Explain three-dimensional close packing from two-dimensional hexagonal close packing. Explain the formation of face-centered cubic $(FCC)$ unit cell and hexagonal close packing $(HCP)$ in three dimensions.
(N/A) $1$. Three-dimensional close packing from two-dimensional hexagonal layers is achieved by stacking layers one above the other.
$2$. When the second layer $(B)$ is placed over the first layer $(A)$,it covers one set of triangular voids (tetrahedral voids). Placing the third layer over the second layer can lead to two patterns:
- $ABAB...$ pattern: The third layer spheres align with the first layer $(A)$,resulting in hexagonal close packing $(HCP)$.
- $ABCABC...$ pattern: The third layer spheres are placed over the octahedral voids of the second layer,resulting in cubic close packing $(CCP)$ or face-centered cubic $(FCC)$ structure.
$2$. When the second layer $(B)$ is placed over the first layer $(A)$,it covers one set of triangular voids (tetrahedral voids). Placing the third layer over the second layer can lead to two patterns:
- $ABAB...$ pattern: The third layer spheres align with the first layer $(A)$,resulting in hexagonal close packing $(HCP)$.
- $ABCABC...$ pattern: The third layer spheres are placed over the octahedral voids of the second layer,resulting in cubic close packing $(CCP)$ or face-centered cubic $(FCC)$ structure.
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