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物理代写|固体物理代写Solid-state 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 invarianee 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 $$V{\mathrm{c}}=\left|\mathbf{a}{1} \cdot \mathbf{a}{2} \times \mathbf{a}_{3}\right|,$$

物理代写|固体物理代写Solid-state 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.

固体物理代写

物理代写|固体物理代写Solid-state physics代考|Direct lattice vectors

$$\mathbf{R} 1=n_{1} \mathbf{a} 1+n 2 \mathbf{a} 2+n 3 \mathbf{a} 3,$$

，则平移向量是原始的。这是图中红色和蓝色 向量集的情况2.4; 相反，洋红色向量代表一个传统的集合。由原始平移向量生成的格称为 Bravais 格。在这种情况下，最接近给定点的格点被命名为它的最近邻。 它们的数量 (由于平移不变特性，对于每个晶格点来说必然相等) 是特定布拉维晶格的一个特征：它被称为配位数。

$$V \mathrm{c}=\left|\mathbf{a} 1 \cdot \mathbf{a} 2 \times \mathbf{a}_{3}\right|,$$

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MATLAB代写

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