<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:

Unit of crystals

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 diffraction and Bragg’s Law

X-ray scattering from parallel planes

X-ray scattering from parallel planes

Where $n=\Z$, above is Bragg’s Law

Properties:

Elasticity

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

Young’s modulus

Young's modulus.png

$$ \begin{aligned} \text{Stress}&=\frac{F}{A} \\ \text{Strain}&=\frac{\Delta l}{l}

\end{aligned} $$

Youngs modulus:

$$ E=\frac{\text{Stress}}{\text{Strain}}=\frac{F/A}{\Delta l/l}=\frac{Fl}{A\Delta l} $$