. Lattices and its unit in 3D:T=ma + nb + pc (m, n, p= 0, ±1, ± 2, ...)The Choice of a Unit Cell:Having the highest symmetry and minimal size
a b c The Choice of a Unit Cell: Having the highest symmetry and minimal size a b c c. Lattices and its unit in 3D: T = ma + nb + pc (m, n, p = 0, 1, 2, .)
The Choice of a Primitive Cell1)Theaxialsystemconsistingofthebasisvectors should be right handed.2)Thebasisvectorsshouldcoincideasmuchaspossiblewithdirections of the highest symmetry3)Should be the smallest volume that satisfies condition 24)Ofall latticevectorsnoneisshorterthana5)Of thosenot directed alonga none is shorterthanb6)Ofthosenotlyinginthea,bplanenoneisshorterthanc7)Thethree angles betweenthebasis vectors a,b,c areeitherallacuteorobtuseConditions4-6define[a| ≤[≤[a]
1) The axial system consisting of the basis vectors should be right handed. 2) The basis vectors should coincide as much as possible with directions of the highest symmetry. 3) Should be the smallest volume that satisfies condition 2. 4) Of all lattice vectors none is shorter than a. 5) Of those not directed along a none is shorter than b. 6) Of those not lying in the a, b plane none is shorter than c. 7) The three angles between the basis vectors a,b,c are either all acute or obtuse. The Choice of a Primitive Cell a b c Conditions 4-6 define
Crystal structure =lattice +structural motif(basis)Structuralmotifaturalndifandl.faesamlelnluilTvtalttu(ngaaitangrinatefmdretks)
Crystal structure = lattice + structural motif (basis) Structural motif
Atomic Coordinates: Fractional coordinatesFractional coordinates:The positions of atoms inside aunit cell are specified using0.5fractional coordinates (x,y,z)CThesecoordinatesspecifythe0.6position as fractions of the unit celledgelengths.t: (1.0, 0.6, 0.5)
Atomic Coordinates: Fractional coordinates 0.5 0.6 i i: (1.0, 0.6, 0.5) Fractional coordinates: • The positions of atoms inside a unit cell are specified using fractional coordinates (x,y,z). • These coordinates specify the position as fractions of the unit cell edge lengths
Example:The Crystal StructureCubicunitcellofCsCla=b=cα=β==90°Cs:(0,0,0)Cl: (1/2,1/2,1/2)Inthis case, the lattice point can be put at the position ofeitherCsorClatom.Eachunitcell containsbothaCsandClatomSingle Crystal: Composed of only one particular type of space latticePolycrystalline matter: Clusters of multiple crystals
Example: Cubic unit cell of CsCl, a=b=c ===90 Cs:(0,0,0) Cl: (1/2,1/2,1/2) Single Crystal: Composed of only one particular type of space lattice. Polycrystalline matter: Clusters of multiple crystals. In this case, the lattice point can be put at the position of either Cs or Cl atom. Each unit cell contains both a Cs and Cl atom