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## 物理代写|固体物理代写Solid-state physics代考|Von Laue scattering conditions

We now perform a detailed analysis of the scattering events occurring in $\mathrm{x}$-ray diffraction. Let $\mathbf{k}{\text {in }}$ be the wavevector of the incoming monochromatic plane wave with amplitude $$\mathcal{A}{\mathrm{in}}(\mathbf{r}, t)=\mathcal{A}{0} \exp \left[i\left(\mathbf{k}{\mathrm{in}} \cdot \mathbf{r}-\omega t\right)\right],$$
where $\omega$ is its angular frequency, while $\mathbf{r}$ and $t$ indicate the position in space and time, respectively. Our goal is to predict the amplitude $\mathcal{A}_{\text {out }}$ of the scattered waves. To this aim we adopt a model originally developed by $\mathrm{M}$ von Laue and based on two simplifying assumptions: (i) the incoming beam is weak enough that its interaction with the sample does not affect the underlying crystal structure and (ii) the scattering events are elastic, that is, $\mathrm{x}$-rays do not lose energy by diffusion (i.e. their intensity is unaffected by scattering).

We now remember that, according to the Huygens-Fresnel principle of elementary optics [13], any point-like object invested by a plane wave becomes the source of a scattered spherical wave. Accordingly, with reference to figure $2.15$, we can write the amplitude $\mathcal{A}{\text {out }}$ of the spherical wave emerging from the atom at position $\mathbf{R}$ and revealed by a detector at a distance $\mathbf{D}$ from the origin of the adopted frame of reference as $$\mathcal{A}{\text {out }}=\underbrace{\mathcal{A}{0} \exp \left[i\left(\mathbf{k}{\text {in }} \cdot \mathbf{R}-\omega t\right)\right]}{\text {incoming plane wave }} \cdot \underbrace{f{\mathbf{R}}}{\text {atomic form factor }} \cdot \underbrace{\frac{\exp \left[i k{\text {in }}|\mathbf{D}-\mathbf{R}|\right]}{|\mathbf{D}-\mathbf{R}|}}{\text {scattered spherical wave }}$$ where the last term on the right-hand side accounts for phase change and amplitude decrease of the scattered wave. Such changes are due to the complex of quantum phenomena occurring in the interaction between the electromagnetic wave and the atom at position $\mathbf{R}$ : their overall effect is summarised by the atomic form factor $f{\mathbf{R}}$ which depends on the atomic number $Z$ of the atomic scatterer, as well as by its electron charge density $\rho(\mathbf{r})$ through the general expression
$$f_{\mathbf{R}}=\int \rho(\mathbf{r}) \exp [i \mathbf{K} \cdot(\mathbf{r}-\mathbf{R})] d \mathbf{r}$$

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

The reciprocal lattice is formally described by the same concepts developed in section 2 for the direct one. More specifically, its points are given by
$$\mathbf{G}=m_{1} \mathbf{b}{1}+m{2} \mathbf{b}{2}+m{3} \mathbf{b}{3},$$ where $\left{\mathbf{b}{1}, \mathbf{b}{2}, \mathbf{b}{3}\right}$ are named reciprocal translation vectors and $m_{1}, m_{2}, m_{3}=$ $\pm 1, \pm 2, \pm 3, \ldots$ The maximum scattering vectors $\mathbf{K}$ entering equation (2.10) lie on this reciprocal lattice and, therefore, they must fulfil equation (2.12); accordingly, by setting $\mathbf{K}=\mathbf{G}$, after some little algebra we obtain that the Laue conditions are satisfied if
$$\mathbf{b}{1}=2 \pi \frac{\mathbf{a}{2} \times \mathbf{a}{3}}{\mathbf{a}{1} \cdot \mathbf{a}{2} \times \mathbf{a}{3}} \quad \mathbf{b}{2}=2 \pi \frac{\mathbf{a}{3} \times \mathbf{a}{1}}{\mathbf{a}{1} \cdot \mathbf{a}{2} \times \mathbf{a}{3}} \quad \mathbf{b}{3}=2 \pi \frac{\mathbf{a}{1} \times \mathbf{a}{2}}{\mathbf{a}{1} \cdot \mathbf{a}{2} \times \mathbf{a}{3}} .$$
This result provides the formal definition of $\mathbf{b}$-vectors and stresses the fact that the reciprocal lattice is closely related to its direct counterpart. A number of formal relations hold, including the following most important ones:
\begin{aligned} \mathbf{a}{i} \cdot \mathbf{b}{j} &=2 \pi \delta_{i j} & & \text { with } \quad i, j=1,2,3 \ \mathbf{G} \cdot \mathbf{a}{i} &=2 \pi m{i} & & \text { with } i=1,2,3 \ \exp \left(i \mathbf{G} \cdot \mathbf{R}{1}\right) &=1 & & \text { for any pair of direct/reciprocal vectors, } \end{aligned} where $\delta{i j}$ is Kroenecker delta-symbol. Furthermore, we observe that $\mathbf{b}{i}$ and $\mathbf{b}{j}$ with $i, j \neq k$ are normal to $\mathbf{a}{k}$, while $\mathbf{b}{i}$ is not in general parallel to $\mathbf{a}_{i}$ : this holds only in crystals with orthogonal axes.

The reciprocal lattice has a number of important formal properties which are very easy to prove by applying the above definitions:

• the reciprocal lattice is a kind of Bravais lattice;
• through equation (2.12) an infinite lattice is generated, once again characterised by translational invariance;

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|Von Laue scattering conditions

$$\mathcal{A i n}(\mathbf{r}, t)=\mathcal{A} 0 \exp [i(\operatorname{kin} \cdot \mathbf{r}-\omega t)],$$

$$f_{\mathbf{R}}=\int \rho(\mathbf{r}) \exp [i \mathbf{K} \cdot(\mathbf{r}-\mathbf{R})] d \mathbf{r}$$

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

$$\mathbf{G}=m_{1} \mathbf{b} 1+m 2 \mathbf{b} 2+m 3 \mathbf{b} 3,$$

$$\mathbf{b} 1=2 \pi \frac{\mathbf{a} 2 \times \mathbf{a} 3}{\mathbf{a} 1 \cdot \mathbf{a} 2 \times \mathbf{a} 3} \quad \mathbf{b} 2=2 \pi \frac{\mathbf{a} 3 \times \mathbf{a} 1}{\mathbf{a} 1 \cdot \mathbf{a} 2 \times \mathbf{a} 3} \quad \mathbf{b} 3=2 \pi \frac{\mathbf{a} 1 \times \mathbf{a} 2}{\mathbf{a} 1 \cdot \mathbf{a} 2 \times \mathbf{a} 3} .$$

• 倒数格是一种布拉维格;
• 通过方程 (2.12) 生成一个无限晶格，再次以平移不变性为特征;

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