7.1.2ThelatticeandunitcellLattice: A periodic pattern of points in space, such thateach lattice point has identical surroundingsCan be reproduced by translational motionalongthevectorbetweenanytwopoints
7.1.2 The lattice and unit cell Lattice: • A periodic pattern of points in space, such that each lattice point has identical surroundings. • Can be reproduced by translational motion along the vector between any two points
a.1Dlatticeandits unit:Eachmotifinthis1Dperiodicpatterncanberepresentedbyapoint.Alatticeof6repeatingpointsisthus obtainedtorepresenttheabove 1D system.Thetranslationvectors connecting anytwo lattice pointsconstituteatranslation group.i.e.,Tm = ma (m = 0, ±1, ±2, ..., ±o0)a:basic vector
a. 1D lattice and its unit: • The translation vectors connecting any two lattice points constitute a translation group. i.e., Tm = ma (m = 0, 1, 2，, ) a: basic vector. Each motif in this 1D periodic pattern can be represented by a point. A lattice of repeating points is thus obtained to represent the above 1D system
Examples ofiDlattice一bStructural motifd)a isa 1Dlatticeitself:b-d arenot lattices,but canbe represented byalatticeApattern with periodicity=a lattice+ structural motif!
Examples of 1D lattice b) a) c) d) b-d are not lattices, a is a 1D lattice itself; but can be represented by a lattice. A pattern with periodicity = a lattice + structural motif! Structural motif
b. Lattice and its unit in 2Di(m, n=0,±1,± 2, ...Tmn = ma + nba&b: independent basicvectors[Tmn } - a translation groupBASISCrystal structure=lattice+structuralmotif(basis)
b. Lattice and its unit in 2D: Tmn = ma + nb（m, n = 0,1, 2， ) •Crystal structure = lattice + structural motif (basis) a & b: independent basic vectors b a {Tmn } – a translation group
Lattice:·Aperiodicpattern of points in space,such that eachlatticepointhasidentical surroundings·Can be reproduced by translational motion along thevectorbetweenanytwopoints2D2D lattice.lattice.This2Dpatternitselfis not a lattice,butcanberepresentedby a2DLattice
Lattice: • A periodic pattern of points in space, such that each lattice point has identical surroundings. • Can be reproduced by translational motion along the vector between any two points. This 2D pattern itself is not a lattice, but can be represented by a 2D Lattice. 2D lattice. 2D lattice