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# 物理代写|固体力学代写Solid Mechanics代考|PHYS881 The classical (Drude) theory of the conduction gas

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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

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

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