# 物理代写|固体物理代写Solid-state physics代考|PHYS7120

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

Debye assumed that a crystalline solid can be represented by an isotropic elastic continuum. Atoms in a solid do not vibrate with the same frequency but have the same dispersion. Because there is finite number of atoms, however, not all values of the frequency are allowed. There is a maximum allowed frequency that any atom can have. Thus, it is possible to propagate wave through solids covering a wavelength region extending from low frequencies (sound waves) up to short waves (infrared absorption). The essential difference between the Debye model and the Einstein model is that Debye considers the vibrational modes of a crystal as a whole, whereas Einstein’s starting point was to consider the vibration of a single atom, assuming the atomic vibrations to be independent of each other.

The solids are by no means continuous but are built up of atoms, that is, discrete mass points. However, the Debye continuum is justified on the following basis. Consider an elastic wave propagated in a crystal of volume $V$. As long as the wavelength of the wave is large compared with interatomic distances, the crystal looks like a continuum from the point of view of the wave. The essential assumption of Debye is now that this continuum model may be employed for all possible vibrational modes of the crystal. Further, the fact that the crystal actually consists of atoms is taken into account by limiting the total number of vibrational modes to $3 N, N$ being the total number of atoms. In other words, the frequency spectrum corresponding to a perfect continuum is cut off so as to comply with a total of $3 \mathrm{~N}$ modes.

In the Debye model, the velocity of sound is taken as constant as it would be for a classical elastic continuum. The dispersion relation is then
$$\omega=v k$$
$v=$ constant velocity of sound.
The average energy of the oscillator of frequency $\omega$ is
$$\bar{E}=\frac{1}{2} \hbar \omega+\frac{\hbar \omega}{\exp \left[\frac{\hbar \omega}{k_B T}\right]-1}$$

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

Classical physics regarded a solid crystal as an assembly of atoms held together in a periodic array by certain attractive forces. The atoms were assumed to be free to vibrate about their equilibrium positions under the constraints of the resulting forces and to a first approximation, forces and atomic displacements would be related by Hook’s law. The effect of thermal energy than would be to set these atoms into vibration as harmonic oscillators about their equilibrium positions. Since the oscillators are purely harmonic, therefore the potential is a parabolic function of position. The minimum of the potential energy curve is the classical equilibrium position of the atom, if it is at rest. If the interatomic forces were such that the atom if set in motion thermally would vibrate about its equilibrium position as an ideal classical harmonic oscillator. The atom would execute vibrations about the equilibrium position, the maximum displacement from the equilibrium position in either direction being equal, and the average distance $\langle x\rangle$ would be equal to value of lattice constant $a$ at zero temperature. There would thus be no thermal expansion.

Let the potential well in which the atoms vibrate have approximately the appearance as shown in Fig. 7.5.

In this case, although nearly parabolic about the minimum point $A$, the actual will deviate from the parabolic form more and more as the distance from the minimum point increases. If the atom has energy $U_0$, it should according to the classical picture vibrate between the extreme amplitude limits $B$ and $C$, the vibrations are somewhat anharmonic in character. But the distance $D C$ between the equilibrium positions is now greater than the distance $B D$ between the equilibrium position and the maximum compression position. The average interatomic distance $\langle x\rangle$ is thus greater than the zero temperature lattice constant, and thermal expansion is observed. Therefore, in order to account for the thermal expansion, it is necessary to consider anharmonic terms in the potential. We approximate the true potential more accurately by adding high-order (anharmonic) terms as follows
$$U(x)=c x^2-g x^3-f x^4$$

# 固体物理代写

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

lomega $=\mathrm{vk}$
$\$ \$$\ \mathrm{v}=\$$ 恒定声速。

U(x)=c x^2-g x^3-f x^4


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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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