Where we use $|\mathbf{k'}| = |\mathbf{k}|$ in the second line and insert the Laue condition in the third line. During the exercises for this week, you will derive an important relation $G_{hkl} = \frac{2}{d_{hkl}}$ where $d_{hkl}$ is the spacing between $(hkl)$ miller planes. With this, one can finally derive **Bragg's Law**:
Where we use $|\mathbf{k'}| = |\mathbf{k}|$ in the second line and insert the Laue condition in the third line. In the exercises this week, you will see that Miller planes (hkl) are normal to $G_{hkl}$ vectors and that the spacing between planes is given by $d_{hkl} = \frac{2 \pi}{G_{hkl}}$. With this, one can finally derive **Bragg's Law**:
$$ \lambda = 2 d_{hkl} \sin(\theta) $$
where $\phi = \theta - \pi/2$.
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@@ -416,6 +416,7 @@ where $\phi = \theta - \pi/2$.
* Diffraction experiments reveal information about crystal structure.
* Laue condition: difference between wavevectors of incoming and diffracted waves matches a reciprocal lattice vector, necessary for constructive interference.
* Structure factor: describes the contribution of the atoms in a unit cell to the diffraction pattern.
* Powder diffraction and relating its experimental results to the crystal structure via Bragg's law.
## Exercises
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@@ -456,9 +457,11 @@ Consider a two-dimensional crystal with a rectangular lattice and lattice vector
2. Consider an X-ray diffraction experiment performed on this crystal using monochromatic X-rays with wavelength $0.166$ nm. By assuming elastic scattering, find the magnitude of the wave vectors of the incident and reflected X-ray beams.
3. On the reciprocal lattice sketched in 3.1, draw the "scattering triangle" corresponding to the diffraction from (210) planes. To do that use the Laue condition $\Delta \mathbf{k} = \mathbf{G}$ for the constructive interference of diffracted beams.
### Exercise 4: Structure factors
### Exercise 4: Structure factors and powder diffraction
1. Compute the structure factor $\mathbf{S}$ of the BCC lattice.
2. Compute which diffraction peaks are missing.
2. How does this structure factor change if the atoms in the center of the conventional unit cell have a different form factor from the atoms at the corner of the conventional unit cell?
5. Explain why X-ray diffraction can be observed in first order from the (110) planes of a crystal with the BCC lattice, but not from the (110) planes of the FCC lattice.
3. How does this structure factor change if the atoms in the center of the conventional unit cell have a different form factor from the atoms at the corner of the conventional unit cell?
4. A student carried out X-ray powder diffraction on Chromium (Cr) which is known to have a BCC structure. The first five diffraction peaks are given below. Furthermore, the student took the liberty of assigning Miller indices to the peaks. Is the assignment correct? Fix any mistakes and explain your reasoning.
5. Calculate the lattice constant, $a$, of the chromium bcc unit cell. Note that X-ray diffraction was carried out using Cu K-$\alpha$ ($1.5406 \AA$) radiation.
