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## 物理代写|固体物理代写Solid-state physics代考|More on relaxation times

In our discussion on transport coefficients $\sigma_{\mathrm{e}}$ and $\kappa_{\mathrm{e}}$ we have twice introduced the notion of relaxation time which, although conceptually different in the two cases, was considered the same for charge and heat currents. It is now necessary to reconsider this aspect in greater detail.

Let us start by readdressing the direct-current conductivity. Electrons, during their drift motion under the action of an external electric field $\mathbf{E}$, undergo scattering with lattice defects and ionic oscillations ${ }^{21}$. The former provide a constant contribution $\tau_{\mathrm{d}}$ to the electron relaxation time, while the effect of the ionic oscillation can be described as electron-phonon scattering events: their contribution $\tau_{\mathrm{ph}}(T)$ is inherently dependent on temperature since the phonon population of each mode is so. If we assume that the two mechanisms are independent (that is, if the number of defects is small enough to leave unaffected the vibrational spectrum of the system), then we can apply the same Matthiessen rule already introduced in section $4.3$ to understand thermal transport and write
$$\frac{1}{\tau_{\mathrm{e}}}=\frac{1}{\tau_{\mathrm{d}}}+\frac{1}{\tau_{\mathrm{ph}}(T)} .$$
By now inserting this expression for the electron relaxation time into equation (7.7), we immediately obtain the resistivity $\rho_{\mathrm{e}}$ of a metal in the form
$$\rho_{\mathrm{e}}=\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{e}}}=\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{d}}}+\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{ph}}(T)}=\rho_{\mathrm{d}}+\rho_{\mathrm{ph}}(T),$$
where the two contributions are referred to as the residual resistivity and ideal resistivity, respectively, since $\rho_{\mathrm{d}}$ is the only one active even at zero temperature, while $\rho_{\mathrm{ph}}(T)$ is the only one found even in a totally defect-free system. The electron-phonon scattering largely affects the relaxation time, which is typically decreased from $10^{-11} \mathrm{~s}$ at $T=0 \mathrm{~K}$ down to $10^{-14} \mathrm{~s}$ at room temperatures. By multiplying the Fermi velocity by $\tau_e$ we can easily estimate the order of magnitude of the electron mean free path $\lambda_e$ to be as large as dozens of $n \mathrm{~m}$ at room temperature or dozens of $\mu \mathrm{m}$ at zero temperature. This is indeed a much more accurate estimation of $\lambda_e$ than provided by the Drude theory and, more importantly, it better proves that the average distance covered between two successive collisions is much larger than the lattice interatomic spacing: as far as charge current phenomena are concerned, the electrons in a metal can be really considered as free, that is not colliding with lattice ions.

## 物理代写|固体物理代写Solid-state physics代考|Failures of the Sommerfeld theory

The Sommerfeld theory outclasses the Drude one by more accurately predicting many physical properties of metals; it also enlightens some important concepts like the difference between the chemical potential and the Fermi energy or the real need to treat the electron conduction gas as a fermion gas obeying the Fermi-Dirac statistics. However, it cannot yet be regarded as the most complete and predictive quantum theory of electron states in a crystal, since its predictions are still not in good agreement with experiments in some important cases.

First of all, we remark that the Sommerfeld theory for the charge current is basically the same as the Drude one and, therefore, it suffers the same limitation, in particular as regards the wrong predictions about the Hall coefficient ${ }^{24}$. This is mainly due to the fact that in deriving such a coefficient the fermionic nature of the charge carriers is not explicitly taken into account ${ }^{25}$. Even the alternate-current conductivity provided by the two free electron models is only grossly adequate in describing metal reflectivity, but it falls short with other optical properties of metals like, notably, their colour. Finally, the Fermi surface of real metals is not a simple sphere with radius $k_{\mathrm{F}}[3,4]$.

A part for these phenomenological failures, the Sommerfeld theory is unable to explain a very fundamental fact: why in Nature do insulators exist? Our basic assumption was to treat the system of valence electrons as a gas of delocalised charge carriers. Why is this a reasonably good approximation in some materials (metals) and not in many others (insulators)? The Sommerfeld theory does not provide an answer to this question. We need a more refined approach to the electronic structure of a crystalline solid.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|更多关于放松时间的内容

.

$$\frac{1}{\tau_{\mathrm{e}}}=\frac{1}{\tau_{\mathrm{d}}}+\frac{1}{\tau_{\mathrm{ph}}(T)} .$$

$$\rho_{\mathrm{e}}=\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{e}}}=\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{d}}}+\frac{m_e}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{ph}}(T)}=\rho_{\mathrm{d}}+\rho_{\mathrm{ph}}(T),$$
，其中这两个贡献分别称为残余电阻率和理想电阻率，因为$\rho_{\mathrm{d}}$是即使在零温度下也唯一活跃的，而$\rho_{\mathrm{ph}}(T)$是即使在完全无缺陷的系统中也唯一发现的。电子-声子散射很大程度上影响弛豫时间，在室温下，弛豫时间通常从$T=0 \mathrm{~K}$处的$10^{-11} \mathrm{~s}$下降到$10^{-14} \mathrm{~s}$。通过将费米速度乘以$\tau_e$，我们可以很容易地估计出电子平均自由程$\lambda_e$的数量级，在室温下可以达到数十个$n \mathrm{~m}$，在零温度下可以达到数十个$\mu \mathrm{m}$。这确实是对$\lambda_e$的一个比德鲁德理论提供的精确得多的估计，更重要的是，它更好地证明了两次连续碰撞之间的平均距离远远大于晶格原子间的间距:就电荷电流现象而言，金属中的电子可以真正地被认为是自由的，即没有与晶格离子碰撞

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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