrm{~T}^{-1},$$
and $\mathbf{N}$ is the muclear spin or, equivalently, the total nuclear angular momentum.
Electrons are also characterised by an orbital magnetic moment, since their orbital motion around the nucleus corresponds to a current or, equivalently, to a magnetic moment $\mathbf{M}_{L}$ defined as $[1,2,10]$

$$\mathbf{M}{L}=-g{L} \frac{\mu_{B}}{\hbar} \mathbf{L},$$
where $g_{L}$ is the orbital $g$-factor and $\mathbf{L}$ is the electron orbital angular momentum obeying the quantisation rules
\begin{aligned} &L^{2}=l(l+1) \hbar^{2} \ &L_{z}=m_{l} \hbar \end{aligned}
cast in terms of the orbital quantum number $l=0,1,2, \ldots$ and of the magnetic quantum number $m_{l}=0, \pm 1, \pm 2, \ldots, \pm l$. The spectroscopic notation is widely adopted to label quantum states differing by $l$ : we will set $l=0 \rightarrow s$-states, $l=1 \rightarrow$ $p$-states, $l=2 \rightarrow d$-states, and so on [2].

## 物理代写|固体物理代写Solid-state physics代考|Electronic configuration

The central problem of the physics of atoms is to determine their ground-state configuration, that is: the distribution of their electrons, among all available quantum states, corresponding to the minimum total energy. For a multi-electron atom this task is accomplished by following a rather complex procedure, qualitatively summarised below. A full account can be found elsewhere $[1,2,10]$.

The first step consists in solving the complete Schrödinger equation ${ }^{13}$ for the atom: a formidable indeed many-body quantum problem. The full scenario contains electrostatic interactions (among electrons and between the nucleus and each clectron) as well as magnetic interactions (among all existing magnetic dipoles). Coulomb interactions are by far the strongest ones and they determine the main features of the atomic energy spectrum which can be calculated, for instance, within the central field approximation $(\mathrm{CFA})^{14}$. Here each electron is treated as a singleparticle undergoing an average central field due to the nucleus and the remaining electrons. In this way the many-body problem is reduced to $Z$ single-particle ones, each separately solved by ordinary methods of atomic physics. The resulting CFA electron wavefunctions $\psi_{n m_{i}}^{\mathrm{CFA}}(\mathbf{r})=\bar{R}{n l}(r) Y{\operatorname{lm}{i}}(\theta, \phi)$ are written in polar coordinates (the central field has by construction a spherical symmetry!) as the product between a radial function $\bar{R}$ and a spherical harmonic function $Y$. Accordingly, each quantum state is labelled by three quantum numbers, namely: the principal quantum number $n=1,2,3, \ldots$ and the $l$ and $m{l}$ ones already introduced in section 1.2.2 where their values have been assigned ${ }^{15}$. A twofold picture emerges that (i) the energy spectrum is discrete and (ii) allowed atomic quantum states are organised in shells and sub-shells, respectively, corresponding to a given value of the $n$ and of $l=0,1,2, \ldots,(n-1)$.

## 物理代写|固体物理代写Solid-state physics代考|Angular and magnetic momenta

$$S^{2}=s(s+1) \hbar^{2} \quad S_{z}=m_{s} \hbar$$

$$\mathbf{M} S=-g_{S} \frac{\mu_{B}}{\hbar} \mathbf{S},$$

$$\mathbf{M} N=g_{N} \frac{\mu_{\mathrm{N}}}{\hbar} \mathbf{N},$$

$$\mu_{\mathrm{N}}=\frac{m_{\mathrm{e}}}{m_{\mathrm{p}}} \mu_{B}=5.05082 \times 10^{-27} \mathrm{~J} \mathrm{~T}^{-1},$$

$$L^{2}=l(l+1) \hbar^{2} \quad L_{z}=m_{l} \hbar$$

## 物理代写|固体物理代写Solid-state physics代考|Electronic configuration

$\psi_{n m_{i}}^{\mathrm{CFA}}(\mathbf{r})=\bar{R} n l(r) Y \operatorname{lm} i(\theta, \phi)$ 写在极坐标中（中心场通过构造具有球对称性!）作为径向函数之间的乘积 $\bar{R}$ 和球谐函数 $Y$. 因此，每个量子态由三个量子数标记，即：主量子数 $n=1,2,3, \ldots$ 和 $l$ 和 $m l$ 已经在 1.2.2 节中介 绍过的，它们的值已经被赋值 ${ }^{15}$. 出现了一个双重画面: (i) 能谱是离散的， (ii) 允许的原子量子态分别组织在 壳和子壳中，对应于给定的 $n$ 和 $l=0,1,2, \ldots,(n-1)$.

## 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|固体物理代写Solid-state physics代考|PHYSICS3544

statistics-lab™ 为您的留学生涯保驾护航 在代写固体物理Solid-state 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。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state 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|固体物理代写Solid-state physics代考|Basic definitions

The first task we must accomplish is defining the physical system we are actually interested in. This semantic exercise is quite important, since it will properly define the topic treated in this textbook: the physics of crystalline solids.

A fruitful constitutive hypothesis to start from is embodied by the atomistic picture [1, 2] which, relying on robust experimental evidence, states that ordinary matter is made by elementary constituents known to be atoms ${ }^{1}$. We agree that condensed matter forms whenever a very large number of such atoms (belonging to just one or more chemical species) tightly bind together by electrostatic interactions. Both features are indeed necessary in order to sharply define the state of aggregation we are interested in: (i) the fact that the number of atoms is very large allows us to exclude single molecules ${ }^{2}$ from the horizon of our interest, while (ii) the strong character of their mutual interactions allows us to neglect the case of gaseous systems.

The definition just given is actually very generic and it does not allow us to distinguish between two paradigmatically different situations. In order to clarify and resolve this ambiguity, let us consider a sample of condensed matter and let us label by $\mathbf{R}{a}(t)$ the position of its $\alpha$ th atom at time $t$. We define the mean square atomic displacement $\Delta^{2} R(t)$ as $$\Delta^{2} R(t)=\frac{1}{N} \sum{a=1}^{N}\left|\mathbf{R}{a}(t)-\mathbf{R}{a}(0)\right|^{2},$$
where $\mathbf{R}{a}(0)$ represents the initial position of the $\alpha$ th atom and $N$ is the total number of particles in the system. It is understood that the system is in equilibrium at temperature $T$. The calculation of $\Delta^{2} R(t)$ for a silicon sample is reported in figure $1.1$ at two different temperatures, respectively, above and below its melting temperature $T{\mathrm{m}}^{\mathrm{Si}}=1685 \mathrm{~K}$. If, according to atomic diffusion theory [3], we now link such a quantity to the corresponding diffusion coefficient $D(T)$
$$D(T)=\lim {t \rightarrow+\infty} \frac{1}{6} \frac{\Delta^{2} R(t)}{t},$$ we immediately realise that below $T{\mathrm{m}}^{\mathrm{Si}}$ the sample does not show any self-diffusion characteristics ${ }^{3}$ while above $T_{\mathrm{m}}^{\mathrm{Si}}$ it flows. In other words, the definition of condensed matter given above allows both solid and liquid systems to be called condensates, despite their physics being largely different. Therefore, we make it clear that from now on we will focus our attention only on the solid state, i.e. only on condensed matter systems that do not show any diffusive behaviour (an introduction to the fascinating physics of liquids can be found elsewhere [4]).

## 物理代写|固体物理代写Solid-state physics代考|Atomic structure

We know that an atom is a bound system consisting of a nucleus with a positive charge $+N_{\mathrm{p}} e$, where $N_{\mathrm{p}}$ is the atomic number, that is the number of protons, and a set of $Z$ electrons, each carrying a charge $-e$. We recall that $e=1.60219 \times 10^{-19} \mathrm{C}$ is the elementary electric charge. If $Z=N_{\mathrm{p}}$ then the atom is in a neutral configuration, while if $Z \neq N_{\mathrm{p}}$ then we say that the atom has been ionised (either positively or negatively provided that $Z$ is smaller or larger than $N_{\mathrm{p}}$, respectively). The nucleus also contains a number $N_{\mathrm{n}}$ of neutrons, carrying no electric charge. While all electrons have the same mass $m_{\mathrm{e}}=9.109 \times 10^{31} \mathrm{~kg}$, the nucleus of each chemical species has instead a specific mass $M$ determined as: $M=\left(N_{\mathrm{p}}+N_{\mathrm{n}}\right) m_{\mathrm{p}}$, where $m_{\mathrm{p}}=1.672 \times 10^{-27} \mathrm{~kg}$ is the proton mass ${ }^{11}$. We remark that $A=N_{\mathrm{p}}+N_{\mathrm{n}}$ is referred to as the atomic mass number. Atoms with the same number of protons, but a different number of neutrons are referred to as isotopes.

As for nuclei, we will further neglect their inner structure by treating them as point-like, massive, and charged objects ${ }^{12}$. This is indeed a very good approximation for any situation described in this volume and, therefore, protons and neutrons will no longer enter as single objects in our theory. On the other hand, electrons will be individually addressed. Nuclei and electrons are inherently non-classical objects and, therefore, they must be duly described in quantum mechanical terms.

## 物理代写|固体物理代写Solid-state physics代考|Basic definitions

$$\Delta^{2} R(t)=\frac{1}{N} \sum a=1^{N}|\mathbf{R} a(t)-\mathbf{R} a(0)|^{2},$$

$$D(T)=\lim t \rightarrow+\infty \frac{1}{6} \frac{\Delta^{2} R(t)}{t}$$

## 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。