2DPrimitiveCellChoice of Unit CellThere is always more than onepossible choice of unit cell.By convention,the chosen unitcellshouldbeassmallaspossible while reflecting thefull symmetry of the lattice.1)Thehighestsymmetry2)The smallest area (or volume)
2D Primitive Cell By convention, the chosen unit cell should be as small as possible while reflecting the full symmetry of the lattice. 1) The highest symmetry 2) The smallest area (or volume) Choice of Unit Cell There is always more than one possible choice of unit cell
Fivetypesof2D latticesThere are literally thousands of crystalline materialsthereareonly5distinctplanarlatticesParallelogramSquareRectangleyanglebetweentwo basicvectors.a=b,=90°ab, ± 90°a#b,=90°i,C4&*120°i,C2Centered Rectanglei)PrimitiveHexagonala=by=120°ii) Centeredi,Ca+b,=90°/l Primiative cell:iCentred cell:i+C
Five types of 2D lattices a=b, =90 ab, =90 a=b, =120 ab, =90 ab, 90 & 120 ii) Centered i) Primitive There are literally thousands of crystalline materials, there are only 5 distinct planar lattices. -angle between two basic vectors. Centred cell: i + C2 // Primiative cell: i i i,C4 i,C2 i,C6
Question:Boththecentredrectangularandsimplehexagonal2Dlattices have a rhombic primitive unit cell.What is thekeydifferencebetweenthem?CenteredRectanglenaa=b,=120°a±b, =90°
Question: Both the centred rectangular and simple hexagonal 2D lattices have a rhombic primitive unit cell. What is the key difference between them? a=b, =120 ab, =90
Example:2D-latticeofGrapheneorWhat's the smallest structure motif of a graphene sheet?What type of lattice does aplanar graphene sheetbelongto?
Example: 2D-lattice of Graphene • What’s the smallest structure motif of a graphene sheet? • What type of lattice does a planar graphene sheet belong to? or
stackinggraphite3-Dlatticecanbeused!C. Lattices and its unit in 3D:T= ma + nb + pc (m, n, p = 0, ±1, ± 2, ...)
c. Lattices and its unit in 3D: T = ma + nb + pc (m, n, p = 0, 1, 2, .) graphite stacking 3-D lattice can be used!