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物理代写|固体物理代写Solid-state physics代考|Experimental measurement of phonon dispersion relations

The experimental determination of the $\omega=\omega_s(\mathbf{q})$ dispersion relations over the entire $1 \mathrm{BZ}$ needs a probe fulfilling two conditions: (i) its wavelength must be comparable with the typical interatomic distances in the crystal structure and (ii) its energy must be of the same order of typical phonon quanta $\hbar \omega_s(\mathbf{q})$, which range mostly in the interval $\left[1,10^2\right] \mathrm{meV}$. Optical probes are unsuitable: $x$-rays have the right wavelength, but a too high energy of the order $\mathcal{O}\left(10^4 \mathrm{eV}\right)$; other kinds of photons, instead, can only explore the $\mathbf{q} \sim 0$ region of the Brillouin zone, i.e. they can only detect (some) zone-centre phonons. A probe consisting in a flux of electrons is also impractical for a twofold reason: (i) their surface scattering is very strong and, therefore, they are unable to probe the bulk region of the crystal; (ii) multiple scattering is likely to occur in the case of electrons and this makes the analysis of the experiment a very challenging task. In contrast, both requirements of suitable wavelength and energy are guaranteed by a flux of thermal neutrons which have typical wavelengths of the order of just a few $\AA$ and energies in between a few and a few tens of meV. Accordingly, neutron spectroscopy is the most powerful technique for measuring the phonon dispersion relations [16,17].

The typical experimental setup for neutron spectroscopy is shown in figure $3.11$. A continuous flux of neutrons is emitted by a source, typically a fission reactor; the beam is thermalised by collisions within the reactor moderator and, therefore, neutrons are produced with a Maxwell-Boltzmann distribution of velocities corresponding to the room temperature (for this reason they are referred to as ‘thermal neutrons’). The thermalised beam emerging from the source is Bragg reflected by a single-crystal monochromator which selects neutrons of same momentum $\hbar \mathbf{p}{\mathrm{in}}$ and energy $E{\text {in }}=\hbar^2 p_{\mathrm{in}}^2 / 2 m_{\mathrm{n}}$ where $m_{\mathrm{n}}$ is the neutron mass. The resulting monochromatic beam is now focussed by a collimator and then it falls on a sample with known crystalline orientation. The momentum vector $\hbar \mathbf{p}{\text {out }}$ of the scattered neutrons is defined by letting the beam pass through a second collimator and eventually arrive at the analyser. Here the magnitude $p{\text {out }}$ and the corresponding energy $E_{\text {out }}=\hbar^2 p_{\text {out }}^2 / 2 m_{\mathrm{n}}$ of the detected neutrons are determined by measuring the Bragg reflection angles from the planes of the crystalline analyser.

物理代写|固体物理代写Solid-state physics代考|The vibrational density of states

In many frameworks it is necessary to explicitly take into account the distribution of phonon frequencies over the full vibrational spectrum. To this aim it is useful to define the quantity $G(\omega)$, hereafter referred to as the vibrational density of states (vDOS), defined such that $G(\omega) d \omega$ represents the number of phonons (that is: vibrational modes) with frequency in the interval $[\omega, \omega+d \omega]$. In order to calculate a general expression for $G(\omega)$ we will proceed by two steps of increasing generality.

Let us preliminarily consider the model case of a monoatomic sc crystal with first nearest neighbours distance $a$ and subject to Born-von Karman periodic boundary conditions. For convenience we choose the crystal in the form of a cube with edge length $L$ and assume that there is only one dispersion relation: basically, this is the three-dimensional counterpart of the monoatomic linear chain. Accordingly, by generalising equation (3.9) we understand that the allowed phonon wavevectors $\mathbf{q}$ have Cartesian components $q_i=2 \pi \xi_i / L$ with $L=N a$, while $i=x, y, z$ and $\xi_i=0,1,2, \ldots(N-1)$ if $N$ is the total number of atoms in the crystal. Their number density in the reciprocal space is straightforwardly calculated as $L^3 /(2 \pi)^3=V /(2 \pi)^3$ where $V=L^3$ is the volume of the sample. The number of allowed wavevectors corresponding to vibrational modes with a frequency in between $\omega$ and $\omega+d \omega$ is given by the product between their number density and phonon branch, this number equals the number of vibrational frequencies in the selected interval. We accordingly write
$$G(\omega) d \omega=\frac{V}{(2 \pi)^3} 4 \pi q^2 d q .$$
We remark that $4 \pi q^2 d q$ is the volume of the spherical shell precisely bounded by the two constant-frequency surfaces corresponding to $\omega$ and $\omega+d \omega$, respectively.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|The vibrational density of states

$$G(\omega) d \omega=\frac{V}{(2 \pi)^3} 4 \pi q^2 d q$$

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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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