- Construct a reciprocal lattice from a given real space lattice
- Compute the intensity of X-ray diffraction of a given crystal
### Reciprocal lattice
### Reciprocal lattice in 2D and 3D.
In ([lecture 7](7_tight_binding.md)) we discussed the reciprocal space (k-space) for a 1D lattice. We found that the points $k$ and $k+G$, where $G=m2\pi/a$ with $m$ an integer, are equivalent to eachother. The reason was that we considered waves of the form
Recall that in ([lecture 7](7_tight_binding.md)) we discussed the reciprocal space (k-space) for a 1D lattice. We found that the points $k$ and $k+G$, where $G=m2\pi/a$ with $m$ an integer, are equivalent to eachother. The reason was that we considered waves of the form
$$
e^{ikx_n} = e^{ikna}
...
...
@@ -24,25 +24,25 @@ $$
e^{iGx_n} = e^{im2\pi n} = 1.
$$
The points $G=m2\pi/a$ form the reciprocal lattice.
The points $G=m2\pi/a$ form the _reciprocal lattice_.
We will now generalize this discussion to describe the reciprocal lattice in 3 dimensions.
For every real-space lattice
$$
{\bf R}=n_1{\bf a_1}+n_2{\bf a_2}+n_3{\bf a_3}
{\bf R}=n_1{\bf a_1}+n_2{\bf a_2}+n_3{\bf a_3},
$$
where $n_1$, $n_2$ and $n_3$ are integers, there exists a reciprocal lattice
$$
{\bf G}=m_1{\bf b_1}+m_2{\bf b_2}+m_3{\bf b_3}
{\bf G}=m_1{\bf b_1}+m_2{\bf b_2}+m_3{\bf b_3},
$$
where $m_1$, $m_2$ and $m_3$ are also integers.
The two lattices are related as follows:
Similar to the 1D case, the two lattices are related as follows:
