Write a short note on Bravais lattices.

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(N/A) The French mathematician Bravais showed that there are only $14$ possible three-dimensional lattices. These are called Bravais lattices.The char

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(N/A) The French mathematician Bravais showed that there are only $14$ possible three-dimensional lattices. These are called Bravais lattices.The characteristics of primitive and centered unit cells are listed below:						$S.No.$ Crystal System			Possible Variations			Edge Lengths			Axial Angles			Examples							$(1)$ Cubic			Primitive,Body-centered,Face-centered			$a=b=c$			$\alpha=\beta=\gamma=90^{\circ}$			$NaCl, Cu, ZnS$							$(2)$ Tetragonal			Primitive,Body-centered			$a=b 
eq c$			$\alpha=\beta=\gamma=90^{\circ}$			White $Sn, SnO_2, TiO_2, CaSO_4$							$(3)$ Orthorhombic			Primitive,Body-centered,Face-centered,End-centered			$a 
eq b 
eq c$			$\alpha=\beta=\gamma=90^{\circ}$			Rhombic sulphur,$KNO_3, BaSO_4$							$(4)$ Hexagonal			Primitive			$a=b 
eq c$			$\alpha=\beta=90^{\circ}, \gamma=120^{\circ}$			Graphite,$ZnO, CdS$							$(5)$ Rhombohedral			Primitive			$a=b=c$			$\alpha=\beta=\gamma 
eq 90^{\circ}$			Calcite $(CaCO_3), HgS$							$(6)$ Monoclinic			Primitive,End-centered			$a 
eq b 
eq c$			$\alpha=\gamma=90^{\circ}, \beta 
eq 90^{\circ}$			Monoclinic sulphur,$Na_2SO_4 \cdot 10H_2O$							$(7)$ Triclinic			Primitive			$a 
eq b 
eq c$			$\alpha 
eq \beta 
eq \gamma 
eq 90^{\circ}$			$K_2Cr_2O_7, CuSO_4 \cdot 5H_2O, H_3BO_3$

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$S.No.$ Crystal System	Possible Variations	Edge Lengths	Axial Angles	Examples
$(1)$ Cubic	Primitive,Body-centered,Face-centered	$a=b=c$	$\alpha=\beta=\gamma=90^{\circ}$	$NaCl, Cu, ZnS$
$(2)$ Tetragonal	Primitive,Body-centered	$a=b \neq c$	$\alpha=\beta=\gamma=90^{\circ}$	White $Sn, SnO_2, TiO_2, CaSO_4$
$(3)$ Orthorhombic	Primitive,Body-centered,Face-centered,End-centered	$a \neq b \neq c$	$\alpha=\beta=\gamma=90^{\circ}$	Rhombic sulphur,$KNO_3, BaSO_4$
$(4)$ Hexagonal	Primitive	$a=b \neq c$	$\alpha=\beta=90^{\circ}, \gamma=120^{\circ}$	Graphite,$ZnO, CdS$
$(5)$ Rhombohedral	Primitive	$a=b=c$	$\alpha=\beta=\gamma \neq 90^{\circ}$	Calcite $(CaCO_3), HgS$
$(6)$ Monoclinic	Primitive,End-centered	$a \neq b \neq c$	$\alpha=\gamma=90^{\circ}, \beta \neq 90^{\circ}$	Monoclinic sulphur,$Na_2SO_4 \cdot 10H_2O$
$(7)$ Triclinic	Primitive	$a \neq b \neq c$	$\alpha \neq \beta \neq \gamma \neq 90^{\circ}$	$K_2Cr_2O_7, CuSO_4 \cdot 5H_2O, H_3BO_3$