Posted on Categories:Solid Mechanics, 固体力学, 物理代写

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## 物理代写|固体力学代写Solid Mechanics代考|Finite temperature properties

Let us now consider a metal in equilibrium at temperature $T>0 \mathrm{~K}$. In this case the eDOS is written as
\begin{aligned} G(E, T) &=G(E) n_{\mathrm{FD}}(E, T) \ &=\frac{V}{2 \pi^2 \hbar^3}\left(2 m_{\mathrm{e}}\right)^{3 / 2} \frac{1}{1+\exp \left[\left(E-\mu_c\right)\right] / k_{\mathrm{B}} T} E^{1 / 2}, \end{aligned}
where we have combined the expression given in equation (7.28), which is a mere counting of states, with the finite-temperature probability $n_{\mathrm{FD}}(E, T)$ that the quantum level $E$ is occupied, a correction entering our theory through equation (6.7). The $G(E, T)$ function is plotted in figure $7.5$ (thick blue line), together with its zero-temperature counterpart (thin black line). We remark that in plotting this figure we have neglected the temperature dependence of the chemical potential and, accordingly, we have set $\mu_{\mathrm{c}}=E_{\mathrm{F}}$ at any $T \geqslant 0$. We will very soon critically re-address this assumption, proving that it is valid to a very good extent.

The number $N$ of electrons is obviously unaffected by temperature and we can therefore cast the normalisation condition (previously expressed as in equation $(7.29))$ in a new form

$$N=\int_0^{+\infty} G(E, T) d E=\int_0^{+\infty} G(E) n_{\mathrm{FD}}(E, T) d E,$$
which allows us to interpret the shaded area of figure $7.5$ as the conserved number of electrons. Since this notion is valid for any selected range of energy, we can develop a new interesting concept.

## 物理代写|固体力学代写Solid Mechanics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
$$\rho_{\mathrm{e}}=\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{e}}}=\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{d}}}+\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{ph}}(T)}=\rho_{\mathrm{d}}+\rho_{\mathrm{ph}}(T),$$
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 $\tau_{\mathrm{e}}$ we can easily estimate the order of magnitude of the electron mean free path $\lambda_{\mathrm{e}}$ to be as large as dozens of nm at room temperature or dozens of $\mu$ m at zero temperature. This is indeed a much more accurate estimation of $\lambda_{\mathrm{e}}$ 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|固体力学代写Solid Mechanics代考|有限温度特性

.

\begin{aligned} G(E, T) &=G(E) n_{\mathrm{FD}}(E, T) \ &=\frac{V}{2 \pi^2 \hbar^3}\left(2 m_{\mathrm{e}}\right)^{3 / 2} \frac{1}{1+\exp \left[\left(E-\mu_c\right)\right] / k_{\mathrm{B}} T} E^{1 / 2}, \end{aligned}
，其中我们将公式(7.28)中给出的表达式(仅仅是状态计数)与量子能级$E$被占据的有限温度概率$n_{\mathrm{FD}}(E, T)$结合起来，通过公式(6.7)进入我们的理论修正。$G(E, T)$函数被绘制在图$7.5$(粗蓝线)中，以及它的零温度对应函数(细黑线)。我们注意到，在绘制这个图时，我们忽略了化学势的温度依赖性，因此，我们将$\mu_{\mathrm{c}}=E_{\mathrm{F}}$设为任何$T \geqslant 0$。我们很快就会批判性地重新处理这个假设，证明它在很大程度上是有效的

$$N=\int_0^{+\infty} G(E, T) d E=\int_0^{+\infty} G(E) n_{\mathrm{FD}}(E, T) d E,$$

## 物理代写|固体力学代写Solid Mechanics代考|更多关于弛缓时间

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

$$\rho_{\mathrm{e}}=\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{e}}}=\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{d}}}+\frac{m_{\mathrm{e}}}{n_{\mathrm{e}} e^2} \frac{1}{\tau_{\mathrm{ph}}(T)}=\rho_{\mathrm{d}}+\rho_{\mathrm{ph}}(T),$$
，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$\tau_{\mathrm{e}}$，我们可以很容易地估计出电子平均自由程$\lambda_{\mathrm{e}}$的数量级，在室温下可以达到几十纳米，在零温度下可以达到几十$\mu$ m。这确实是对$\lambda_{\mathrm{e}}$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

## 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:Solid Mechanics, 固体力学, 物理代写

## avatest™帮您通过考试

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## 物理代写|固体力学代写Solid Mechanics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 $\tau_{\mathrm{e}}$ (thus averaging among all possible scattering mechanisms) defined such that $1 / \tau_{\mathrm{e}}$ 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|固体力学代写Solid Mechanics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 $\mathbf{J}{\mathrm{q}}$ $$\mathbf{J}{\mathrm{q}}=-n_{\mathrm{e}} e \mathbf{V}{\mathrm{d}}=\frac{n{\mathrm{e}} e^2 \tau_{\mathrm{e}}}{m_{\mathrm{e}}} \mathbf{E},$$
and the Drude expression for the direct-current conductivity $\sigma_{\mathrm{e}}$
$$\sigma_{\mathrm{e}}=\frac{n_{\mathrm{e}} e^2 \tau_{\mathrm{e}}}{m_{\mathrm{e}}},$$
which links this quantity to few microscopic physical parameters associated either with the charge carriers ( $e$ and $m_{\mathrm{e}}$ ) or to the specific material $\left(n_{\mathrm{e}}\right.$ and $\tau_{\mathrm{e}}$ ). 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 $\lambda_{\mathrm{e}} \sim 1-10 \AA$ 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|固体力学代写固体力学代考|电导率

Drude理论的第一个应用是预测金属的直流电导率。设$\mathbf{v}{\mathrm{d}}$为在外加均匀恒定电场$\mathbf{E}$作用下的电子漂移速度。加速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$$-e \mathbf{E}=m{\mathrm{e}} \dot{\mathbf{V}}{\mathrm{d}}+\beta \mathbf{v}{\mathrm{d}},$$
，其中$\beta$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，我们有$d \mathbf{v}{\mathrm{d}} / d t=0$，因此有$$-\frac{e}{m{\mathrm{e}}} \mathbf{E}=\frac{\beta}{m_{\mathrm{e}}} \mathbf{v}{\mathrm{d}}$$，很自然地，${ }^4$导致了$\beta=m{\mathrm{e}} / \tau_{\mathrm{e}}$的定义。这允许我们计算电子漂移速度为

$$\mathbf{v}{\mathrm{d}}=-\frac{e \tau{\mathrm{e}}}{m_{\mathrm{e}}} \mathbf{E},$$

## 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。