Describing condensed phase structures Describing the structure of an isolated small molecule is easy to do Just specify the bond distances and angles How do we describe the structure of a condensed phase? we have ~ Avogadro s number of atoms to locate we should either give up on specifying the position of every atom or find a trick to help us out The structure of liquids and glasses We can use pair distribution functions to describe the structure of such systems The structure of crystalline materials The lattice and unit cell in 1D We can use the symmetry of a crystal to reduce the number of unique atom positions we have to specify The most important type of symmetry is translational this can be described by a lattice The structure associated with the lattice can be carved up into boxes (unit cells) that pack together to reproduce the whole crystal structure Lattices and unit cells 2 D Unit cell choice There is always more than possible choice of unit cell By convention the unit cell is chosen so that it is as small as possible while reflecting the full symmetry of the lattice Page 1
Unit cell choice in 2D Picking a unit cell for NaCl Other types of symmetry ❽Crystallographers make use of all the symmetry in a crystal to minimize the number of independent coordinates ❽Lattice symmetry ❽Point symmetry ❽Other translational symmetry elements screw axes and glide planes Point symmetry elements ❽A point symmetry operation does not alter at least one point that it operates on rotation axes mirror planes rotation-inversion axes ❽Screw axes and glide planes are not point symmetry elements!!! Benzene A two fold rotation Page 2
A mirror plane An inversion center A rotation inversion axis Point symmetry elements compatible with 3D translations Symmetry element Symbol Mirror plane Rotation axis Inversion axis Center of symmetry m n = 2,3,4,6 n (= 1,2,3,4,6) 1 υνδερσχορε Point symmetry and packing Unit cells in 3D Page 3
The seven crystal systems System Unit Cell Minimum Symmetry The symmetry elements of a cube Triclinic α β γ 90º a b c Monoclinic α = γ = 90º β 90º a b c Orthorhombic α = β = γ = 90º a b c Trigonal Hexagonal Tetragonal Cubic α = β = γ 90º a = b = c α = β = 90º γ = 120º a = b c α = β = γ = 90º a = b c α = β = γ = 90º a = b = c None One two-fold axis or one symmetry plane Any combination of three mutually perpendicular two-fold axes or planes of symmetry One three-fold axis One six-fold axis or one six-fold improper axis One four-fold axis or one fourfold improper-axis Four three-fold axes at 109º 23 to each other Centering Bravais Lattices Screw axes and glide planes A two fold screw ❽Crystalline solids often posses symmetry that can be described as a combination of a rotation and a translation a screw axis or a combination of a reflection and a translation a glide plane Page 4
An a glide Lattice planes ❽ It is possible to describe certain directions and planes with respect to the crystal lattice using a set of three integers referred to as Miller Indices Miller indices (hkl) Examples of Miller indices ❽ Miller Indices are the reciprocal intercepts of the plane on the unit cell axes ❽ Identify plane adjacent to origin can not determine for plane passing through origin ❽ Find intersection of plane on all three axes ❽ Take reciprocal of intercepts ❽ If plane runs parallel to axis, intercept is at, so Miller index is 0 Families of planes d-spacing formulae ❽Miller indices describe the orientation a spacing of a family of planes The spacing between adjacent planes in a family is referred to as a d-spacing Three different families of planes d-spacing between (300) planes is one third of the (100) spacing For a unit cell with orthogonal axes (1 / d 2 hkl) = (h 2 /a 2 ) + (k 2 /b 2 ) + (l 2 /c 2 ) Hexagonal unit cells (1 / d 2 hkl) = (4/3)([h 2 + k 2 + hk]/ a 2 ) + (l 2 /c 2 ) Page 5
Unit cells and d hkl Bragg s law 2d sinθ =nλ Consider crystal to contain repeating reflecting planes (lattice planes) Fractional coordinates ❽The 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 Specifying orientation and direction ❽ Miller indices (hkl) are used to specify the orientation and spacing of a family of planes. {hkl} are used to specify all symmetry equivalent sets of planes ❽ Miller indices [hkl] are used to specify a direction in space with respect to the unit cell axes and <hkl> are used to specify a set of symmetry equivalent directions To specify a direction parallel to a line joining the origin and a point with coordinates x,y,z in the unit cell multiply x,y,z by the smallest number that will result in three integers» these are the Miller indices specifying the direction» Passes through 0.3333,0.6667,1 so Miller indices are [123] Density ❽Density measurement and calculation can be used to determine number of formula units in unit cell check that your supposed formula is correct establish defect mechanism Density = (FW x Z) /(V x N) = (FW x Z x 1.66) /V V unit cell volume Z formula units in cell FW formula weight Page 6