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## 物理代写|固体力学代写Solid Mechanics代考|Elastic moduli

We have so far introduced four different elastic parameters, namely the Young modulus, the Poisson ratio and the two Lamé coefficients. We previously introduced also the bulk modulus $B$ when we investigated thermal expansion in section 4.2.1. Since $B$ was defined as the inverse of the isothermal compressibility (see appendix $C$ ), that is it deals with volume variations, it is expected to be related to some elastic parameter. In order to elucidate this issue, let us consider a hydrostatic stress $T_{i j}=P \delta_{i j}$ (where $P$ is the macroscopic hydrostatic pressure) and insert it into the constitutive equation (5.39) so as to get
$$\mathbb{S}=\frac{1}{3} \frac{1}{\lambda+\frac{2}{3} \mu} P \mathbb{.} .$$
The connection with equation (C.11) is established by defining
$$B=\lambda+\frac{2}{3} \mu,$$

so that
$$\mathbb{S}=\frac{1}{3 B} P \rrbracket \quad \rightarrow \quad \operatorname{Tr}(\mathbb{S})=\sum_i \epsilon_{i i}=\frac{\Delta V}{V}=\frac{P}{B},$$
which leads to the following definition
$$\frac{1}{B}=\frac{1}{V} \frac{\Delta V}{P},$$
representing the finite difference counterpart of equation (C.11). This result reconciles the thermodynamical and elastic treatment of deformations affecting the system volume and it allows us to recast the stress-strain relation of a homogeneous and isotropic linear elastic medium in the form
\begin{aligned} \mathbb{U} &\left.=2 \mu \mathbb{S}+\left(B-\frac{2}{3} \mu\right) \operatorname{Tr}(\mathbb{S})\right] \ &\left.=3 B\left[\frac{1}{3} \operatorname{Tr}(\mathbb{S})\right]\right]+2 \mu\left[\mathbb{S}-\frac{1}{3} \operatorname{Tr}(\mathbb{S}) \mathbb{]},\right. \end{aligned}
where the first and second term on the right-hand side are, respectively, named spherical part and deviatoric part of the stress tensor: they describe the hydrostatic volume variation and the change in shape of the solid body subject to $\mathbb{I}$.

## 物理代写|固体力学代写Solid Mechanics代考|Thermoelasticity

We have so far implicitly assumed that the deformations are imposed to the system at zero temperature. While this assumption was useful to define a clean purely-elastic problem, we must duly generalise our theory to include stress actions applied at $T>0 \mathrm{~K}$ as well [6].

The starting point is of course the energy balance stated by the first law of thermodynamics (see equation (C.5)) which for a system with volume $V$ in equilibrium at temperature $T$ under some elastic action is written as
$$d \mathcal{U}=V \sum_{i j} T_{i j} d \epsilon_{i j}+T d S,$$
where the mechanical work contributing to the internal energy $\mathcal{U}$ has been written in terms of the stress tensor since we know that this latter describes any possible kind of volume and shape variation of the system. It is easy to reconcile equation (5.47) with the standard thermodynamical formulation by simply considering the case of a hydrostatic stress $T_{i j}=-P \delta_{i j}$, where $P$ is the applied pressure whose negative sign indicates that the mechanical action is compressive. We assume it to operate quasistatically, like anywhere else in the remaining of this chapter. By inserting the hydrostatic stress into equation (5.47) we get
\begin{aligned} d \mathcal{U} &=V \sum_{i j}\left(-P \delta_{i j}\right) d \epsilon_{i j}+T d S \ &=-P V \sum_i d \epsilon_{i i}+T d S \ &=-P V \frac{d V}{V}+T d S \ &=-P d V+T d S \end{aligned}

consistently with the first law of thermodynamics. Equation (5.47) is valid for any arbitrary deformation and, therefore, it allows for a thermodynamical definition of the stress tensor
$$T_{i j}=\left.\frac{1}{V} \frac{\partial \mathcal{U}}{\partial \epsilon_{i j}}\right|S=\left.\frac{1}{V} \frac{\partial \mathcal{F}}{\partial \epsilon{i j}}\right|T,$$ where $\mathcal{F}=\mathcal{U}-T S$ is the Helmholtz free energy, corresponding to the work exchanged quasi-statically during the constant-temperature deformation (see appendix C). This result defines the next task to accomplish, namely: deriving the explicit dependence $\mathcal{F}=\mathcal{F}\left(\epsilon{i j}\right)$.

## 物理代写|固体力学代写Solid Mechanics代考|弹性模量

$$\mathbb{S}=\frac{1}{3} \frac{1}{\lambda+\frac{2}{3} \mu} P \mathbb{.} .$$
。通过定义
$$B=\lambda+\frac{2}{3} \mu,$$ ，建立了与方程(C.11)的联系

，使
$$\mathbb{S}=\frac{1}{3 B} P \rrbracket \quad \rightarrow \quad \operatorname{Tr}(\mathbb{S})=\sum_i \epsilon_{i i}=\frac{\Delta V}{V}=\frac{P}{B},$$
，从而得到以下定义
$$\frac{1}{B}=\frac{1}{V} \frac{\Delta V}{P},$$

\begin{aligned} \mathbb{U} &\left.=2 \mu \mathbb{S}+\left(B-\frac{2}{3} \mu\right) \operatorname{Tr}(\mathbb{S})\right] \ &\left.=3 B\left[\frac{1}{3} \operatorname{Tr}(\mathbb{S})\right]\right]+2 \mu\left[\mathbb{S}-\frac{1}{3} \operatorname{Tr}(\mathbb{S}) \mathbb{]},\right. \end{aligned}
，其中右侧的第一项和第二项分别称为应力张量的球形部分和偏离部分:它们描述了静压体积变化和固体形状的变化$\mathbb{I}$。

## 物理代写|固体力学代写Solid Mechanics代考|热弹性

.

$$d \mathcal{U}=V \sum_{i j} T_{i j} d \epsilon_{i j}+T d S,$$
，其中贡献于内能$\mathcal{U}$的机械功被写成应力张量的形式，因为我们知道后者描述了任何可能的体积和形状变化系统。通过简单地考虑静水应力$T_{i j}=-P \delta_{i j}$的情况，很容易使式(5.47)与标准热力学公式相一致，其中$P$是施加的压力，其负号表示机械作用为压缩。我们假设它是准静态的，就像本章其余部分的其他内容一样。将静水应力代入(5.47)式，得到
\begin{aligned} d \mathcal{U} &=V \sum_{i j}\left(-P \delta_{i j}\right) d \epsilon_{i j}+T d S \ &=-P V \sum_i d \epsilon_{i i}+T d S \ &=-P V \frac{d V}{V}+T d S \ &=-P d V+T d S \end{aligned}

$$T_{i j}=\left.\frac{1}{V} \frac{\partial \mathcal{U}}{\partial \epsilon_{i j}}\right|S=\left.\frac{1}{V} \frac{\partial \mathcal{F}}{\partial \epsilon{i j}}\right|T,$$的热力学定义，其中$\mathcal{F}=\mathcal{U}-T S$是亥姆霍尔兹自由能，对应于在恒温变形过程中准静态交换的功(见附录C)。这个结果定义了下一个要完成的任务，即:导出显式依赖$\mathcal{F}=\mathcal{F}\left(\epsilon{i j}\right)$ .

.

## 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。

Posted on Categories:固体物理, 物理代写

## 物理代写|固体物理代写Solid Physics代考|PHYS440 Quantum theory of harmonic crystals

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## 物理代写|固体物理代写SOLID 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 $\left(n_{s \mathbf{q}}+1 / 2\right) \hbar \omega_{s}(\mathbf{q})$ where $n_{s \mathbf{q}}=0,1,2, \ldots$ 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 $\left(n_{s \mathbf{q}}+1 / 2\right) \hbar \omega_{s}(\mathbf{q})$ 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 .

## 物理代写|固体物理代写SOLID PHYSICS代考|Experimental measurement of phonon dispersion relations

The experimental determination of the $\omega=\omega_{s}(\mathbf{q})$ dispersion relations over the entire $1 \mathrm{BZ}$ needs a probe fulfilling two conditions: (i) its wavelength must be comparable with the typical interatomic distances in the crystal structure and (ii) its energy must be of the same order of typical phonon quanta $\hbar \omega_{s}(\mathbf{q})$, which range mostly in the interval $\left[1,10^{2}\right] \mathrm{meV}$. Optical probes are unsuitable: $x$-rays have the right wavelength, but a too high energy of the order $\mathcal{O}\left(10^{4} \mathrm{eV}\right)$; other kinds of photons, instead, can only explore the $\mathbf{q} \sim 0$ region of the Brillouin zone, i.e. they can only detect (some) zone-centre phonons. A probe consisting in a flux of electrons is also impractical for a twofold reason: (i) their surface scattering is very strong and, therefore, they are unable to probe the bulk region of the crystal; (ii) multiple scattering is likely to occur in the case of electrons and this makes the analysis of the experiment a very challenging task. In contrast, both requirements of suitable wavelength and energy are guaranteed by a flux of thermal neutrons which have typical wavelengths of the order of just a few $\AA$ and energies in between a few and a few tens of meV. Accordingly, neutron spectroscopy is the most powerful technique for measuring the phonon dispersion relations $[16,17]$.

## 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。