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# 物理代写|固体物理代写solid physics代考|Dynamics of one-dimensional crystals

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## 物理代写|固体物理代写solid 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 $l$ th 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
$$q=\frac{2 \pi}{a} \frac{\xi}{N} \quad \text { with } \quad \xi=0,1,2,3, \ldots, N-1$$
This is a very important result: in an $N$-atom monoatomic chain there are only $N$ independent values of the wavevector $q$ associated with as many independent solutions of the equations of motion. In more physical terms: if we consider a one-dimensional chain containing $N$ identical ions, there are only $N$ different ways in which they can longitudinally oscillate around their equilibrium positions.

## 物理代写|固体物理代写solid 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.

## 物理代写|固体物理代写SOLID PHYSICS代考|MONOATOMIC LINEAR CHAIN

$$M \ddot{u}{l}=\gamma^{(L)}\left(u{l+1}+u_{l-1}-2 u_{l}\right),$$
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
$$q=\frac{2 \pi}{a} \frac{\xi}{N} \quad \text { with } \quad \xi=0,1,2,3, \ldots, N-1$$ 不同的方式，它们可以围绕它们的平衡位置纵向振荡。

## 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。