## Reciprocal lattice and Bragg's law Bragg's law is a consequence of periodicity of the [[Lattice and periodic system|lattice]], i.e., $2d \sin \theta=n\lambda$, $\lambda$ is wavelength, $\theta$ is angle between plane and incident light, $d$ is spacing between layers (periodic constructive interference). >[!Notice] >The spacing $d$ does not to be the minimum. Due to the periodicity, the property (locally, no changes) after translations, i.e., by $\vec{T}=\sum \vec{a_i}n_i$, remains unchanged. We still have $n(\vec{r}+\vec{T})=n(\vec{r})$. And into this idea to represent this is put it in Fourier space. So we get the so-called reciprocal lattice. $b_1=2\pi\frac{a_2\times a_3}{a_1\cdot a_2\times a_3};b_2=2\pi\frac{a_3\times a_1}{a_1\cdot a_2\times a_3};b_3=2\pi\frac{a_1\times a_2}{a_1\cdot a_2\times a_3}$ >[!Note] > The denominator is just the volume of selected cell. Let $G=\sum \vec{a_i}b_i$, then $n(\vec{r}+\vec{T})=n(\vec{r}) = \sum_G n_Ge^{iGr}$ This is an expansion with respect to $G$. The set of reciprocal lattice vector $G$ determines the possible x-ray reflection. That is, $G=\Delta k = k'-k$, $d=\frac{2\pi}{G}$. This is Bragg's law. And from this periodicity, one could deduce [[Bloch's theorem and Born-von Karman boundary condition|Bloch's theorem]] considered the potential $V$ is also periodic, i.e., $V(\vec{r}+\vec{T})=V(\vec{r})$. Under this condition, we have $\Psi_{nk}(\vec{r}) = u_{nk}(r)e^{ikr}$ ^98c7c6 >[!Info] >More on reciprocal lattice: https://en.wikipedia.org/wiki/Reciprocal_lattice >Bragg's Law: https://en.wikipedia.org/wiki/Bragg%27s_law >Bloch's theorem: https://en.wikipedia.org/wiki/Bloch%27s_theorem > >These will also be included and extensive discussed in other lectures. See [[Bloch function and Bloch theorem]].