分类：固体物理代写

物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

Posted on 2023年6月1日2023年6月1日 by statistics-lab_01, statistics-lab_01

如果你也在怎样代写固体物理Solid Physics PHYS881这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。固体物理Solid Physics是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。固体物理学研究固体材料的大尺度特性是如何产生于其原子尺度特性的。因此，固态物理学构成了材料科学的理论基础。它也有直接的应用，例如在晶体管和半导体的技术中。

固体物理Solid Physics是由密密麻麻的原子形成的，这些原子之间有强烈的相互作用。这些相互作用产生了固体的机械（如硬度和弹性）、热、电、磁和光学特性。根据所涉及的材料及其形成的条件，原子可能以有规律的几何模式排列（晶体固体，包括金属和普通水冰）或不规则地排列（非晶体固体，如普通窗玻璃）。

statistics-lab™ 为您的留学生涯保驾护航在代写固体物理Solid-state physics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写固体物理Solid-state physics代写方面经验极为丰富，各种代写固体物理Solid-state physics相关的作业也就用不着说。

物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

One of the most powerful tools for determining the band structure of a material is the photoemission process, by which an incoming photon kicks an electron out of the solid.

In vacuum, the electron will travel ballistically with the momentum and energy it had when it left the material. A current of electrons ejected in this way from the material can then be analyzed for their direction of motion and kinetic energy. This measurement is known as angle-resolved photoemission spectroscopy(ARPES).

Typically, the momentum of the photon is negligible compared to the momentum of the electron. The absorption of the photon can therefore be viewed as a “vertical” process, in which the electron moves to higher energy while staying at nearly the same $k$-vector. The high-energy electron can then have enough energy to overcome the work function of the material and leave the crystal.

In thinking of the process by which the electron leaves the solid, the question immediately arises of what conservation rules to apply. We have already seen that $\hbar k$ is not the true momentum of an electron; this is given by $(1.6 .10)$,
$$
\left\langle\psi_{\vec{k}}|\vec{p}| \psi_{\vec{k}}\right\rangle=\hbar \vec{k}-i \hbar \int d^3 r u_{n \vec{k}}^* \nabla u_{n \vec{k}} .
$$
When the electron crosses the boundary of the solid, do we conserve momentum, or do we conserve $\hbar k$ ? The answer is that we conserve $\hbar k$ in the direction parallel to the surface, not the total electron momentum. This can be understood as a consequence of the wave nature of the electrons, in analogy with Snell’s law, which is discussed in detail in Chapter 3. We write $\vec{k}=\vec{k}{|}+k{\perp} \hat{z}$, where $\vec{k}{|}$is the wave vector component parallel to the surface and $k{\perp}$ is the component perpendicular to the surface. The spacing of the wave fronts along a direction $\vec{x}$ on the surface is given by the condition $\vec{k} \cdot \vec{x}=k_{|} x=2 \pi n$, where $n$ is an integer. The distance between points of phase $2 \pi$ is therefore $\Delta x=2 \pi / k_{|}$. This spacing must be the same for the wave both inside and outside of the solid, a condition generally known as phase matching Although $\vec{k}_{|}$is conserved, the total momentum of the electron is in general not conserved. Therefore, the crystal must recoil slightly, taking up the difference

物理代写|固体物理代写Solid-state physics代考|Molecular Bonds

In general, electronic bands and molecular bonds are intimately related. Harrison (1980) gives an excellent discussion of the relation of chemical bonds and bands in detail.

As discussed in Section 1.1.2, when two electron orbitals overlap, two new states are created that can be approximated as symmetric and antisymmetric linear combinations of the original orbitals. As mentioned in Section 1.1.2, these are called the bonding and antibonding states, respectively.

Figure 1.34 shows how bonding occurs in the case of two atoms with overlapping $s$-orbitals, in the LCAO approximation. If there is one electron per atom, then the electrons from both atoms can fall into the lowest state, thus reducing the total energy of the pair of atoms. This is why we say they are bonded. Separating the two atoms would increase the total energy of the system, that is, would require work.

If the two $s$-orbitals were filled, in other words, if each original orbital had two electrons, then the energy splitting due to the wave function overlap would not lead to bonding. Two of the electrons would fall into the lower, bonding state, while the remaining two would have to go into the higher, antibonding state, because of the Pauli exclusion principle. Since, according to (1.1.12), the average energy of the two states remains the same, there is no decrease of the total energy of the atoms. This is why, for example, helium atoms do not form homo atomic molecules.

Figure 1.35 shows the case of two atoms with partially filled, overlapping $s$ – and p-orbitals. Without knowing the exact location of the atoms, we cannot say how the orbitals will split when the atomic states overlap, but we can say that, in general, there will be an equal number of states shifting upward and downward by the same amounts. This follows from the general mathematical theorem that the sum of the eigenvalues of a matrix is equal to the trace of the matrix, no matter how large the off-diagonal elements are. In the LCAO approximation, we construct a square matrix as in (1.1.11), in which the diagonal elements are the unperturbed atomic state energies, and the off-diagonal elements are the coupling integrals. If these are nonzero, the energy eigenvalues will shift, but the sum of all the shifted energies will remain the same. This means that if some states are shifted to lower energy, other states must shift upward by the same amount.

If the total number of electrons in the atomic states is less than half of the total number of states, then all of the electrons can lower their energy when the atoms get near to each other. This leads to stable bonding. As illustrated in Figure 1.35, if the total number of electrons in the two types of orbitals is equal to eight, then the lower, bonding orbitals will be completely full and the upper, antibonding orbitals will be completely empty. This is known as a “full shell” in chemistry terminology.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Angle-Resolved Photoemission Spectroscopy

确定材料能带结构最有力的工具之一是光电过程，通过该过程，入射光子将电子踢出固体。

在真空中，电子将带着离开材料时的动量和能量进行弹道运动。以这种方式从材料中射出的电子电流，然后可以分析它们的运动方向和动能。这种测量被称为角分辨光发射光谱(ARPES)。

通常，光子的动量与电子的动量相比可以忽略不计。因此，光子的吸收可以看作是一个“垂直”过程，在这个过程中，电子移动到更高的能量，同时保持在几乎相同的$k$矢量上。高能电子可以有足够的能量来克服材料的功函数并离开晶体。

在考虑电子离开固体的过程时，立即产生了应用什么守恒规则的问题。我们已经知道$\hbar k$不是电子的真正动量;这是由$(1.6 .10)$给出的，
$$
\left\langle\psi_{\vec{k}}|\vec{p}| \psi_{\vec{k}}\right\rangle=\hbar \vec{k}-i \hbar \int d^3 r u_{n \vec{k}}^* \nabla u_{n \vec{k}} .
$$
当电子穿过固体边界时，我们是保持动量，还是保持$\hbar k$ ?答案是，我们保持$\hbar k$平行于表面的方向，而不是总电子动量。这可以理解为电子波动性质的结果，类似于斯涅尔定律，这将在第三章详细讨论。我们写$\vec{k}=\vec{k}{|}+k{\perp} \hat{z}$，其中$\vec{k}{|}$是平行于表面的波矢量分量$k{\perp}$是垂直于表面的分量。沿表面方向$\vec{x}$的波阵面间距由条件$\vec{k} \cdot \vec{x}=k_{|} x=2 \pi n$给出，其中$n$是一个整数。因此，相位$2 \pi$点之间的距离为$\Delta x=2 \pi / k_{|}$。这个间隔对于固体内部和外部的波来说必须是相同的，这个条件通常被称为相位匹配，虽然$\vec{k}_{|}$是守恒的，但电子的总动量通常是不守恒的。因此，晶体必须有轻微的反冲，以弥补差异

物理代写|固体物理代写Solid-state physics代考|Molecular Bonds

一般来说，电子带和分子键密切相关。哈里森(1980)对化学键和化学带的关系作了极好的详细讨论。

如第1.1.2节所述，当两个电子轨道重叠时，会产生两个新的状态，它们可以近似为原始轨道的对称和反对称线性组合。如第1.1.2节所述，这两种状态分别称为成键状态和反键状态。

图1.34显示了在LCAO近似中，两个$s$轨道重叠的原子是如何成键的。如果每个原子有一个电子，那么来自两个原子的电子可以落入最低的状态，从而降低了这对原子的总能量。这就是我们说它们成键的原因。分离这两个原子会增加系统的总能量，也就是说，需要做功。

如果两个s轨道被填满，换句话说，如果每个原始轨道都有两个电子，那么由波函数重叠引起的能量分裂就不会导致成键。根据泡利不相容原理，两个电子会进入较低的成键态，而剩下的两个电子会进入较高的反键态。由于根据(1.1.12)，两种状态的平均能量保持不变，所以原子的总能量没有减少。这就是为什么，例如，氦原子不形成同属原子分子。

图1.35显示了两个原子部分填充、重叠的s轨道和p轨道的情况。在不知道原子的确切位置的情况下，我们不能说当原子状态重叠时，轨道是如何分裂的，但我们可以说，一般来说，会有相同数量的状态向上和向下移动，移动的量是一样的。这是由一般数学定理得出的，即矩阵的特征值和等于矩阵的迹，无论非对角线元素有多大。在LCAO近似中，我们构造了一个如(1.1.11)所示的方阵，其中对角元素是无摄动原子态能量，非对角元素是耦合积分。如果这些是非零的，能量特征值将会移动，但是所有移动的能量的总和将保持不变。这意味着，如果一些状态转移到较低的能量，其他状态必须向上移动相同的量。

如果处于原子状态的电子总数少于状态总数的一半，那么当原子彼此靠近时，所有的电子都可以降低它们的能量。这导致了稳定的键合。如图1.35所示，如果两种轨道的电子总数等于8个，则较低的成键轨道将完全满，而较高的反键轨道将完全空。这在化学术语中被称为“全壳”。

物理代写|固体物理代写Solid-state physics代考| 请认准statistics-lab™

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金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

Posted on 2023年6月1日2023年6月1日 by statistics-lab_01, statistics-lab_01

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

So far we have looked at the wave function of one electron in fixed external potential. In solids, however, the potential energy felt by an electron arises from the interaction of each electron with all the nuclei and all the other electrons. The potential energy felt by an electron is not only given by the positively charged nuclei, but also by the average negative charge of all the other electrons.

If a crystal has a certain periodicity, then the potential created from all these particles will have that periodicity, so all of the above theorems for periodic potentials still apply. Determining the exact nature of the electron bands, however, is a difficult task. We cannot simply solve for the eigenstate of a single electron in a fixed potential; we must solve for the eigenstates of the whole set of electrons, taking into account exchange between the identical electrons, which must be treated according to Fermi statistics. Methods of treating the Fermi statistics of electrons will be discussed in Chapter 8 .

In general, the calculation of band structure is not an exact science. Typically, determining the band structure of a given solid involves interaction between experiment and theory – the crystal symmetry is determined by x-ray scattering, a band structure is calculated, this is corrected by other experiments such as optical absorption and reflectivity, etc. Many band structure calculations use experimental inputs from chemistry such as the electronegativity of ions, bond lengths, and so on. Calculating band structures from first principles, using nothing but the charge and masses of the nuclei, is still an area of frontier research, involving high-level math and supercomputers.

In this book, we will typically treat the electron bands as known functions for a given solid. Band diagrams have been published for many solids and tabulated in books, for example, Madelung (1996). At the same time, there are several useful approximation methods which allow us to write simple mathematical formulas for the bands without needing to go through all of the calculations to generate a band structure. The value of these approximations is not so much to predict the actual band structures quantitatively; rather, these models help to give us physical intuition about the nature of electron bands.

物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

In a three-dimensional crystal, the full calculation of all the band energies involves finding the energy of each band at every point in the three-dimensional Brillouin zone. This is a large amount of information, which we need to present in a simple fashion for it to be useful.

As discussed in Section 1.6, there are certain critical points in the Brillouin zone which correspond to the points on the surfaces of the zone that are half way between the origin and another reciprocal lattice vector. Figure 1.25 gives the standard labeling of these critical points for common lattice structures. Typically, band structure calculations give the band energies along lines from the center of the Brillouin zone to one of these points, or from one of these points to another one. Figure 1.26(a) shows a typical band structure plot for silicon, a cubic crystal. The critical points are labeled according to the drawing shown in Figure $1.25(\mathrm{~b})$. Note that the diagram is not symmetric about the $\Gamma$ point because two different paths away from this point are plotted. Note also that the $\mathrm{U} / \mathrm{K}$ point is not the midpoint between two reciprocal lattice points, and therefore the slope of the bands is not zero in every direction there; in particular, the slope is not zero along lines that are not normal to the zone boundary. The $\mathrm{L}$ and $\mathrm{X}$ points are critical points, and therefore if the bands are plotted with enough detail, one will see that they have zero slope at those points.

Figure 1.26(b) shows the density of states for the same crystal. As seen in this figure, van Hove singularities (discontinuities in the slope) correspond to critical points in the band structure. When bands overlap in energy, the density of states is just the sum of the density of states of the two bands. Notice that there is a gap in the density of states which corresponds to the energy gap in one band at the $\Gamma$ point (zone center) and the minimum of the next higher band at the $X$ point (the zone boundary in the [100] direction).

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

到目前为止，我们已经研究了一个电子在固定外电位下的波函数。然而，在固体中，电子感受到的势能来自于每个电子与所有原子核和所有其他电子的相互作用。电子感受到的势能不仅是由带正电的原子核给出的，而且是由所有其他电子的平均负电荷给出的。

如果晶体具有一定的周期性，那么所有这些粒子形成的势也具有周期性，所以上面所有关于周期势的定理仍然适用。然而，确定电子带的确切性质是一项艰巨的任务。我们不能简单地解出固定势下单个电子的本征态;我们必须解出整个电子组的本征态，考虑到相同电子之间的交换，这必须根据费米统计来处理。处理电子费米统计量的方法将在第8章讨论。

一般来说，谱带结构的计算并不是一门精确的科学。通常，确定给定固体的能带结构涉及实验和理论之间的相互作用-通过x射线散射确定晶体对称性，计算能带结构，并通过其他实验如光吸收和反射率等进行校正。许多能带结构计算使用化学实验输入，如离子的电负性、键长等。从第一性原理计算能带结构，只使用原子核的电荷和质量，仍然是一个前沿研究领域，涉及高级数学和超级计算机。

在本书中，我们通常将电子带视为给定固体的已知函数。已经出版了许多实体的带图，并将其制成表格，例如《Madelung》(1996年)。同时，有几种有用的近似方法，使我们能够写出简单的带的数学公式，而不需要通过所有的计算来生成带结构。这些近似值并不能定量地预测实际的能带结构;相反，这些模型帮助我们对电子带的性质有了物理上的直观认识。

物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

在三维晶体中，所有能带能量的完整计算包括在三维布里渊带的每个点上找到每个能带的能量。这是大量的信息，我们需要以一种简单的方式来呈现它，以使其有用。

如1.6节所述，在布里渊区存在某些临界点，这些临界点对应于该区域表面上位于原点和另一个倒晶格向量中间的点。图1.25给出了常见晶格结构的这些临界点的标准标记。典型地，能带结构计算给出从布里渊带中心到其中一个点的能带能量，或者从其中一个点到另一个点的能带能量。图1.26(a)显示了立方体晶体硅的典型能带结构图。根据图$1.25(\ mathm {~b})$所示的绘图标记临界点。注意，该图对于$\Gamma$点不是对称的，因为从该点出发的两条不同路径被绘制出来。还要注意$\mathrm{U} / \mathrm{K}$点不是两个互反晶格点之间的中点，因此带的斜率在每个方向上都不是零;特别是，在不垂直于区域边界的直线上，斜率不为零。$\mathrm{L}$和$\mathrm{X}$点是临界点，因此，如果波段被绘制得足够详细，人们会看到它们在这些点上的斜率为零。

图1.26(b)显示了同一晶体的态密度。如图所示，van Hove奇点(斜率中的不连续点)对应于带结构中的临界点。当能带在能量上重叠时，态密度就是两个能带的态密度之和。请注意，态密度中存在一个缺口，对应于$\Gamma$点(区域中心)的一个能带的能量缺口，以及$X$点([100]方向的区域边界)的下一个更高能带的最小值。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

Posted on 2023年6月1日2023年6月1日 by statistics-lab_01, statistics-lab_01

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Electron Band Calculationsin Three Dimensions

物理代写|固体物理代写Solid-state physics代考|How to Read a Band Diagram

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|X-rayScattering

Posted on 2023年6月1日2023年6月1日 by statistics-lab_01, statistics-lab_01

物理代写|固体物理代写Solid-state physics代考|X-rayScattering

The reciprocal lattice has a natural connection to $\mathrm{x}$-ray scattering. Suppose a plane wave with wave vector $\vec{k}0$ impinges on a crystal, as shown in Figure 1.17. We write this plane wave as $$ A{\text {in }}=e^{i\left(\vec{k}_0 \cdot \vec{r}-\omega t\right)}
$$
Atoms in the crystal will lead to scattering of the incoming wave. In the Fraunhofer limit, a scattered wave far away can also be approximated by a plane wave with wave vector $\vec{k}$. We define the scattering vectoras the difference between the incoming and outgoing (scattered) wave vectors:
$$
\vec{s}=\vec{k}-\vec{k}_0 .
$$

If $\vec{a}$ is the vector from one atom to another, then the phase difference between the scattered waves from these two atoms will be
$$
\delta=\vec{k} \cdot \vec{a}-\overrightarrow{k_0} \cdot \vec{a}=\left(\vec{k}-\overrightarrow{k_0}\right) \cdot \vec{a}=\vec{s} \cdot \vec{a} .
$$
The amplitude of the scattered wave from these two atoms will be proportional to
$$
A_{\mathrm{sum}}=\left(e^{i(0)}+e^{i \vec{s} \cdot \vec{a}}\right) e^{-i \omega t}
$$
and therefore the intensity will be proportional to
$$
\begin{aligned}
I=A_{\mathrm{sum}}^* A_{\mathrm{sum}} & =\left(1+e^{-i \vec{s} \cdot \vec{a}}\right)\left(1+e^{i \vec{s} \cdot \vec{a}}\right) \
& =2(1+\cos \vec{s} \cdot \vec{a}) .
\end{aligned}
$$

物理代写|固体物理代写Solid-state physics代考|General Properties of Bloch Functions

Even without knowing anything about the periodic potential in a particular crystal, Bloch’s theorem allows us to make several general statements about the eigenstates of the system.

Bloch theorem as a Fourier serißse can use Fourier transform theory to express the cell function $u_{n \vec{k}}(\vec{r})$ in terms of the reciprocal lattice vectors. Since this function has the same periodicity as the lattice, we can use the Fourier transform formula
$$
\begin{aligned}
F_u(\vec{q}) & =\int_{-\infty}^{\infty} d^3 r u_{n \vec{k}}(\vec{r}) e^{i \vec{q} \cdot \vec{r}} \
& =\sum_{\vec{R}} e^{i \vec{q} \cdot \vec{R}}\left(\int_{\text {cell }} d^3 r_b u_{n \vec{k}}\left(\vec{r}_b\right) e^{i \vec{q} \cdot \vec{r}_b}\right),
\end{aligned}
$$
where in the second line we have written $\vec{r}=\vec{R}+\vec{r}_b$, and have broken the integral over all space into a sum over all Bravais lattice positions $\vec{R}$ and an integral over the relative coordinate $\vec{r}_b$ within each primitive cell. We have introduced a new reciprocal-space variable $\vec{q}$ because we are leaving $\vec{k}$ constant.

As discussed in Section 1.4, the Fourier transform (1.6.1) has nonzero values only when $\vec{q}$ is equal to a reciprocal lattice vector $\vec{Q}$. The cell function $u_{n \vec{k}}(\vec{r})$, which is the inverse Fourier transform of $F_u$, can therefore be written as a sum over the full set of reciprocal lattice vectors,
$$
u_{n \vec{k}}(\vec{r})=\sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{-i \vec{Q} \cdot \vec{r}},
$$
where $C_{n \vec{k}}(\vec{Q})$ is a weight factor. ${ }^2$ The dependence of $C_{n \vec{k}}(\vec{Q})$ on $\vec{k}$ gives an overall multiplier for the whole set of weight factors.
The full Bloch functions are then given by
$$
\begin{aligned}
\psi_{n \vec{k}}(\vec{r}) & =\frac{1}{\sqrt{V}} u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}} \
& =\frac{1}{\sqrt{V}} \sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{i(\vec{k}-\vec{Q}) \cdot \vec{r}}
\end{aligned}
$$

固体物理代写

物理代写|固体物理代写Solid-state physics代考|X-rayScattering

倒易晶格与$\mathrm{x}$射线散射有自然的联系。假设波矢量$\vec{k}0$的平面波撞击晶体，如图1.17所示。我们把这个平面波写成$$ A{\text {in }}=e^{i\left(\vec{k}_0 \cdot \vec{r}-\omega t\right)}
$$
晶体中的原子将导致入射波的散射。在弗劳恩霍夫极限下，远处的散射波也可以近似为波矢量$\vec{k}$的平面波。我们将散射矢量定义为入射和出射(散射)波矢量之差:
$$
\vec{s}=\vec{k}-\vec{k}_0 .
$$

如果$\vec{a}$是从一个原子到另一个原子的矢量，那么这两个原子散射波之间的相位差将是
$$
\delta=\vec{k} \cdot \vec{a}-\overrightarrow{k_0} \cdot \vec{a}=\left(\vec{k}-\overrightarrow{k_0}\right) \cdot \vec{a}=\vec{s} \cdot \vec{a} .
$$
这两个原子发出的散射波的振幅将成正比
$$
A_{\mathrm{sum}}=\left(e^{i(0)}+e^{i \vec{s} \cdot \vec{a}}\right) e^{-i \omega t}
$$
因此强度与
$$
\begin{aligned}
I=A_{\mathrm{sum}}^* A_{\mathrm{sum}} & =\left(1+e^{-i \vec{s} \cdot \vec{a}}\right)\left(1+e^{i \vec{s} \cdot \vec{a}}\right) \
& =2(1+\cos \vec{s} \cdot \vec{a}) .
\end{aligned}
$$

物理代写|固体物理代写Solid-state physics代考|General Properties of Bloch Functions

即使不知道特定晶体的周期势，布洛赫定理也能让我们对系统的特征态做出几个一般的表述。

作为傅里叶级数的布洛赫定理ßse可以用傅里叶变换理论来表示细胞函数$u_{n \vec{k}}(\vec{r})$用互反晶格向量表示。因为这个函数和晶格有相同的周期性，我们可以用傅里叶变换公式
$$
\begin{aligned}
F_u(\vec{q}) & =\int_{-\infty}^{\infty} d^3 r u_{n \vec{k}}(\vec{r}) e^{i \vec{q} \cdot \vec{r}} \
& =\sum_{\vec{R}} e^{i \vec{q} \cdot \vec{R}}\left(\int_{\text {cell }} d^3 r_b u_{n \vec{k}}\left(\vec{r}_b\right) e^{i \vec{q} \cdot \vec{r}_b}\right),
\end{aligned}
$$
在第二行中，我们写了$\vec{r}=\vec{R}+\vec{r}_b$，并将整个空间的积分分解为对所有Bravais晶格位置的和$\vec{R}$和对每个原始单元内的相对坐标$\vec{r}_b$的积分。我们引入了一个新的往复式空间变量$\vec{q}$因为我们让$\vec{k}$保持不变。

正如第1.4节所讨论的，傅里叶变换(1.6.1)只有在$\vec{q}$等于倒易晶格向量$\vec{Q}$时才具有非零值。细胞函数$u_{n \vec{k}}(\vec{r})$，也就是$F_u$的傅里叶反变换，因此可以写成对整个互反晶格向量集合的和，
$$
u_{n \vec{k}}(\vec{r})=\sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{-i \vec{Q} \cdot \vec{r}},
$$
$C_{n \vec{k}}(\vec{Q})$是一个权重因子。${ }^2$$C_{n \vec{k}}(\vec{Q})$对$\vec{k}$的依赖给出了一组权重因子的总体乘数。
完整的布洛赫函数由
$$
\begin{aligned}
\psi_{n \vec{k}}(\vec{r}) & =\frac{1}{\sqrt{V}} u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}} \
& =\frac{1}{\sqrt{V}} \sum_{\vec{Q}} C_{n \vec{k}}(\vec{Q}) e^{i(\vec{k}-\vec{Q}) \cdot \vec{r}}
\end{aligned}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|KYA322

Posted on 2023年4月4日2023年4月4日 by statistics-lab

如果你也在怎样代写固体物理Solid-state physics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

固态物理学是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。

我们提供的固体物理Solid-state physics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|固体物理代写Solid-state physics代考|Reciprocal Lattice

The vectors $a^, b^$ and $c^$ can be used as a basis for a new lattice whose vectors are given by $$ \boldsymbol{G}=u a^+v \boldsymbol{b}^+w \boldsymbol{c}^
$$
where $u, v$ and $w$ are any set of integers. The lattice defined by $\boldsymbol{G}$ is known as reciprocal lattice, and $\boldsymbol{a}^, \boldsymbol{b}^$ and $\boldsymbol{c}^$ are called reciprocal basis vectors. The relations (5.7)-(5.9) are still valid if we replace lattice vectors by reciprocal lattice vectors and inversely reciprocal lattice vectors by lattice vectors. If $V^$ is the volume of the reciprocal unit cell, then $V * V$ $=1$ is also valid.

The vector $\boldsymbol{a}^$ is normal to the plane defined by the vectors $\boldsymbol{b}$ and $\boldsymbol{c} ; \boldsymbol{b}^$ is normal to the plane defined by the vectors $c$ and $a$, and $c^$ is normal to the plane defined by $a$ and $\boldsymbol{b}$. The relation between reciprocal lattice vectors and primitive lattice vectors may be obtained as follows. Let us consider $$ \begin{gathered} a^ \cdot \boldsymbol{a}=\frac{2 \pi(b \times c)}{V} \cdot \boldsymbol{a}=2 \pi \frac{a \cdot(b \times c)}{\boldsymbol{a} \cdot(\boldsymbol{b} \times \boldsymbol{c})}=2 \pi \
a^* \cdot b=\frac{2 \pi(b \times c)}{V} \cdot b=2 \pi \frac{\boldsymbol{b} \cdot(\boldsymbol{b} \times \boldsymbol{c})}{\boldsymbol{a} \cdot(\boldsymbol{b} \times \boldsymbol{c})}=0
\end{gathered}
$$
Similarly
$$
\begin{gathered}
b^* \cdot b=c^* \cdot c=2 \pi \
a^* \cdot c=b^* \cdot a=b^* \cdot c=c^* \cdot a=c^* \cdot b=0
\end{gathered}
$$
The reciprocal lattice possesses the same rotational symmetry as the direct lattice. The reciprocal lattice always falls in the same crystal system as its direct lattice. The reciprocal lattice for hexagonal, monoclinic, …, triclinic lattices is also hexagonal, monoclinic,…. triclinic, respectively.

物理代写|固体物理代写Solid-state physics代考|BRAGG’s Law

For diffraction, this path difference must be equal to an integral multiple of wavelength, that is,
$$
S Q+Q T=n \lambda
$$
From Fig. 5.2
$$
S Q=Q T=d \sin \theta
$$
Substituting Eqs. (5.18) in (5.17)
$$
2 d \sin \theta=n \lambda
$$
This relation is known as Bragg’s law, $n$ is order of reflection which may be integer ( $n$ $=1,2,3, \ldots)$ consistent with $\sin \theta$ not exceeding unity. From Eq. (5.19), it is seen that diffraction intensities can be built only at certain values of $\theta$, corresponding to a specific value of $\lambda$ and $d$. From Eq. (5.19), we have
$$
\theta=\sin ^{-1} \frac{n \lambda}{2 d}
$$
From this, it is seen that rays diffracted by a crystal are given off in different directions corresponding to different values of the interplanar spacing $d$. From the experimentally observed diffraction angles, it is possible to determine the $d$ of a crystal. From a list of such spacing, it is then possible to determine the lattice of the crystal.
The highest possible order can be determined by the condition $$
\sin \theta_{\max .}=1 \text { or } \frac{n \lambda}{2 d} \leq 1
$$
This indicates that $\lambda$ must not be greater than twice the interplanar spacing otherwise no diffraction will occur. Since each plane reflects $10^{-3}$ to $10^{-5}$ of the incident radiation so that $10^3$ to $10^5$ planes may contribute to the formation of Bragg reflected beam in a perfect crystal. Bragg’s law is a consequence of the periodicity of the lattice.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Reciprocal Lattice

载体^、人和齐可以用作新格的基础，其向量由下式给出
$$
\left.\backslash \text { boldsymbol }{G}=u a^{\wedge}+\mathrm{v} \backslash \text { boldsymbol }{b}^{\wedge}+\mathrm{W} \backslash \text { boldsymbol{c }\right}^{\wedge}
$$
在哪里 $u, v$ 和 $w$ 是任意整数集。由定义的晶格 $G$ 被称为倒易格，并且 \boldsymbol{a}^, \boldsymbol{b $}^{\wedge}$ 和 \boldsymbol{c}^ 称为倒数基向量。如果我们用倒数点阵向量代替点阵向量，用点阵向量代替倒数点阵向量，则关系式 (5.7) – (5.9) 仍然有效。如果 $\mathrm{V}^{\wedge}$ 是倒数晶胞的体积，则 $V * V=1$ 也是有效的。
载体 $\backslash$ boldsymbol ${a}^{\wedge}$ 垂直于向量定义的平面 $b$ 和 $\backslash b o l d s y m b o l{c} ; \backslash b o l d s y m b o \mid{b}^{\wedge}$ 垂直于向量定义的平面 $c$ 和 $a$ ，和 $\mathrm{c}^{\wedge}$ 垂直于由定义的平面 $a$ 和 $b$. 倒数点阵向量和本原点阵向量之间的关系可以如下获得。让我们考虑一下
$$
a \cdot \frac{2 \pi(b \times c)}{V} \cdot \boldsymbol{a}=2 \pi \frac{a \cdot(b \times c)}{\boldsymbol{a} \cdot(\boldsymbol{b} \times \boldsymbol{c})}=2 \pi a^* \cdot b=\frac{2 \pi(b \times c)}{V} \cdot b=2 \pi \frac{\boldsymbol{b} \cdot(\boldsymbol{b} \times \boldsymbol{c})}{\boldsymbol{a} \cdot(\boldsymbol{b} \times \boldsymbol{c})}=0
$$
相似地
$$
b^* \cdot b=c^* \cdot c=2 \pi a^* \cdot c=b^* \cdot a=b^* \cdot c=c^* \cdot a=c^* \cdot b=0
$$
倒易晶格与正晶格具有相同的旋转对称性。倒晶格总是与其正晶格属于同一晶系。六角、单斜、……、三斜晶格的倒易点阵也是六角、单斜、……。分别为三斜晶系。

物理代写|固体物理代写Solid-state physics代考|BRAGG’s Law

对于衍射，这个路径差必须等于波长的整数倍，即
$$
S Q+Q T=n \lambda
$$
从图 5.2
$$
S Q=Q T=d \sin \theta
$$
代入方程式。(5.18) 在 (5.17)
$$
2 d \sin \theta=n \lambda
$$
这种关系被称为布拉格定律， $n$ 是反射阶数，可以是整数 $(n=1,2,3, \ldots)$ 是一致的 $\sin \theta$ 不超过统一。从等式。(5.19)，可以看出衍射强度只能建立在某些值 $\theta$ ，对应于特定值 $\lambda$ 和 $d$. 从等式。(5.19)，我们有
$$
\theta=\sin ^{-1} \frac{n \lambda}{2 d}
$$
由此可见，晶面间距的不同值对应于晶体的衍射射线在不同的方向发出 $d$. 从实验观察到的衍射角，可以确定 $d$ 的一个水晶。从这样的间距列表中，可以确定晶体的晶格。最高可能的顺序可以由条件确定
$$
\sin \theta_{\max .}=1 \text { or } \frac{n \lambda}{2 d} \leq 1
$$
这表明 $\lambda$ 不得大于晶面间距的两倍，否则不会发生衍射。由于每个平面反映 $10^{-3}$ 到 $10^{-5}$ 入射辐射使得 $10^3$ 到 $10^5$ 平面可能有助于在完美晶体中形成布拉格反射光束。布拉格定律是晶格周期性的结果。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|PHYSICS7544

Posted on 2023年4月4日2023年4月4日 by statistics-lab

如果你也在怎样代写固体物理Solid-state physics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

固态物理学是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。

我们提供的固体物理Solid-state physics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|固体物理代写Solid-state physics代考|Pauli Exclusion Principle

In 1925, W. Pauli discovered the fundamental principle that governs the electron configurations of multielectron atoms. His exclusion principle states that ‘in a multielectron atom there can never be more than one electron in the same quantum state. Each electron must have a different set of quantum numbers $n, l, m_l$ and $m_{\mathrm{s}}$ ‘. He established from the analysis of experimental data that the exclusion principle represents a property of electrons and not, particularly, of atoms. The exclusion principle operates in any system containing electrons.

It is seen that the complete wave function $\psi$ of the hydrogen atom can be expressed as the product of three separate wave functions, each describing that part of $\psi$, which is a function of one of the three, coordinates $r, \theta$ and $\varphi$. A multielectron system consisting of $\mathrm{n}$ non-interacting electrons can be expressed as the product of wave functions $\psi(1)$, $\psi(2), \ldots \psi(n)$ of the individual electrons, that is $$
\Psi(1,2, \ldots, n)=\psi(1) \psi(2) \ldots \psi(n)
$$
Each of the eigenfunction describing the electron require quantum numbers $n, l, m l$ to specify the mathematical form of its dependence on the three coordinates. In addition each require one more quantum number $m_s$ to specify the orientation of the spin of the electron. To designate a particular set of four quantum numbers, the symbols such as $a$, $b, c$, . etc. are used. Let us consider a wave function used to describe a system of two electrons. Suppose electron number 1 is in quantum state $a$ and electron number 2 is in state $b$. The wave function is
$$
\Psi_I=\psi_a(1) \psi_b(2)
$$
Because the electrons are identical, there is no physical way to distinguish the electron wave function given by Eq. (4.175) from the wave function
$$
\Psi_{\mathrm{II}}=\psi_a(2) \psi_b(1)
$$
in which electron number 2 now has quantum state $a$, etc. Similarly no conceivable physical experiment could distinguish the six three electron wave functions:
$$
\begin{aligned}
& \psi_a(1) \psi_b(2) \psi_c(3) ; \psi_a(3) \psi_b(1) \psi_c(2) ; \psi_a(2) \psi_b(3) \psi_c(1) \
& \psi_a(2) \psi_b(1) \psi_c(3) ; \psi_a(3) \psi_b(2) \psi_c(1) ; \psi_a(1) \psi_b(3) \psi_c(2)
\end{aligned}
$$

物理代写|固体物理代写Solid-state physics代考|X-Rays

X-rays are electromagnetic radiations of wavelength between $\sim 10 \mathrm{pm}$ and $\sim 10 \mathrm{~nm}$. X-rays are characterized by index of refraction very close to unity for all materials. X-rays are produced when a beam of highly accelerated particles such as electrons are allowed to strike a metal target. In the process, electrons suffer energy loss and this loss is emitted in the form of electromagnetic radiation. The X-rays are produced both by deceleration of electrons in the metal target and by the excitation of the core electrons in the atom of the target. The first process gives a broad continuous spectrum and the second gives sharp lines. When a moving electron is stopped suddenly, all its energy appears as photon of frequency $v$ of X-rays. The energy of an electron of charge $e$ in dropping through a potential difference $V$ is $e V$ and
$$
\begin{gathered}
E=h v=\frac{h c}{\lambda}=e V \
\lambda=\frac{h c}{E}=\frac{h c}{e V}
\end{gathered}
$$
An electron will not lose all its energy in this way; it will have a number of glancing collisions with the atoms that it collides and causing them to vibrate. As a result of this, the temperature of the target increases. Equation (5.1), therefore, gives the minimum value $\lambda$ can possibly have and accounts for the short wavelength cut-off. Larger wavelengths are more probable and so the rapid increase in the intensity. The intensity falls off gradually indicating that there is no upper limit. Figure 5.1 shows the X-ray spectrum that results when molybdenum target is bombarded by electron at $35 \mathrm{keV}$. The electron beam on striking the target not only gets decelerated but also a small fraction of electrons of the beam strikes the target and ejects the inner shell’s electrons. The atom is then unstable, and outer shell electrons in the same atom will drop into the hole (vacancy) caused by the ejection of the electron. In doing so, it loses energy and a photon is emitted. If $E$ is the energy lost, we have
$$
\lambda=\frac{h c}{E}
$$
$E$ is a definite quantity associated with the electron energy change in the atom. Therefore, the wavelength concerned is specific. Several wavelengths are possible, and they constitute the characteristic X-ray line spectrum shown as peaks in Fig. 5.1. The energy of the characteristic X-ray produced is very weakly dependent on the chemical structure in which the atom is bound indicating that non-bonding shells of atoms are the characteristic X-ray source. The resulting characteristic spectrum is superimposed on the continuum. An atom remains ionized for a very short time $\left(\sim 10^{-14} \mathrm{~s}\right)$, and thus, the incident electrons that arrive about every $\sim 10^{-17} \mathrm{~s}$ can repeatedly ionize an atom. However, not all outer electrons can fall into holes to provide X-rays.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Pauli Exclusion Principle

1925 年，W. 泡利 (W. Pauli) 发现了支配多电子原子电子构型的基本原理。他的不相容原理指出，“在一个多电子原子中，处于同一量子态的电子永远不会超过一个”。每个电子必须有一组不同的量子数 $n, l, m_l$ 和 $m_{\mathrm{s}}$ ‘. 他通过对实验数据的分析确定，不相容原理代表了电子的特性，而不是原子的特性。排斥原理适用于任何包含电子的系统。
可见完整的波函数 $\psi$ 氢原子的一部分可以表示为三个独立波函数的乘积，每个波函数都描述了氢原子的那一部分 $\psi$, 这是三个坐标之一的函数 $r, \theta$ 和 $\varphi$. 多电子系统由 $\mathrm{n}$ 非相互作用的电子可以表示为波函数的乘积 $\psi(1), \psi(2), \ldots \psi(n)$ 单个电子的，即
$$
\Psi(1,2, \ldots, n)=\psi(1) \psi(2) \ldots \psi(n)
$$
描述电子的每个特征函数都需要量子数 $n, l, m l$ 指定其依赖于三个坐标的数学形式。另外每一个都需要多一个量子数 $m_s$ 指定电子自旋的方向。为了指定一组特定的四个量子数，符号如 $a, b, c_r$. 等被使用。让我们考虑用于描述两个电子系统的波函数。假设 1 号电子处于量子态 $a$ 电子数 2 处于状态 $b$. 波函数是
$$
\Psi_I=\psi_a(1) \psi_b(2)
$$
因为电子是相同的，所以没有物理方法来区分方程式给出的电子波函数。(4.175) 从波函数
$$
\Psi_{\mathrm{II}}=\psi_a(2) \psi_b(1)
$$
其中 2 号电子现在具有量子态 $a$ 等。同样，没有任何可以想象的物理实验可以区分六个三电子波函数:
$$
\psi_a(1) \psi_b(2) \psi_c(3) ; \psi_a(3) \psi_b(1) \psi_c(2) ; \psi_a(2) \psi_b(3) \psi_c(1) \quad \psi_a(2) \psi_b(1) \psi_c(3) ; \psi_a(3) \psi_b(2) \psi_c(1)
$$

物理代写|固体物理代写Solid-state physics代考|X-Rays

X射线是波长介于 $~ 10 \mathrm{pm}$ 和 $10 \mathrm{~nm}$. X射线的特征是所有材料的折射率都非常接近统一。当一束高度加速的粒子 (例如电子) 撞击金属目标时，就会产生 X 射线。在此过程中，电子遭受能量损失，这种损失以电磁辐射的形式发射。X射线是通过金属靶中电子的减速和靶原子中核心电子的激发产生的。第一个过程给出了广泛的连续光谱，第二个过程给出了清晰的线条。当一个运动的电子突然停止时，它的所有能量都表现为频率为光子 $v$ X射线。一个电荷电子的能量 $e$ 通过电位差下降 $V$ 是 $e V$ 和
$$
E=h v=\frac{h c}{\lambda}=e V \lambda=\frac{h c}{E}=\frac{h c}{e V}
$$
电子不会以这种方式失去所有能量；它会与它碰撞并导致它们振动的原子发生多次擦肩而过的碰撞。结果，目标的温度升高。因此，等式 (5.1) 给出了最小值 $\lambda$ 可能具有并解释短波长截止。更大的波长更有可能，因此强度会迅速增加。强度逐渐下降表明没有上限。图 5.1 显示了钼靶在 $35 \mathrm{keV}$. 撞击目标的电子束不仅会减速，而且电子束中的一小部分电子会撞击目标并射出内壳的电子。然后原子不稳定，同一原子中的外壳电子将落入由电子喷射引起的空穴 (空位) 中。这样做时，它会失去能量并发射光子。如果 $E$ 是能量损失，我们有
$$
\lambda=\frac{h c}{E}
$$
$E$ 是与原子中电子能量变化相关的确定量。因此，所涉及的波长是特定的。几种波长是可能的，它们构成了特征 $X$ 射线线谱，如图 5.1 中的峰值所示。产生的特征 $X$ 射线的能量非常微弱地依赖于原子所结合的化学结构，这表明原子的非键合壳层是特征 X射线源。得到的特征光谱叠加在连续谱上。原子保持电离状态的时间很短 $\left(\sim 10^{-14} \mathrm{~s}\right)$ ，因此，大约每个到达的入射电子 $~ 10^{-17} \mathrm{~s}$ 可以反复电离一个原子。然而，并非所有的外层电子都能落入空穴以提供 X 射线。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|PHYS7635 Solid-state physics

Posted on 2023年4月4日2023年4月4日 by statistics-lab

Statistics-lab™可以为您提供cornell.edu PHYS7635 Solid-state physics固体物理的代写代考和辅导服务！

PHYS7635 Solid-state physics课程简介

Prerequisite: good undergraduate solid-state physics course (e.g., PHYS 4454), undergraduate statistical mechanics, and familiarity with graduate-level quantum mechanics.
D. Ralph.
Survey of the physics of solids: crystal structures, X-ray diffraction, phonons, and electrons. Selected topics from semiconductors, magnetism, superconductivity, disordered materials, dielectric properties, and mesoscopic physics. The focus is to enable graduate research at the current frontiers of condensed matter physics.

PREREQUISITES

PHYS7635 Solid-state physics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Name the three most important probes used in diffraction experiments on crystals. What is the one essential condition that they must all satisfy? Describe briefly what each probe is suitable for.

The three most important probes used in diffraction experiments on crystals are:

X-rays
Neutrons
Electrons

The one essential condition that they all must satisfy is that the wavelength of the probe must be similar in size to the interatomic spacing in the crystal. This is known as the Bragg condition and is expressed as nλ = 2d sinθ, where n is an integer, λ is the wavelength of the probe, d is the interatomic spacing, and θ is the angle of incidence.

Brief descriptions of each probe and what they are suitable for:

X-rays: X-rays are the most commonly used probe in crystallography. They have a short wavelength and high penetration power, allowing them to diffract off the atoms in the crystal and produce a diffraction pattern. X-ray diffraction is suitable for determining the crystal structure of a wide range of materials, including small molecules, proteins, and minerals.
Neutrons: Neutrons have a longer wavelength than X-rays and can penetrate deeper into the sample, making them useful for studying the structure of materials that are difficult to study with X-rays, such as hydrogen-containing compounds. Neutron diffraction is also sensitive to the positions of light atoms, such as hydrogen, which are often invisible to X-rays.
Electrons: Electron diffraction is a powerful technique for studying the structure of small crystals and thin films. Electrons have a much shorter wavelength than X-rays and neutrons, allowing them to diffract off smaller structures. Electron diffraction is also highly sensitive to the arrangement of atoms in a crystal, making it useful for studying defects and disorder in crystal structures.

问题 2.

Calculate the geometrical structure factor for a $b c c$ lattice. Name some important planes which will be missing from the X-ray diffraction pattern.

In a $b c c$ lattice, the basis consists of two atoms located at $(0,0,0)$ and $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ of the unit cell. The structure factor for a Bragg reflection from a plane with Miller indices $(h,k,l)$ is given by:

F_{hkl} = \sum_j f_j e^{2 \pi i (hx_j + ky_j + lz_j)}Fhkl=∑jfje2πi(hxj+kyj+lzj)

where $f_j$ is the atomic scattering factor for atom $j$, and $x_j$, $y_j$, and $z_j$ are the fractional coordinates of atom $j$ within the unit cell.

For the $b c c$ lattice, we have $f_1 = f_2 = f$, where $f$ is the atomic scattering factor for the atoms in the basis. Using the coordinates for the two atoms in the basis, we have:

F_{hkl} = f\left[ e^{2 \pi i (0\times h + 0 \times k + 0 \times l)} + e^{2 \pi i (\frac{1}{2} \times h + \frac{1}{2} \times k + \frac{1}{2} \times l)}\right]Fhkl=f[e2πi(0×h+0×k+0×l)+e2πi(21×h+21×k+21×l)]

Simplifying this expression, we get:

F_{hkl} = f\left[ 1 + (-1)^{h+k+l}\right]Fhkl=f[1+(−1)h+k+l]

Therefore, the geometrical structure factor for a $b c c$ lattice is given by:

|F_{hkl}|^2 = f^2 \left[ 2 + 2(-1)^{h+k+l}\right]∣Fhkl∣2=f2[2+2(−1)h+k+l]

This expression shows that the intensity of the diffraction peaks will depend on whether $h+k+l$ is even or odd. If $h+k+l$ is odd, the structure factor will be zero, and no diffraction peak will be observed for that plane.

Some important planes which will be missing from the X-ray diffraction pattern for a $b c c$ lattice are those with odd values of $h+k+l$, such as $(1,1,3)$, $(2,3,1)$, etc. These planes are called “systematically absent” planes, and their absence is a characteristic feature of the $b c c$ lattice.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

此图像的alt属性为空；文件名为%E7%B2%89%E7%AC%94%E5%AD%97%E6%B5%B7%E6%8A%A5-1024x575-10.png — PHYS7635 Solid-state physics

Statistics-lab™可以为您提供cornell.edu PHYS7635 Solid-state physics固体物理的代写代考和辅导服务！请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

物理代写|固体物理代写Solid-state physics代考|PHYSICS7544

Posted on 2023年3月29日2023年3月29日 by statistics-lab

如果你也在怎样代写固体物理Solid-state physics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

固态物理学是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。

我们提供的固体物理Solid-state physics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|固体物理代写Solid-state physics代考|PHYSICS7544

物理代写|固体物理代写Solid-state physics代考|Quantum Numbers for the Hydrogen Atom Wave Function

In the process of solving the Schrödinger wave equation for hydrogen like atoms, the integer $\mathrm{n}, l$ and $\mathrm{m}$ are introduced in a logical way. These integers are called quantum numbers. In Bohr’s theory the quantum number $n$ is introduced arbitrarily in the form of the quantized condition. The wave function $\Psi_{\operatorname{nlm}}(\theta, \varphi)$ is description of states of the system. They are related to $n, l$ and $m$. Thus the quantum number themselves may be said to describe the state of the system. The values of these quantum numbers are
$$
\begin{aligned}
n & =1,2,3, \ldots \
l & =0,1,2, \ldots, n-1 \
m & =0, \pm 1, \pm 2, \ldots, \pm l
\end{aligned}
$$
It is now shown that the quantum numbers $l$ and $m$ are related to the magnitude $L$ of the orbital angular momentum and $m$ is the $z$ component $L_z . Y_{l m}(\theta, \varphi)$ is simultaneously the eigenfunction of $L^2$ and $L_z$ operator. Therefore, $L^2$ and $L_z$ commute, i.e. corresponding observable can be precisely measured simultaneously. The magnitude of the orbital angular momentum is $\hbar \sqrt{l(l+1)}$ while that of $L_z$ is $m \mathrm{~h}$. Since $L^2$ and $L_z$ do not operate on the radial part of the wave function, $\Psi_{n l m}(\theta, \varphi)$ itself is a simultaneous eigenfunction of $L^2$ and $L_z$. The quantum number $l$ gives the magnitude of the angular momentum, $m$ the orientation of the angular momentum and $\mathrm{n}$ gives the quantization of energy. The quantum numbers $l$ and $m$ can be equal only for $l=0$.

物理代写|固体物理代写Solid-state physics代考|Harmonic Oscillator

The harmonic oscillator is a system in which a particle of mass $m$ subject to a linear restoring force $\mathbf{F}$ proportional to the displacement $x$ from the equilibrium position
$$
F=-k x
$$
The proportionality constant $k$ is known as force constant. The minus sign indicates that force is in the direction opposite to the direction of the displacement.
The potential energy is given by
$$
\begin{gathered}
V=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2 \
\omega=\sqrt{\frac{k}{m}}
\end{gathered}
$$
One Dimensional Harmonic Oscillator
The Schrödinger wave equation in one dimension for a particle of mass $m$ is
$$
\frac{\mathrm{d}^2 \psi}{\mathrm{d} x^2}+\frac{2 m}{\hbar^2}[E-V] \psi=0
$$ where $E$ is the energy and $V$ is given by Eq. (4.124). Substituting the value of $V$ in Eq. $(4.126)$
$$
\frac{\mathrm{d}^2 \psi}{\mathrm{d} x^2}+\frac{2 m}{\hbar^2}\left[E-\frac{1}{2} k x^2\right] \psi=0
$$
Let
$$
\begin{gathered}
\lambda=\frac{2 m}{\hbar^2} E \
\alpha^2=\frac{m k}{\hbar^2}=\frac{m^2 \omega^2}{\hbar^2}
\end{gathered}
$$
Equation (4.127) is then written as
$$
\frac{d^2 \psi}{d x^2}+\left(\lambda^2-\alpha^2 x^2\right) \psi=0
$$
with the boundary condition $\psi \rightarrow 0$ as $|x| \rightarrow \infty$. Let us suppose $\alpha x$ to be very large in particular $\alpha x>>1$ and $\alpha x \gg \lambda$.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Quantum Numbers for the Hydrogen Atom Wave Function

在求解类氢原子薛定谔波动方程的过程中，整数 $\mathrm{n}, l$ 和 $\mathrm{m}$ 以合乎逻辑的方式介绍。这些整数称为量子数。在玻尔的理论中，量子数 $n$ 以量化条件的形式任意引入。波函数 $\Psi_{\mathrm{nlm}}(\theta, \varphi)$ 是系统状态的描述。它们与 $n, l$ 和 $m$. 因此，可以说量子数本身描述了系统的状态。这些量子数的值是
$$
n=1,2,3, \ldots l=0,1,2, \ldots, n-1 m=0, \pm 1, \pm 2, \ldots, \pm l
$$
现在证明量子数l和 $m$ 与幅度有关 $L$ 轨道角动量和 $m$ 是个 $z$ 成分 $L_z . Y_{l m}(\theta, \varphi)$ 同时是的特征函数 $L^2$ 和 $L_z$ 操作员。所以， $L^2$ 和 $L_z$ 通勤，即可以同时精确测量相应的可观察值。轨道角动量的大小是 $\hbar \sqrt{l(l+1)}$ 而那个 $L_z$ 是 $m \mathrm{~h}$. 自从 $L^2$ 和 $L_z$ 不对波函数的径向部分进行操作， $\Psi_{n l m}(\theta, \varphi)$ 本身是同时的特征函数 $L^2$ 和 $L_z$. 量子数 $l$ 给出角动量的大小， $m$ 角动量的方向和 $\mathrm{n}$ 给出能量的量子化。量子数 $l$ 和 $m$ 只能等于 $l=0$.

物理代写|固体物理代写Solid-state physics代考|Harmonic Oscillator

谐振子是一个系统，其中一个质量粒子 $m$ 受到线性恢复力 $\mathbf{F}$ 与位移成正比 $x$ 从平衡位置
$$
F=-k x
$$
比例常数 $k$ 被称为力常数。负号表示力的方向与位移的方向相反。势能由下式给出
$$
V=\frac{1}{2} k x^2=\frac{1}{2} m \omega^2 x^2 \omega=\sqrt{\frac{k}{m}}
$$
一维谐振子质量
粒子的一维薛定谔波动方程 $m$ 是
$$
\frac{\mathrm{d}^2 \psi}{\mathrm{d} x^2}+\frac{2 m}{\hbar^2}[E-V] \psi=0
$$
在哪里 $E$ 是能量和 $V$ 由方程式给出。(4.124)。代入价值 $V$ 在等式中 (4.126)
$$
\frac{\mathrm{d}^2 \psi}{\mathrm{d} x^2}+\frac{2 m}{\hbar^2}\left[E-\frac{1}{2} k x^2\right] \psi=0
$$
让
$$
\lambda=\frac{2 m}{\hbar^2} E \alpha^2=\frac{m k}{\hbar^2}=\frac{m^2 \omega^2}{\hbar^2}
$$
方程 (4.127) 可以写成
$$
\frac{d^2 \psi}{d x^2}+\left(\lambda^2-\alpha^2 x^2\right) \psi=0
$$
与边界条件 $\psi \rightarrow 0$ 作为 $|x| \rightarrow \infty$. 让我们假设 $\alpha x$ 特别是非常大 $\alpha x>>1$ 和 $\alpha x \gg \lambda$.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|PHYSICS3544

Posted on 2023年3月29日2023年3月29日 by statistics-lab

如果你也在怎样代写固体物理Solid-state physics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

固态物理学是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。

我们提供的固体物理Solid-state physics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|固体物理代写Solid-state physics代考|PHYSICS3544

物理代写|固体物理代写Solid-state physics代考|Boundary Conditions

The wave function itself has no physical interpretation, however, the square of its absolute magnitude $|\Psi(x, t)|^2$ evaluated at a particular place and at a particular time is proportional to the possibility of finding the particle at that time. The probability density $|\Psi(x, t)|^2$ is positive and real and is taken equal to $\Psi^*(\boldsymbol{r}, t) \Psi(\boldsymbol{r}, t)$. The wave function $\Psi$ can take on negative values but probability density is always be positive. Besides fulfilling the normalization condition a solution of the time independent Schrödinger equation must obey the following boundary conditions.

The wave function must be continuous and single valued.
$\frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y}$ and $\frac{\partial \psi}{\partial z}$ must be continuous and single valued everywhere.
The integral of the square modulus of the wave function over all values $x$ must be finite
$$
\int \psi^* \psi \mathrm{d} \tau=\text { finite }
$$
that is the wave function must be square integrable. This condition means that wave function must be normalizable that is wave function must go to zero as $x(y, z) \rightarrow \pm \infty$ in order that $\int|\Psi|^2 \mathrm{~d} \tau$ over all space is finite constant.

The boundary conditions ensure that the probability of finding the particle in the vicinity of any point is unambiguously defined rather than having two or more possible values. Thus the wave function is single valued and continuous. If $\psi(x)$ and $(\mathrm{d} \psi / \mathrm{d} x)$ are not single valued, finite then the same is true for $\Psi(x, t)$. Since the given formula for calculating the expectation values of position and momentum contains $\Psi(x, t)$ and $\frac{\partial \Psi}{\partial t}$. We observe that in any of these cases we might not obtain finite and definite values when we evaluate measured quantities.

The first derivative of the wave function with respect to position coordinates must be continuous every where except where there is an infinite discontinuity in the potential. We know any function always has an infinite derivative whenever it has a discontinuity. Let us consider the time independent Schrödinger Eq. (4.67) in one dimension
$$
\frac{\mathrm{d}^2 \Psi}{\mathrm{d} x^2}=\frac{2 m}{\hbar^2}(V-E) \psi
$$
for finite $V, E$ and $\psi,\left(\mathrm{d}^2 \psi / \mathrm{d} x^2\right)$ is finite. This in turn requires (d $\psi / \mathrm{d} x$ ) to be continuous. A finite discontinuity in $(\mathrm{d} \psi / \mathrm{d} x)$ implies an infinite discontinuity in $\left(\mathrm{d}^2 \psi / \mathrm{d} x^2\right)$ and from the Schrödinger equation in $V(x)$.

物理代写|固体物理代写Solid-state physics代考|Hydrogen Atom

Consider the hydrogen atom as a system of two interacting particles, the interaction being due to Coulomb attraction of their electrical charges. Let the charge on the nucleus is $Z q$ and the charge on the electron is $-q$. The potential energy of the system in the absence of the external field is
$$
V(r)=-\frac{Z q^2}{\left(4 \pi \varepsilon_0\right) r}
$$
in which $r$ is the distance between the electron and the nucleus.
Let $m_1$ and $m_2$ are the masses of nucleus and the electron, respectively. If we write for the Cartesian coordinates of the nucleus and the electrons $x_1, y_1, z_1$ and $x_2, y_2, z_2$, respectively, the Hamiltonian of the hydrogenic atoms has the form
$$
H=\frac{p_1^2}{2 m_1}+\frac{p_2^2}{2 m_2}+V(r)=E
$$
The Schrödinger wave equation is $$
\frac{1}{m_1}\left(\frac{\partial^2 \Psi}{\partial x_1^2}+\frac{\partial^2 \Psi}{\partial y_1^2}+\frac{\partial^2 \Psi}{\partial z_1^2}\right)+\frac{1}{m_2}\left(\frac{\partial^2 \Psi}{\partial x_2^2}+\frac{\partial^2 \Psi}{\partial y_2^2}+\frac{\partial^2 \Psi}{\partial z_2^2}\right)+\frac{2}{\hbar^2}[E-V] \Psi=0
$$
Here wave function $\Psi$ refers to the complete system with six coordinates. Equation (4.73) can be separated into two, one of which represents the translational motion of a molecule as a whole and the other, the relative motion of the two particles. For this, consider new variables $X, Y, Z$ which are Cartesian coordinates of the centre of mass of the system and $r, \theta$ and $\varphi$ of the polar coordinates of the second particle relative to the first. These coordinates are related to the Cartesian coordinates of the two particles by the equations
$$
\begin{gathered}
X=\frac{m_1 x_1+m_2 x_2}{m_1+m_2} \
Y=\frac{m_1 y_1+m_2 y_2}{m_1+m_2} \
Z=\frac{m_1 z+m_2 z_2}{m_1+m_2} \
x=x_2-x_1=r \sin \theta \cos \varphi \
y=y_2-y_1=r \sin \theta \sin \varphi \
z=z_2-z_1=r \cos \theta
\end{gathered}
$$

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Boundary Conditions

波函数本身没有物理解释，但是，它的绝对大小的平方 $|\Psi(x, t)|^2$ 在特定地点和特定时间评估的值与在该时间找到粒子的可能性成正比。概率密度 $|\Psi(x, t)|^2$ 是正实数，取等于 $\Psi^*(\boldsymbol{r}, t) \Psi(\boldsymbol{r}, t)$. 波函数 $\Psi$ 可以取负值，但概率密度始终为正。除了满足归一化条件外，与时间无关的薛定谔方程的解还必须遵守以下边界条件。

波函数必须是连续的和单值的。
$\frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y}$ 和 $\frac{\partial \psi}{\partial z}$ 必须是连续的并且处处都是单值的。
波函数的平方模对所有值的积分 $x$ 必须是有限的
$$
\int \psi^* \psi \mathrm{d} \tau=\text { finite }
$$
即波函数必须是平方可积的。这个条件意味着波函数必须是可归一化的，即波函数必须归零为 $x(y, z) \rightarrow \pm \infty$ 为了使 $\int|\Psi|^2 \mathrm{~d} \tau$ 在所有空间上都是有限常数。
边界条件确保在任何点附近找到粒子的概率被明确定义，而不是有两个或更多可能的值。因此波函数是单值连续的。如果 $\psi(x)$ 和 $(\mathrm{d} \psi / \mathrm{d} x)$ 不是单值的，有限的那么同样适用于 $\Psi(x, t)$. 由于用于计算位置和动量期望值的给定公式包含 $\Psi(x, t)$ 和 $\frac{\partial \Psi}{\partial t}$. 我们观察到，在任何这些情况下，当我们评估测量量时，我们可能无法获得有限和确定的值。
波函数相对于位置坐标的一阶导数在任何地方都必须是连续的，除了势能中存在无限不连续的地方。我们知道，任何函数只要有不连续点，就总是有无穷导数。让我们考虑时间无关的薛定谔方程。(4.67) 一维
$$
\frac{\mathrm{d}^2 \Psi}{\mathrm{d} x^2}=\frac{2 m}{\hbar^2}(V-E) \psi
$$
对于有限 $V, E$ 和 $\psi,\left(\mathrm{d}^2 \psi / \mathrm{d} x^2\right)$ 是有限的。这又需要 $(\mathrm{d} \psi / \mathrm{d} x)$ 是连续的。中的有限不连续性 $(\mathrm{d} \psi / \mathrm{d} x)$ 意味着无限不连续 $\left(\mathrm{d}^2 \psi / \mathrm{d} x^2\right)$ 从薛定谔方程 $V(x)$.

物理代写|固体物理代写Solid-state physics代考|Hydrogen Atom

将氢原子视为两个相互作用粒子的系统，相互作用是由于它们的电荷的库仑吸引。让原子核上的电荷是 $Z q$ 电子上的电荷是 $-q$. 在没有外场的情况下系统的势能是
$$
V(r)=-\frac{Z q^2}{\left(4 \pi \varepsilon_0\right) r}
$$
其中 $r$ 是电子与原子核之间的距离。
让 $m_1$ 和 $m_2$ 分别是原子核和电子的质量。如果我们写下原子核和电子的笛卡尔坐标 $x_1, y_1, z_1$ 和 $x_2, y_2, z_2$ ，氢原子的哈密顿量分别具有以下形式
$$
H=\frac{p_1^2}{2 m_1}+\frac{p_2^2}{2 m_2}+V(r)=E
$$
辠定谔波动方程是
$$
\frac{1}{m_1}\left(\frac{\partial^2 \Psi}{\partial x_1^2}+\frac{\partial^2 \Psi}{\partial y_1^2}+\frac{\partial^2 \Psi}{\partial z_1^2}\right)+\frac{1}{m_2}\left(\frac{\partial^2 \Psi}{\partial x_2^2}+\frac{\partial^2 \Psi}{\partial y_2^2}+\frac{\partial^2 \Psi}{\partial z_2^2}\right)+\frac{2}{\hbar^2}[E-V] \Psi=0
$$
这里的波函数 $\Psi$ 指具有六个坐标的完整系统。方程 (4.73) 可以分为两个，一个表示分子整体的平移运动，另一个表示两个粒子的相对运动。为此，考虑新变量 $X, Y, Z$ 这是系统质心的笛卡尔坐标和 $r, \theta$ 和 $\varphi$ 第二个粒子相对于第一个粒子的极坐标。这些坐标通过方程式与两个粒子的笛卡尔坐标相关
$$
X=\frac{m_1 x_1+m_2 x_2}{m_1+m_2} Y=\frac{m_1 y_1+m_2 y_2}{m_1+m_2} Z=\frac{m_1 z+m_2 z_2}{m_1+m_2} x=x_2-x_1=r \sin \theta \cos \varphi y=y_2
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|固体物理代写Solid-state physics代考|PHYSICS7544

Posted on 2023年3月3日2023年3月3日 by statistics-lab

如果你也在怎样代写固体物理Solid-state physics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

固态物理学是通过量子力学、晶体学、电磁学和冶金学等方法研究刚性物质或固体。它是凝聚态物理学的最大分支。

我们提供的固体物理Solid-state physics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|固体物理代写Solid-state physics代考|Parabolic bands approximation

If we concentrate on considering only the band portions close to the VB top and the CB bottom, we recognise that in these regions the bands have a trend that, in very good approximation, can be described as parabolic (see figures $8.7$ and $8.8$ at the $\Gamma$ point). Therefore, whenever we are interested in studying the physics of electrons in the proximity of the forbidden gap, we can meaningfully use the so called parabolic bands approximation.

This approximation is straightforwardly applied in the one-dimensional case by taking the limit of $k \rightarrow 0$ of equations (8.20) and (8.21)
$$
\begin{aligned}
& \lim {k \rightarrow 0} E{\mathrm{VB}}(k)=E_{\mathrm{a}}+2 \gamma\left[1-\frac{(k a)^2}{2}\right] \
& \lim {k \rightarrow 0} E{\mathrm{CB}}(k)=E_{\mathrm{a}}^{\prime}-2 \delta\left[1-\frac{(k a)^2}{2}\right] .
\end{aligned}
$$
Taking this limit is justified by the fact that at $k=0$ we found both the VB top and the CB bottom, as shown in figure 8.6. A general remarkable feature is drawn: the thermal excitation of electrons basically occurs within a parabolic band scheme.

It is very convenient to address electron dynamics within a semi-classical scheme according to which: (i) electron energy states are described quantum mechanically,but (ii) their equations of motion are classical. This approximated scheme is trustworthy when one wants to study the motion of the electrons over a length scale much larger than the interatomic distances. This is, for instance, the relevant case of motion under the action of an externally applied and slowly varying electric field, that is an electric field which is practically constant over the length scale of interatomic lattice distances. On the other hand, the results of this approximation can hardly be extended to the case of nanostructures [16, 17], that is to solid state systems whose structural features display on the $10^{-9} \mathrm{~m}$ scale: here a full quantum theory of electron transport is needed, as detailed elsewhere [12, 17-19].

物理代写|固体物理代写Solid-state physics代考|Electric field effects

Let us now apply a constant and uniform electric field $\mathbf{E}$ along a one-dimensional crystal. Within the semi-classical scheme a driving force
$$
F=-e|\mathbf{E}|=\hbar \frac{d k}{d t},
$$
is calculated, governing the drift motion of the electron. The solution of this equation of motion is
$$
k(t)=k_0-\frac{e|\mathbf{E}|}{\hbar} t,
$$
where $k_0$ is the electron wavevector at time $t=0$, that is, when the electric field is turned on. By making use of equation (8.29) this result reflects in a time-dependent electron velocity
$$
v_n(k, t)=\frac{1}{\hbar} \frac{d E_n}{d k(t)}
$$

suggesting the practical rule that under the action of an electric field, the electron velocity at time $t$ is calculated by evaluating the slope of band tangent at the point $k(t)$ given in equation (8.32). This result has a quite interesting implication, as we easily understand by considering the case of an electron in the valence band: under the action of the electric field, which we consider oriented to the left with no loss of generality, the wavevector varies linearly with time, assuming gradually increasing values and, therefore, it will sooner or later end up reaching the right edge of the 1BL. However, given the crystalline periodicity, the $k=+\pi / a$ value defines a quantum state equivalent to the one described by $k^{\prime}=k+G$ with $G=-2 \pi / a$ a reciprocal lattice vector. This is tantamount to saying that the electron, once it reaches the right edge, is flipped back to a state corresponding the left one. Next, as time goes by, the electron will again assume increasing wavevector values, as before eventually reaching the right edge of $1 \mathrm{BZ}$ : here its wavevector will be flipped back once more. And so on … This periodic back-and-forth variation of $k(t)$ in the Brillouin zone will continue as long as the electric field is present. This phenomenon is described by saying that under the action of an electric field a band electron is subjected to Bloch oscillations: their graphical rendering is reported in figure $8.10$.
We remark that this result has been obtained by guessing the equation of motion (8.31) where no scattering phenomena appear, contrarily to what we discussed in section 7.1. This is of course a very crude approximation: in practice, it is very difficult to experimentally observe Bloch oscillations in real materials just because ionic motions and defects disturb the electron motion. Such oscillations are only detected at low temperature and in chemically pure systems, since the occurrence of such circumstances makes the periodic variation of $k(t)$ only marginally affected by electron-phonon and electron-defect scattering events or, equivalently, the friction term appearing in equation (7.3) to play a marginal role.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Parabolic bands approximation

如果我们只专注于考虑靠近 VB 顶部和 CB 底部的波段部分，我们会认识到在这些区域中，波段有一个趋势，在非常近似的情况下，可以描述为抛物线 (见图 $8.7$ 和 $8.8$ 在 $\Gamma$ 观点) 。因此，每当我们有兴趣研究禁带附近的电子物理时，我们都可以有意义地使用所谓的抛物线近似。
该近似值直接应用于一维情况，取的极限是 $k \rightarrow 0$ 等式 (8.20) 和 (8.21)
$$
\lim k \rightarrow 0 E \operatorname{VB}(k)=E_{\mathrm{a}}+2 \gamma\left[1-\frac{(k a)^2}{2}\right] \quad \lim k \rightarrow 0 E \operatorname{CB}(k)=E_{\mathrm{a}}^{\prime}-2 \delta[1
$$
采取这个限制是合理的，因为在 $k=0$ 我们找到了 $\mathrm{VB}$ 顶部和 $\mathrm{CB}$ 底部，如图 $8.6$ 所示。得出一个普㴜的显着特征: 电子的热激发基本上发生在抛物线带方案内。
在半经典方案中解决电子动力学非常方便，根据该方案: (i) 电子能态用量子力学描述，但 (ii) 它们的运动方程是经典的。当人们想要研究电子在比原子间距离大得多的长度尺度上的运动时，这种近似方案是值得信赖的。例如，这是在外部施加的缓曼变化的电场作用下运动的相关情况，该电场在原子间晶格距离的长度尺度上实际上是恒定的。另一方面，这种近似的结果很难扩展到纳米结构的情况 [16，17]，即结构特征显示在 $10^{-9} \mathrm{~m}$ 规模: 这里需要一个完整的电子传输量子理论，正如其他地方详述的那样 [12, 17-19]。

物理代写|固体物理代写Solid-state physics代考|Electric field effects

现在让我们施加一个恒定且均匀的电场E沿着一维晶体。在半经典方案中，驱动力
$$
F=-e|\mathbf{E}|=\hbar \frac{d k}{d t},
$$
计算，控制电子的漂移运动。这个运动方程的解是
$$
k(t)=k_0-\frac{e|\mathbf{E}|}{\hbar} t
$$
在哪里 $k_0$ 是时刻的电子波矢 $t=0$ ，也就是说，当电场打开时。通过使用等式 (8.29)，该结果反映在时间相关的电子速度中
$$
v_n(k, t)=\frac{1}{\hbar} \frac{d E_n}{d k(t)}
$$
表明在电场作用下，电子速度随时间变化的实际规律 $t$ 通过评估带切线在该点的斜率来计算 $k(t)$ 在等式 (8.32) 中给出。这个结果有一个非常有趣的含义，正如我们通过考虑价带中电子的情况很容易理解的那样: 在不失一般性的情况下我们认为向左取向的电场的作用下，波矢量线性变化时间，假设值逐渐增加，因此迟早会到达 1BL 的右边缘。然而，考虑到晶体的周期性， $k=+\pi / a$ value 定义了一个量子态，等价于由 $k^{\prime}=k+G$ 和 $G=-2 \pi / a$ 倒数点阵向量。这无异于说电子一旦到达右边缘，就会翻转回与左边缘对应的状态。接下来，随着时间的流逝，电子将再次呈现增加的波矢值，就像之前最终到达右边缘一样 $1 \mathrm{BZ}$ ：这里它的波向量将再次翻转回来。依此类推……这种周期性的来回变化 $k(t)$ 只要存在电场，布里渊区的光就会继续。这种现象的描述是，在电扬的作用下，带电子受到布洛赫振荡：它们的图形洹染如图所示 $8.10$.
我们注意到这个结果是通过猜测没有出现散射现象的运动方程 (8.31) 获得的，这与我们在 $7.1$ 节中讨论的相反。这当然是一个非常粗略的近似值：在实践中，很难仅仅因为离子运动和缺陷干扰电子运动而通过实验观察真实材料中的布洛赫振荡。这种振荡仅在低温和化学纯系统中检测到，因为这种情况的发生使得 $k(t)$ 仅受电子声子和电子缺陷散射事件的轻微影响，或者等效地，等式 (7.3) 中出现的摩擦项起着微不足道的作用。

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