## 物理代写|固体物理代写Solid-state physics代考|PHYS4210

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## 物理代写|固体物理代写Solid-state physics代考|Classical Lattice Heat Capacity

The first law of thermodynamics states that
$$Q=\mathrm{d} U-p \mathrm{~d} V$$
$\mathrm{d} Q$ is the amount of heat added to the system and is equal to increase in energy $\mathrm{d} U$ of the system plus the amount of work done by the system and $T$ is temperature. The specific heat at constant volume is
$$C_V=\left(\frac{\partial W}{\partial t}\right)_V=\left(\frac{\partial U}{\partial t}\right)_V$$
where partial derivatives are used because the energy may be a function of other variables in addition to temperature and the subscript denotes that volume is kept constant.

Thus to calculate the specific heat, one should have an expression for the internal energy, that is, total energy of all the particles. The contribution of the phonons to the heat capacity of a crystal is called the lattice heat capacity and is denoted by $C_{\text {lat }}$.

Let us evaluate the value of specific heat. For this consider, a model in which it is assumed that each atom of solid is bound to its site by a harmonic force. When the solid is heated, the atom vibrates about their equilibrium position like a set of harmonic oscillators. The average energy, $\bar{E}$, for a one-dimensional oscillator is equal to $k_B T$, where $k_B$ is Boltzmann constant. Therefore, the average energy per atom, regarded as a threedimensional harmonic oscillator is $3 k_B T$. The energy per mole is
$$\bar{E}=3 N_A k_B T=3 R T$$
where is Avogadro number and $R=N_A k_{\mathrm{B}}$.

## 物理代写|固体物理代写Solid-state physics代考|Einstein Model of Specific Heat

In this model, the atoms are treated as independent oscillator. Einstein assumed a rigid lattice of identical oscillators, all having the same resonance frequencies. From the discussion of lattice vibrations, it is clear that the optical branch has a single resonant frequency when one of the masses becomes very large. The light masses then vibrate entirely independent of each other with their own resonant frequency. This is essentially the Einstein model.

According to quantum mechanics, the energy of an isolated oscillator is restricted to the value
$$E_n=\left(n+\frac{1}{2}\right) \hbar \omega$$

where $n$ is positive integer or zero. That is $n=0,1,2,3, \ldots$ The constant $\omega$ is the frequency of the oscillator. Thus, the energy of the oscillator is quantized. Equation (7.4) refers to an isolated oscillator but the atomic oscillators in a solid are not isolated. They are continually exchanging energy with the thermal bath surrounding the solid. The energy of the oscillator is therefore continually changing, but the average value at thermal equilibrium is given by
$$\bar{E}=\frac{\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right) \hbar \omega \exp \left[-\frac{\left(n+\frac{1}{2}\right) \hbar \omega}{k_B T}\right]}{\sum_{n=0}^{\infty} \exp \left[-\frac{\left(n+\frac{1}{2}\right) \hbar \omega}{k_B^T}\right]}$$
The $\exp \left[-\frac{\left(n+\frac{1}{2}\right) \hbar \omega}{k_B T}\right]=\exp \left[-\frac{E_n}{k_B T}\right]$ is Boltzmann factor, which gives the probability that the energy state $E_n$ is occupied and the sum in the denominator is inserted for correct normalization. $n=0$ is included in the summation to get the right average we need to include those oscillators that are not oscillating at all.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|Classical Lattice Heat Capacity

$\$ \$$Q=I m a t h r m{d} Up \backslash mathrm {\sim d} \vee \ \$$

## 有限元方法代写

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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。