In the beginning of the lecture we considered waves of the form
$$
e^{i \mathbf{k} \cdot \mathbf{r}}.
$$
These waves are **periodic** with respect to reciprocal lattice $\mathbf{G}$: shifting the wavevector $\mathbf{k}$ by an amount of $\mathbf{G}$ does not change the wave itself.
This means that a primitive unit cell of the reciprocal lattice contains a set of unique set of $\mathbf{k}$ vectors!
Last lecture we learned that the choice of a primitive unit cell is not unique and we also introduced the Wigner-Seitz cell.
Because the Wigner-Seitz cell is primitive, the Wigner-Seitz cell of the reciprocal lattice also contains a set of unique $\mathbf{k}$ vectors!
We now introduce a very important convention.
The Wigner-Seitz cell of the reciprocal lattice is the **1st Brillouin zone**!
In order to describe a reciprocal lattice, we need to define a primitive unit cell in reciprocal space.
Previously, we learned that the choice of a primitive unit cell is not unique.
However, a general convention in reciprocal space is to use the Wigner-Seitz cell which is called the **1st Brillouin zone**.
Because the Wigner-Seitz cell is primitive, the 1st Brillouin zone (1BZ) contains a set of unique $\mathbf{k}$ vectors.
This means that all $\mathbf{k}$ vectors outside the 1st Brillouin zone are a copy of those inside the 1st Brillouin zone.
So any wave with arbitrary $\mathbf{k}$ can be described by the same wave present in the 1st Brillouin zone.
For example, any $\mathbf{k'}$ outside the 1BZ is related to a wave vector inside 1BZ $\mathbf{k}$ by shifting it by reciprocal lattice vectors: $\mathbf{k'} = \mathbf{k}+\mathbf{G}$
Below we show the 1st Brillouin zone of the triangular reciprocal lattice.
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### Diffraction: Laue condition
Having introduced the basic notion of the reciprocal lattice, we use it to study the scattering of waves of a crystal structure and use it to describe simple diffraction experiments.
Such waves can be x-rays, neutrons or electrons.
We will derive under what conditions scattered wave will cause constructive interference.
For now we assume that each atom is also a lattice point, we drop this assumption in the next section.
Let us consider atoms that are separated by a lattice vector $\mathbf{R}$, and an incoming wave with wave vector $\mathbf{k}$.
While the atoms can scatter the incoming wave into all directions, we will find that only waves scattered in certain directions will interfere constructively and lead to a signal on the detector.
These directions are directly related to the reciprocal lattice vectors and we try to find how these are related to eachother.
After scattering we are left with a scattered wave with a wave vector $\mathbf{k'}$.
We assume only elastic scattering, meaning $|\mathbf{k'}|=|\mathbf{k}|$.
Below we show a very simple system with atoms seperated by a distance $\mathbf{R}$ and an incoming and scattered wave.
Even though the reciprocal lattice might seem like a far-fetched concept, we can actually probe it directly through diffraction experiments.
A diffraction experiment uses the crystal structure as a target and scatters waves off of it.
The commonly used waves are x-rays, neutrons, or electrons.
As a result of scattering, the outgoing waves form an interference pattern that is closely related to the reciprocal lattice of the crystal.
In order to find the relationship between the two, consider a lattice of atoms separated by a lattice vector $\mathbf{R}$.
An incoming wave with wave vector $\mathbf{k}$ is incident upon the lattice.
After scattering, the outgoing wave's wave vector is $\mathbf{k'}$.
We assume that the atomic scattering is elastic, such that $|\mathbf{k'}|=|\mathbf{k}|$.
Below we present a simple sketch of two different atoms scattering an incoming wave.
We see that the lower part of the wave wave travels a larger distance than the upper part.
This results in a phase difference $\Delta \phi$.
With a bit of geometry, we find that the total extra distance traveled by the lower part relative to the upper part is
It is clear from the figure that the bottom ray travels a larger distance than the upper ray.
The difference is travel distance results in a relative phase shift between the rays $\Delta \phi$.
With a bit of geometry, we find that the extra distance traveled by the lower ray relative to the upper one is
$$
x_{\mathrm{extra}} = \Delta x_1+\Delta x_2 = \cos(\theta) \lvert R \rvert + \cos(\theta') \lvert R \rvert.
$$
We use this to calculate the phase difference:
As a result of the travel distance, the phase difference is:
$$
\begin{align}
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\end{align}
$$
The scattered wave will only interfere constructively if this phase difference equals $2\pi n$, where $n$ is an integer.
In the beginning of the lecture we saw that
However, that is only a phase difference between waves scattered off of two atoms.
To find the outgoing wave's amplitude, we must sum over scattered waves from each and every atom in the lattice:
Therefore, the the phase difference will only equal $2\pi n$ if, and only if
The above sum is non-zero if and only if the scattered waves interfere constructively i.e. the phase difference equals $2\pi n$, where $n$ is an integer.
Furthermore, we know that real and reciprocal lattice vectors are related by $\mathbf{G} \cdot \mathbf{R} = 2 \pi n$.
Therefore, we conclude that the difference between incoming and outgoing waves must be:
$$
\mathbf{k'}-\mathbf{k}=\mathbf{G}.
$$
In other words, if the difference of the wavevector between the incoming and outgoing wave vectors coïncides with a reciprocal lattice point, we expect constructive interference.
This requirement is known as the **Laue condition**.
But how does this affect measurements in a diffraction experiment?
Suppose we perform a diffraction experiment by shooting a x-ray at a crystal structure.
We measure the intensity of the scattered wave and obtain an intensity profile.
The measured intensity of the scattered wave is proportional the to $A^2$, where $A$ is the amplitude of the scattered wave.
Considering the complete lattice, the amplitude of a scattered wave is
This sum will only yield a finite value if $\mathbf{k'}-\mathbf{k} = \mathbf{G}$.
Therefore, we will only measure a non-zero intensity if the incoming wave vector coïncides with a reciprocal lattice point!
Thus only those waves that satisfy the Laue condition are measurable.
This requirement is known as the _Laue condition_.
As a result, the interference pattern produced in diffraction experiments is a direct measurement of the reciprocal lattice!
### Diffraction: Structure factor
Above we have assumed that each atom is also a lattice point.
But what if drop this assumption and now consider multiple atoms per unit cell?
Above we assumed that the unit cell houses only a single atom.
But what if we drop this assumption and consider multiple atoms per unit cell?
In the figure below we show a simple lattice which contains multiple atoms in the unit cell.
Note, the unit cell does not have to be primitive!
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The distance $\mathbf{r}_j$ is taken with respect to lattice point from which we construct the unit cell.
Similar to before, we calculate the amplitude of the scattered wave.
However, now there are multiple atoms in the unit cell and each of these atoms acquires a phase shift of its own.
Therefore, we not only sum over all lattice points, but also over the atoms in a single unit cell::
In order to keep track of the atoms, we define $\mathbf{r}_j$ to be the location of atom $j$ in the unit cell.
The distance $\mathbf{r}_j$ is defined with respect to the lattice point from which we construct the unit cell.
In order to calculate the amplitude of the scattered wave, we must sum not only over all the lattice points but also over the atoms in a single unit cell:
The structure factor tells us a lot about how a material scatters incident waves.
As mentioned before, the intensity of a scattered wave $I \propto A^2$.
We directly see that the intensity of a scattered wave $I \propto S(\mathbf{G})^2$.
Thus the structure factor influences the intensity of the measured wave!
In diffraction experiments, the intensity of the scattered wave is $I \propto A^2$
Therefore, the intensity of a scattered wave depends on the structure factor $I \propto S(\mathbf{G})^2$!
One important feature of the structure factor is that it can cause destructive interference of scattered waves even though the Laue condition is met.
Because $\mathbf{r}_j$ is not necessarily a lattice point, the structure factor can be equal to zero.
This is thus independent of the Laue condition.
The destructive interference caused by the structure factor causes interference peaks in the interference pattern to dissapear!
One important feature of the structure factor is that it can lead to destructive interference of scattered waves even though the Laue condition is met.
The structure factor depends on the arrangement of the atoms inside the unit cell.
Some arrangments allow the structure factor to vanish, resulting in an absent signal peak in a diffraction pattern.
??? Question "Calculate the structure factor in which there is a single atom the unit cell located at the lattice point. Do any diffraction peaks dissapear?"
$\mathbf{r}_1=(0,0,0)\rightarrow S=f_1$.
In this case each reciprocal lattice point gives one interference peak, none of which is cancelled.
In this case, each reciprocal lattice point gives one interference peak, none of which are absent.
### Example: the FCC lattice.
In the last lecture we studied the FCC lattice.
Let us now calculate the structure factor for its conventional unit cell.
Let us now calculate the structure factor of its conventional unit cell.
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The basis of the conventional unit cell of the FCC lattice contains four identical atoms.
The basis of the conventional FCC unit cell contains four identical atoms.
With respect to the reference lattice point, these attoms are located at
