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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

� 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

The lattice and unit cell in 1D

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

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Unit cell choice in 2D Picking a unit cell for NaCl

Other types of symmetry�Crystallographers make use of all the

symmetry in a crystal to minimize the number of independent coordinates

�Lattice symmetry�Point symmetry�Other translational symmetry elements

– screw axes and glide planes

Point symmetry elements�A point symmetry operation does not alter

at least one point that it operates on– rotation axes– mirror planes– rotation-inversion axes

�Screw axes and glide planes are not point symmetry elements !!!

Benzene A two fold rotation

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A mirror plane An inversion center

A rotation inversion axisPoint symmetry elements

compatible with 3D translations

1Center of symmetry

n (= 1,2,3,4,6)Inversion axis

n = 2,3,4,6Rotation axis

mMirror plane

SymbolSymmetry element

υνδερσχορε

Point symmetry and packing Unit cells in 3D

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The seven crystal systems

Four three-fold axes at 109º 23’ to each other

α = β = γ = 90ºa = b = c

Cubic

One four-fold axis or one four-fold improper-axis

α = β = γ = 90ºa = b ≠ c

Tetragonal

One six-fold axis or one six-fold improper axis

α = β = 90ºγ = 120ºa = b ≠ c

Hexagonal

One three-fold axisα = β = γ ≠ 90ºa = b = c

Trigonal

Any combination of three mutually perpendicular two-fold axes or planes of symmetry

α = β = γ = 90ºa ≠ b ≠ c

Orthorhombic

One two-fold axis or one symmetry plane

α = γ = 90ºβ ≠ 90ºa ≠ b ≠ c

Monoclinic

Noneα ≠ β ≠ γ ≠ 90ºa ≠ b ≠ c

Triclinic

Minimum SymmetryUnit CellSystem

The symmetry elements of a cube

Centering Bravais Lattices

Screw axes and glide planes

�Crystalline 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”

A two fold screw

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An “a glide” Lattice planes� 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)� Miller Indices are the

reciprocal intercepts of the plane on the unit cell axes

� Identify plane adjacent to origin– can not determine for plane

passing through origin� Find intersection of plane on

all three axes� Take reciprocal of intercepts � If plane runs parallel to axis,

intercept is at ∞, so Miller index is 0

Examples of Miller indices

Families of planes�Miller 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

d-spacing formulae

� For a unit cell with orthogonal axes– (1 / d2

hkl) = (h2/a2) + (k2/b2) + (l2/c2)

� Hexagonal unit cells– (1 / d2

hkl) = (4/3)([h2 + k2 + hk]/ a2) + (l2/c2)

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Unit cells and dhklBragg’s law

� 2d sinθ = nλ� Consider crystal to contain repeating ‘reflecting’

planes (lattice planes)

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

Specifying orientation and direction

� 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

� 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�Density 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