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Structures of SolidsStructures of Solids& X& X--ray Diffractionray Diffraction

Chemistry 123Chemistry 123Dr. Patrick WoodwardDr. Patrick WoodwardSupplemental Lecture 4Supplemental Lecture 4

Crystalline SolidsCrystalline Solids

Crystal LatticeCrystal Lattice A 3D array of points where each pointA 3D array of points where each pointhas an identical environment.has an identical environment.

Unit CellUnit Cell The repeating unit (a unit cell is to a crystal,The repeating unit (a unit cell is to a crystal,like alike a brickbrick is in a house). In a given crystal all unitis in a house). In a given crystal all unitcells are identical.cells are identical.

CaFCaF22 Unit CellUnit Cell


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CrystalCrystal

SystemsSystemsCubicCubicTetragonalTetragonalHexagonalHexagonal

RhombohedralRhombohedralOrthorhombicOrthorhombic

MonoclinicMonoclinicTriclinicTriclinic

The crystal systems eachhave distinctive symmetry

and unit cell dimensions

Close Packed Array of SpheresClose Packed Array of Spheres

The gray spheres represent a 2DThe gray spheres represent a 2DClose Packed Array.Close Packed Array.

AABB StackingStacking AACC StackingStacking

In 3D the next layer of spheresIn 3D the next layer of spherescould sit on the depressionscould sit on the depressionsmarked inmarked in red (B)red (B) or thoseor those

marked inmarked in blue (C)blue (C)..


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Cubic and Hexagonal Close PackingCubic and Hexagonal Close Packing

Hexagonal Close PackingHexagonal Close Packing(ABAB(ABAB))

AABBAABB StackingStacking

AABBCCAABBCC StackingStacking

Cubic Close PackingCubic Close Packing(ABCABC(ABCABC))

Close Packed SpheresClose Packed Spheres

Hexagonal Close PackingHexagonal Close Packing(ABAB(ABAB))

Cubic Close PackingCubic Close Packing(ABCABC(ABCABC))


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Hexagonal CloseHexagonal ClosePackingPacking

Cubic CloseCubic ClosePackingPacking

Body CenteredBody Centered

Cubic PackingCubic Packing

Cubic or Hexagonal Close PackingCubic or Hexagonal Close PackingCoordination Number = 12Coordination Number = 12Packing Efficiency = 74%Packing Efficiency = 74%

Body Centered Cubic PackingBody Centered Cubic PackingCoordination Number = 8Coordination Number = 8Packing Efficiency = 68%Packing Efficiency = 68%

CsClCsCl StructureStructure


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EutacticEutactic StructuresStructures

Many ionic structure types can be described as aclose packing of anions with cations filling voidsor holes in the structure. Generally we willconsider two types of holes (for the cations)

Octahedral holes - Voids are surrounded by 6 anions andlead to octahedral coordination of the cation

Tetrahedral holes - Voids are surrounded by 4 anionsand lead to tetrahedral coordination of the cation

Octahedral HolesOctahedral HolesStart with a close packed layer of anions (A)Start with a close packed layer of anions (A)

Insert cations in the triangular depressions (c)Insert cations in the triangular depressions (c)

Add another anion layer (B)Add another anion layer (B)

The resulting cation coordination is an octahedronThe resulting cation coordination is an octahedron


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Tetrahedral HolesTetrahedral HolesStart with a close packed layer of anions (A)Start with a close packed layer of anions (A)

Insert cations in the triangular depressions (b)Insert cations in the triangular depressions (b)

Add another anion layer (B)Add another anion layer (B)

The resulting cation coordination is a tetrahedronThe resulting cation coordination is a tetrahedron

EutacticEutactic StructuresStructures

hcphcp2/32/3AlAl22OO33

hcphcp1/21/2TiOTiO22

hcphcp1/21/2CdICdI22

ccpccp1/21/2CdClCdCl22

hcphcp11NiAsNiAs

ccpccp11NaClNaCl

PackingPackingFractionFraction

HolesHolesFilledFilled

StructureStructure

hcphcp1/21/2WurtziteWurtzite

ccpccp1/21/2SphaleriteSphalerite

ccpccp11FluoriteFluorite

PackingPackingFractionFraction

HolesHolesFilledFilled

StructureStructure

Structures obtained byfilling Octahedral Holes

Structures obtained byfilling Tetrahedral Holes

In fluorite (i.e. CaFIn fluorite (i.e. CaF22) the cations are) the cations areclose packed and the anions fill theclose packed and the anions fill the

tetrahedral holes. The opposite is truetetrahedral holes. The opposite is trueof theof the antifluoriteantifluorite structure (Nastructure (Na22O)O)

TheThe hcphcp anion layers are buckled inanion layers are buckled in rutilerutile..


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Cubic close packed (CCP) anion arrayCubic close packed (CCP) anion array

Space GroupSpace Group = Fm3m= Fm3mAtomAtom Site x y zSite x y zAnionAnion 4a4a 0 0 00 0 0TetrTetr HoleHole 8c8c

Rock salt structure (Rock salt structure (NaClNaCl))(Octahedral Holes)(Octahedral Holes)

Space GroupSpace Group = Fm3m= Fm3mAtomAtom Site x y zSite x y zAnionAnion 4a4a 0 0 00 0 0Oct HoleOct Hole 4b4b

AntifluoriteAntifluorite structure (Nastructure (Na22O)O)

(Tetrahedral Holes)(Tetrahedral Holes)

CCP Anion Array & Tetrahedral HolesCCP Anion Array & Tetrahedral Holes

Space GroupSpace Group = Fm3m= Fm3mAtomAtom Site x y zSite x y zAnionAnion 4a4a 0 0 00 0 0TetrTetr HoleHole 8c8c

ZincZinc BlendeBlende Structure (Structure (ZnSZnS))(50% Tetrahedral Holes)(50% Tetrahedral Holes)

Space GroupSpace Group = F43m= F43mAtomAtom Site x y zSite x y zAnionAnion 4a4a 0 0 00 0 0TetrTetr HoleHole 4b4b

AntifluoriteAntifluorite structure (Nastructure (Na22O)O)(100% Tetrahedral Holes)(100% Tetrahedral Holes)


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SphaleriteSphalerite ((ZnSZnS))((ccpccp, 50%, 50% TetrTetr. Holes Filled). Holes Filled)

Space GroupSpace Group = F= F--43m43mAtomAtom Site x y zSite x y z

ZnZn 4a4a 0 0 00 0 0SS 4c4c

CationCation CoordCoord.. TetrahedronTetrahedronAnionAnion CoordCoord.. TetrahedronTetrahedronConnectivityConnectivity Corner sharing Oct.Corner sharing Oct.

WurtziteWurtzite (ZnO)(ZnO)((hcphcp, 50%, 50% TetrTetr. Holes Filled). Holes Filled)

Space GroupSpace Group = P6= P633mcmcAtomAtom Site x y zSite x y z

ZnZn 2b2b 1/3 2/3 01/3 2/3 0OO 2b2b 1/3 2/3 0.381/3 2/3 0.38

CationCation CoordCoord.. TetrahedronTetrahedronAnionAnion CoordCoord.. TetrahedronTetrahedronConnectivityConnectivity Corner sharing Oct.Corner sharing Oct.

Space GroupSpace Group = P6= P633/mmc/mmcAtomAtom Site x y zSite x y zAnionAnion 2c2c 1/3 2/3 1/41/3 2/3 1/4Oct HoleOct Hole 2a 0 0 02a 0 0 0

Hexagonal Close Packed Anion ArrayHexagonal Close Packed Anion Array

Nickel Arsenide Structure (Octahedral Holes)Nickel Arsenide Structure (Octahedral Holes)


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Space GroupSpace Group = P6= P633/mmc/mmcAtomAtom Site x y zSite x y zAnionAnion 2c2c 1/3 2/3 1/41/3 2/3 1/4TetrTetr HoleHole 4f4f 1/3 2/3 z1/3 2/3 z

z ~ 0.632z ~ 0.632

HCP Anion ArrayHCP Anion Array -- Tetrahedral HolesTetrahedral HolesNo such structure existsNo such structure exists

XX--ray Diffractionray Diffraction


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Diffraction DemoDiffraction Demo

Take home messageTake home message The diffraction pattern isThe diffraction pattern is

related but not equal to therelated but not equal to thegrid patterngrid pattern

Diffraction is most effectiveDiffraction is most effectivefor monochromatic light whosefor monochromatic light whosewavelength is similar to thewavelength is similar to thespacing ofspacing of slitsslits

For crystals XFor crystals X--rays have arays have awavelength comparable towavelength comparable to

spacingsspacings of atomsof atoms

PowderPowder DiffractometerDiffractometer

HorizontalDiffraction

Circle

DivergentX-ray

Source

Detector

DivergenceSlit

AntiscatterSlit

Receiving Slit

HorizontalSoller Slits

Sample (VerticalFlat Plate)


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Single CrystalSingle CrystalDiffractionDiffraction

IncidentIncidentBeamBeam

DiffractedDiffractedBeamBeam

IncidentIncidentBeamBeam

DiffractedDiffractedBeamBeam

PowderPowderDiffractionDiffraction

In powder diffraction only asmall fraction of the crystals(shown in blue) are correctly

oriented to diffract.

BraggBraggs Laws Law


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XX--ray Powder Patternray Powder Pattern

Bond Distance from XRD PatternBond Distance from XRD Pattern

(Ex.(Ex. PbSPbS, Rock Salt Structure), Rock Salt Structure)

Braggs Law = 2dhkl sin hkl

1. Determine the 2-theta value, 2, and hkl values for a diffractionpeak

20 25 30 35 40 45 50 55 60 652Theta

0

0

0

0

0

0

0

Int.

2 = 38.7

h=2, k=2, l=0


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Bond Distance from XRD (Cont.)Bond Distance from XRD (Cont.)(Ex.(Ex. PbSPbS, Rock Salt Structure), Rock Salt Structure)

2. Use Braggs Law and the wavelength of radiation (typically =1.541 ) to calculate dhkl

= 2dhkl sin hkldhkl = /(2 sin hkl)

dhkl = 1.541 /{2 sin (38.7/2)} = 2.10

3. The interplanar spacing, dhkl, is related to the unit cell size. For acubic crystal:

a = (h2 + k2 + l2)1/2 dhkl

a = (22 + 22 + 02)1/2(2.10 ) = 5.94

Bond Distance from XRD (Cont.)Bond Distance from XRD (Cont.)

(Ex.(Ex. PbSPbS, Rock Salt Structure), Rock Salt Structure)

4. Now that we know the unit cell size, the Pb-S distance can bedetermined from the unit cell using simple geometry.

dist (Pb-S) = a/2

dist (Pb-S) = 5.94 /2

dist (Pb-S) = 2.97


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Peak PositionsPeak Positions

Braggs Law = 2dhkl sin hklThe distance between different planes of atoms in a crystal, dhkl,where h, k and l are integers that correspond to different planesCubic:

1/d2 = (h2 + k2 + l2)/a2

Tetragonal:

1/d2 = {(h2 + k2)/a2} + (l2/c2)

Orthorhombic:

1/d2 = (h2/a2) + (k2/b2) + (l2/c2)

Hexagonal:

1/d2 = (4/3){(h2 + hk + k2)/a2} + (l2/c2)

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