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# 物理代写|固体物理代写solid physics代考|The direct lattice

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## 物理代写|固体物理代写solid physics代考|Basic definitions

The existence of a discrete regular distribution of matter within a crystal suggests that its organisation consists in a space periodic repetition of identical structural units, each of which may contain one or more atoms ${ }^{1}$. In this latter case, the fundamental structural unit could be formed by atoms belonging to the same or to different chemical species: we will refer to elemental or compound systems in the two cases, respectively.

By some formal rules discussed below we can distinguish between the geometry and the structure of a crystal. Its geometry is described by introducing a suitable discrete grid of points in the space fulfilling the fundamental requirement of translational invariance. In doing that, we have in general many different options and, in particular, we could define the grid such that all its points are equivalent in any respect to the origin of the frame of reference set to define positions. This requirement imposes that the arrangement and the orientation of the points on the grid must appear the same from whichever site is selected to look at their distribution.
In general, not all atoms of a crystal occupy a position with these characteristics, as illustrated in figure $2.1$ for graphene ${ }^{2}$ : a single atomic plane of carbon atoms placed at the corners of regular hexagons. It is clear from figure 2.1(left) that atoms labelled by capital letters $A, B, C, \cdots$ lie at positions fully equivalent (under both the arrangement and orientation criteria) to the position of the origin $O$, while atoms labelled by lower letters $a, b, c, \cdots$ do not have this property. The grid of points $A, B$, $\mathrm{C}, \cdots$ shown in figure $2.1$ (right) is named the lattice of graphene.

## 物理代写|固体物理代写solid physics代考|Direct lattice vectors

The vector positions $\mathbf{R}{1}$ of the lattice points are defined as $$\mathbf{R}{1}=n_{1} \mathbf{a}{1}+n{2} \mathbf{a}{2}+n{3} \mathbf{a}{3},$$ where $\left{\mathbf{a}{1}, \mathbf{a}{2}, \mathbf{a}{3}\right}$ are named translation vectors and $n_{1}, n_{2}, n_{3}=0, \pm 1, \pm 2, \pm 3, \ldots$ Translation vectors must not all lie on the same plane. Through equation (2.1) an infinite lattice is generated (for this reason $\mathbf{R}{1}$ is also referred to as lattice vector), with translational invariance: the geometrical situation is just the same if viewed from any two positions $\mathbf{r}$ and $\mathbf{r}^{\prime}$ such that $\mathbf{r}^{\prime}=\mathbf{r}+\mathbf{R}{1}$ as illustrated in figure $2.3$ in the case of a two-dimensional square lattice.

The choice of translation vectors is not unique, as shown in figure $2.4$ : the same lattice can be equivalently spanned by different sets of translation vectors. We accordingly distinguish between primitive translation vectors and conventional translation vectors following a very simple criterion: if lattice points are found only at the corners of the parallelepiped whose edges are defined by $\left{\mathbf{a}{1}, \mathbf{a}{2}, \mathbf{a}_{3}\right}$, then the translation vectors are primitive. This is the case of the red and blue sets of vectors in figure $2.4$; conversely, the magenta vectors represent a conventional set. Lattices generated by primitive translation vectors are referred to as Bravais lattices. In this case, lattice points closest to a given point are named its nearest neighbours. Their number (necessarily equal for each lattice point because of the translational invariance property) is a characteristic of the specific Bravais lattice: it is called the coordination number.

The volume $V_{\mathrm{c}}$ of the parallelepiped defined by the translation vectors $\left{\mathbf{a}{1}, \mathbf{a}{2}, \mathbf{a}{3}\right}$ is $$V{\mathrm{c}}=\left|\mathbf{a}{1} \cdot \mathbf{a}{2} \times \mathbf{a}_{3}\right|,$$

as imposed by vector algebra, and the corresponding portion of space is referred to as the unit cell of the crystal. In the case where it is defined by primitive translation vectors, it is more precisely labelled as a primitive unit cell; otherwise it is named a conventional unit cell. Such a volume has the remarkable property that it fills all space without overlapping and without leaving voids when translated through equation (2.1).

## 物理代写|固体物理代写SOLID PHYSICS代考|Bravais lattices

Translational invariance represents the dominant structural feature of any crystal, largely dictating its physics. Nevertheless, it is not the only operation taking the lattice in itself. For instance, let us consider the face-centred cubic lattice shown in figure 2.5: it is easy to recognise that any rotation of a $\pi / 2$ angle about a line normal to a face and passing through its centre leaves the lattice unchanged. Similarly, a reflection in any plane defined by the cube faces takes the lattice in itself. These are just simple examples of non-translational symmetry operations: their full description is the core business of crystallography [4-7]. Here we limit ourselves to defining some general features allowing for the classification of the Bravais lattices.

First of all, we understand that all the operations we are dealing with are rigid, that is, they do not change the distance between lattice points. In other words, we are not considering deformations. Under this constraint, we can distinguish between pure translations and other operations that leave just one lattice point fixed. For example, imagine a two-dimensional square lattice and a rotation of a $\pi / 2$ angle about a line normal to the plane and passing through a lattice point. It is a key result of crystallography that by combining a translation with an action leaving just one lattice point fixed we get a symmetry operation for the selected lattice. We do not formally prove this result, but the graphical example shown in figure $2.6$ makes it plausible. In summary, all operations taking a lattice in itself are either pure translations or leave a particular lattice point fixed or are a combination of the two.

## 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。