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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Lattice planes and directions

Within a crystal lattice we can identify sets of planes with the threefold property of (i) containing lattice points, and being (ii) parallel and (iii) equally spaced. Such lattice planes play an important role in determining the diffractions of whatever waves are travelling within the crystal.

Since any plane is determined by three non collinear points in the space, the conventional procedure to identify a family of parallel lattice planes is based on the intercepts made on the crystallographic axes by the nearest plane to the origin (not considering the plane that possibly contains the origin for the reason that will be immediately clear). The first step consists in finding such intercepts in terms of the lattice constants $(a, b, c)$ : this defines a set of three integer numbers; next, the reciprocal of these numbers is taken; finally, these reciprocals are reduced to the smallest three integer numbers $(h, k, l)$ having the same ratio. For instance, let us consider the case shown in figure 2.9: the plane has intercepts with the crystallographic axes given by 3,1 , and 2 in units of the $(a, b, c)$ lattice constants. Their reciprocal are $1 / 3,1$ ad $1 / 2$. The smallest three integers with the same ratio are (263): they label the plane. The three numbers $(h, k, l)$ are known as the Miller indices of the plane (in fact, they identify the family of its parallel lattice planes). In figure $2.10$ some important planes in Bravais lattices of the cubic crystal system are shown. We finally remark that whenever a plane intercepts an axis on its negative side with respect to the origin, the corresponding Miller index is negative: in order to keep this information, this index will be labelled by placing a bar on it.

Similarly to planes, even directions can be identified within a lattice. In this case, a set of three integers is used and put in square parenthesis as $[u, v, w]$ : they represent the set of smallest integers with the same ratio as the components of a vector pointing along the selected direction, referred to the crystallographic axes. In figure $2.11$ we show some important directions in Bravais lattices of the cubic crystal system. We remark that only in this special case is the $[u, v, w]$ direction always normal to the $(u, v, w)$ plane.

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

There is still a remaining criterion for classifying atomic architectures to be discussed. It is based on the assumption to treat atoms as attracting hard spheres. While this is clearly a very crude approximation, it is reasonably well satisfied by metals and this represents the phenomenological foundation for its applicability.

Since atoms are looked at as ‘hard’ spheres, they cannot overlap; however, because of their mutual attraction, they tend to assume an arrangement that minimises the total energy of the system. This implies that they tend to pack as closely as possible 7 .

Let us start by arranging identical hard spheres on a plane: in the closest packing configuration the centres of the spheres lic on a two-dimensional triangular lattice, in positions marked by A letters in figure 2.13. By looking at this configuration from the top, we can identify for the second layer the new triangular lattice (lying on a plane parallel to the first layer) marked by B letters. While this choice is unique, when adding a third layer we can add spheres on the triangular lattice marked by $\mathrm{C}$ letters or, alternatively, on the triangular lattice once again marked by A letters (in both cases the third layer lies on a plane parallel to the two previous ones). The corresponding stacking sequence is $\mathrm{ABCABCABC} \cdots$ and $A \mathrm{ABBABAB} \cdots$, respectively: they are named fcc structure and hexagonal close-packed (hpc) structure. The atomic layers so generated correspond to (111) planes of the fcc lattice or to the basal plane of the hexagonal lattice. In both configurations each atom has 12 nearest neighbours: any model according to which the total energy of a crystal only depends on the number of nearest neighbours must necessarily predict the very same energy for fcc and hpc structures.

The number of different ways to pack hard spheres in arrangement other than the close packed one is actually infinite. In table $2.1$ we summarise some properties of the packing in cubic lattices. The packing fraction is the volume fraction occupied by the hard spheres: the labelling ‘close packing’ for fcc is justified by the fact that its packing fraction is maximum.

# 固体物理代写

## 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。

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