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**.
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@@ -368,29 +368,27 @@ Because the Wigner-Seitz cell is primitive, the 1st Brillouin zone (1BZ) contain
This means that all $\mathbf{k}$ vectors outside the 1st Brillouin zone are a copy of those inside 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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Deze figure nog omzetten naar python.
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We have already learned how to construct Wigner-Seitz cells, however here is a reminder of how a Brillouin zone looks like:
### Diffraction: Laue condition
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 $\mathbf{R}$.
## Diffraction
### Reciprocal lattice Laue conditions
Reciprocal lattice manifests directly in the diffraction experiments.
A diffraction experiment uses the crystal as a target and scatters high energy particles (X-rays, neutrons, or electrons) off of it.
As a result of interference between mutiple waves, the scattered radiation reveals the reciprocal lattice of the crystal.
In order to find the relationship between the incoming wave and the scattered one, let us 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 $|\mathbf{k'}|=|\mathbf{k}|$.
We assume that the atomic scattering is elastic (does not cause an energy loss), such that $|\mathbf{k'}|=|\mathbf{k}|$.
Below we present a simple sketch of two different atoms scattering an incoming wave.
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 $\Delta \phi$.
Observe that the bottom ray travels a larger distance compared to the upper ray.
The difference in 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
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@@ -426,9 +424,9 @@ In other words, if the difference of the wavevector between the incoming and out
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 assumed that the unit cell houses only a single atom.
But what if we drop this assumption and consider multiple atoms per unit cell?
### Structure factor
Above we assumed that the unit cell contains only a single atom.
What if the basis contains more atoms though?
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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@@ -449,32 +447,33 @@ A &\propto \sum_\mathbf{R} \sum_j f_j \mathrm{e}^{i\left(\mathbf{G}\cdot(\mathbf
\end{align}
$$
where $f_j$ is a phenomenological form factor that may be different for atoms of different elements.
The second part in the equation above is called the **structure factor**:
where $f_j$ is the scattering amplitude off of a single atom, and it is called the *form factor*.
The form factor mainly depends on the chemical element, nature of the scattered wave, and finer details like the electrical environment of the atom.
The first part of the equation above is the familiar Laue condition, and it requires that the scattered wave satisfies the Laue condition.
The second part gives the amplitude of the scattered wave, and it is called the **structure factor**:
The structure factor tells us a lot about how a material scatters incident waves.
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$!
Therefore, the intensity of a scattered wave depends on the structure factor $I \propto S(\mathbf{G})^2$.
Because the structure factor depends on the form factors and the positions of the basis atoms, by studying the visibility of different diffraction peaks we may learn the locations of atoms within the unit cell.
### Non-primitive unit cell
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.
Laue conditions allow scattering as long as the scattering wave vector is a reciprocal lattice vector.
However if we consider a non-primitive unit cell of the direct lattice, the reciprocal lattice contains more lattice points, seemingly leading to additional interference peaks.
Computing the structure factor allows us to resolve this apparent contradiction.
??? 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 are absent.
### Example: the FCC lattice.
In the last lecture we studied the FCC lattice.
Let us now calculate the structure factor of its conventional unit cell.
As a demonstration of how it happesn, let us compute the structure factor of the FCC lattice using the conventional unit cell in the real space.
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The basis of the conventional FCC unit cell contains four identical atoms.
With respect to the reference lattice point, these attoms are located at
Because $h$, $k$ and $l$ are integers, all exponents are either $+1$ or $-1$.
Solving this is a bit of a puzzle (try it for yourself!), but the result is:
Because $h$, $k$ and $l$ are integers, all exponents are either $+1$ or $-1$, and the interference is only present if
$$
S =
\begin{cases}
&4f, \:\mathrm{if} \: h, \: k, \:\mathrm{and} \: l \:\mathrm{are \: all \: even \: or \: odd}\\
&0, \:\mathrm{in \: all \: other \: cases}
4f, \:\mathrm{if} \: h, \: k, \:\mathrm{and} \: l \:\mathrm{are \: all \: even \: or \: odd,}\\
0, \:\mathrm{in \: all \: other \: cases}.
\end{cases}
$$.
$$
We now see that the reciprocal lattice points with nonzero amplitude exactly form the reciprocal lattice of the FCC lattice.
### Powder Diffraction
The easiest way to do diffraction measurements is to take a crystal, shoot an X-ray beam through it and measure the direction of outgoing waves.
However, it is highly unlikely that you will fulfil the Laue condition for any set of crystal planes.
There does exist a more practical experiment: **powder diffraction**.
However growing a single crystal may be hard because many materials are polycrystalline
A simple alternative is to perform **powder diffraction**.
By crushing the crystal into a powder, the small crystallites are now orientated in random directions.
This improves the chances of fulfilling the Laue condition for a fixed direction incoming beam.
The experiment is illustrated in the figure above.
The result is that the diffracted beam exits the sample via concentric circles at discrete **deflection angles** $2 \theta$.
```python
defadd_patch(ax,patches,*args,**kwargs):
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@@ -556,10 +560,6 @@ plt.axis('off')
plt.show()
```
By crushing the crystal into a powder, the small crystallites are now orientated in random directions.
This improves the chances of fulfilling the Laue condition for a fixed direction incoming beam.
The experiment is illustrated in the figure above.
The result is that the diffracted beam exits the sample via concentric circles at discrete **deflection angles** $2 \theta$.
In order to deduce the values of $\theta$ of a specific crystal, let us put the Laue condition into a more practical form.
We first take the modulus squared of both sides:
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@@ -588,10 +588,10 @@ $$
Note, $\phi$ is the angle between the vector $\mathbf{k}$ and $\mathbf{G}$, which is not the same as the angle between the incoming and scattered waves $\theta$.
We are nearly there, but we are left with finding out the relation between $\phi$ and $\theta$.
Last lecture we discussed the concept of Miller planes.
These are planes designated by Miller indices $(h,k,l)$ which intersect the lattice vectors at $\mathbf{a}_1 / h$, $\mathbf{a}_22 / k$ and $\mathbf{a}_3 / l$.
It turns out that these miller planes are normal to the reciprocal lattice vector $\mathbf{G} = h \mathbf{b}_1 + k \mathbf{b}_2 + l \mathbf{b}_3$ and that the distance between subsequent Miller planes is given by $d_{hkl} = \frac{2 \pi}{\lvert \mathbf{G} \rvert}$ (you will derive this in [today's exercise](https://solidstate.quantumtinkerer.tudelft.nl/10_xray/#exercise-2-miller-planes-and-reciprocal-lattice-vectors)).
We substitute the expression of $\lvert \mathbf{G} \rvert$ into the equation of the distance:
Recall the concept of Miller planes.
These are sets of planes identified by their Miller indices $(h,k,l)$ which intersect the lattice vectors at $\mathbf{a}_1 / h$, $\mathbf{a}_22 / k$ and $\mathbf{a}_3 / l$.
It turns out that Miller planes are normal to the reciprocal lattice vector $\mathbf{G} = h \mathbf{b}_1 + k \mathbf{b}_2 + l \mathbf{b}_3$ and the distance between subsequent Miller planes is $d_{hkl} = 2 \pi/\lvert \mathbf{G} \rvert$ (you will derive this in [today's exercise](https://solidstate.quantumtinkerer.tudelft.nl/10_xray/#exercise-2-miller-planes-and-reciprocal-lattice-vectors)).
Substituting the expression for $\lvert \mathbf{G} \rvert$ into the expression for the distance between Miller planes we get:
