# 物理代写|固体物理代写Solid-state physics代考|PHYS3702

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## 物理代写|固体物理代写Solid-state physics代考|The Linear (One Dimensional) Diatomic Lattice

Consider a diatomic lattice in one dimension (Fig. 6.4). The distance between nearest neighbours is denoted by $a$. The particles are numbered in such a way that even numbers have a mass $M_1$ and odd ones have $M_2$.

Suppose that each atom interacts only with its nearest neighbours and force constants are identical between all pairs of nearest neighbour atoms. The equation of motion is
$$\begin{gathered} M_1 \frac{\mathrm{d}^2 U_{2 n}}{\mathrm{~d} t^2}=C\left(U_{2 n+1}+U_{2 n-1}-2 U_{2 n}\right) \ M_2 \frac{\mathrm{d}^2 U_{2 n+1}}{\mathrm{~d} t^2}-C\left(U_{2 n}+U_{2 n+2}-2 U_{2 n+1}\right) \end{gathered}$$
Let the solution is in the form of a travelling wave given by
$$\begin{gathered} U_{2 n}=A \exp [i(2 n k a-\omega t)] \ \left.U_{2 n+1}=B \exp [i(2 n+1) k a-\omega t)\right] \ \left.U_{2 n-1}=B \exp [i(2 n-1) k a-\omega t)\right] \end{gathered}$$
where $k$ is the wave vector of a particular mode of vibration; $A$ and $B$ are the amplitude corresponding to particles of mass $M_1$ and $M_2$, respectively. Substituting Eqs. (6.27)(6.29) in Eqs. (6.25) and (6.26)
$$\begin{gathered} -M_1 A \omega^2 \exp [i(2 n k a-\omega t)]=C \exp [i(2 n k a-\omega t)][B \exp (i k a)+B \exp (-i k a)-2 A] \ \left.-M_2 B \omega^2 \exp [i(2 n+1) k a-\omega t]=C \exp [i(2 n+1) k a-\omega t)\right] \ {[A \exp (-i k a)+A \exp (i k a)-2 B]} \end{gathered}$$ or
\begin{aligned} & -M_1 A \omega^2=C[2 B \cos k a-2 A] \ & -M_2 B \omega^2=C[2 A \cos k a-2 B] \end{aligned}
or
\begin{aligned} & \left(M_1 \omega^2-2 C\right) A+2 B C \cos k a=0 \ & \left(M_2 \omega^2-2 C\right) B+2 A C \cos k a=0 \end{aligned}

## 物理代写|固体物理代写Solid-state physics代考|Density of States

Consider the vibrational modes of a continuous elastic medium. Each normal mode of vibration of the medium has a characteristic wavelength. The medium is assumed to be a continuum if the wavelength $\lambda$ is much larger than the interatomic separation $a$, that is, $\lambda \gg a$.

Consider the case in one dimension. For simplicity consider the vibrational modes of a string of length $L$ whose both ends are fixed. When a continuous succession of waves, such as sinusoidal waves arrive at the fixed end of the string, a corresponding continuous successive wave is generated which are reflected back. Thus, the wave is reflected and rereflected. Since both ends of the string are fixed, therefore, there will be nodes at the fixed ends. The different nodes are separated by $\frac{\lambda}{2}$. Suppose $u(z, t)$ represents the deflection of string at point $z$ at any instant $t$. The wave on the string is then given by one-dimensional wave equation $$\frac{\partial^2 u(z, t)}{\partial t^2}=v_s^2 \frac{\partial^2 u(z, t)}{\partial z^2}$$
where $v_s$ is the velocity of propagation of the wave on the string. Since the string is fixed at both the ends, therefore
$$u(0, t)=u(L, t)=0$$
for all values of $t$.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|The Linear (One Dimensional) Diatomic Lattice

$$M_1 \frac{\mathrm{d}^2 U_{2 n}}{\mathrm{~d} t^2}=C\left(U_{2 n+1}+U_{2 n-1}-2 U_{2 n}\right) M_2 \frac{\mathrm{d}^2 U_{2 n+}}{\mathrm{d} t^2}$$

$$U_{2 n}=A \exp [i(2 n k a-\omega t)] U_{2 n+1}=B \exp [i(2 n+1$$

$$-M_1 A \omega^2 \exp [i(2 n k a-\omega t)]=C \exp [i(2 n k a-\omega t)]$$

$$-M_1 A \omega^2=C[2 B \cos k a-2 A] \quad-M_2 B \omega^2=C$$

$$\left(M_1 \omega^2-2 C\right) A+2 B C \cos k a=0 \quad\left(M_2 \omega^2-20\right.$$

## 物理代写|固体物理代写Solid-state physics代考|Density of States

$$\frac{\partial^2 u(z, t)}{\partial t^2}=v_s^2 \frac{\partial^2 u(z, t)}{\partial z^2}$$

$$u(0, t)=u(L, t)=0$$

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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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