<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/817dec9c-7cf6-4968-ab57-8132695ce6ae/crystal.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/817dec9c-7cf6-4968-ab57-8132695ce6ae/crystal.png" width="40px" /> A crystal is a solid where the arrangement of atoms ordered and have symmetrical arrangements of 4 atoms
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Properties of crystals:
- Stable at low $T$ because thermal energy is lower than the depth of the potential well ($K_BT\ll \mathcal E$)
- Binding energy is maximised
- Translational, rotational, reflection, inversion and combination of those symmetries
The crystal lattice is the underlying structure of crystals#
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/ce3b4338-784f-43a4-9541-144071b86617/unti_cell.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/ce3b4338-784f-43a4-9541-144071b86617/unti_cell.png" width="40px" /> Unit cell: smallest unit which can be used to generate the whole structure by tessellation
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🗞️ Example:
$$ \circ \;\; \circ \;\; \circ \;\; \circ \\ \circ \;\; \circ \;\; \circ \;\; \circ \\ \circ \;\; \circ \;\; \circ \;\; \circ \\ \underbrace{\normalsize \circ \;\;\; \circ}_{a} \;\; \circ \;\; \circ \\ $$
Binding energy per unit cell
$$ \mathcal E _{\text{cell}}=\frac{n\mathcal E}{2} $$
where $n$ is the number of nearest neighbours
Total biding energy
$$ E_{\text{tot}}=N \mathcal E_\text{cell} $$
where $N$ is the total amount of unit cells
X-ray scattering from parallel planes
Path difference = $2 d\sin\theta$
Constructive Interference: $\boxed{2d\sin\theta=n\lambda}$
Where $n=\Z$, above is Bragg’s Law
Properties:
Scattering angle is $2\theta$
Need correction of orientation of the crystal planes (the x-ray emitter has to be at the right angle to get the right constructive interference)
or use powdered sample (all orientations at the same time)
Momentum changed, but energy is still the change (elastic scattering / diffraction)
X-rays because $\lambda \simeq d \simeq 1 \text{\AA}$
We can use other probes than X-rays with $\lambda \simeq d$
🗞️ Example: particles such as electrons or neutrons $\lambda=h/p$
Crystals are rigid due to strength and interatomic bonds
$$ \left . \begin{aligned} \text{Applying stress} & \rightarrow \text{Shape changes slightly} \\ \hookrightarrow \;\;\text{Remove stress}&\rightarrow \text{Usually returns to original shape} \end{aligned} \right \} \begin{aligned} &\text{Elastic} \\ &\text{behavior} \end{aligned} $$
Stress = Force / Area
$$ \begin{aligned} \text{Stress}&=\frac{F}{A} \\ \text{Strain}&=\frac{\Delta l}{l}
\end{aligned} $$
Youngs modulus:
$$ E=\frac{\text{Stress}}{\text{Strain}}=\frac{F/A}{\Delta l/l}=\frac{Fl}{A\Delta l} $$