Zumdahls Chapter 10 and Crystal Symmetries

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Zumdahls Chapter 10 and Crystal Symmetries

• Liquids
• Solids

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Contents

• Intermolecular Forces
• The Liquid State
• Types of Solids
• X-Ray analysis
• Metal Bonding
• Network Atomic Solids
• Semiconductors

• Molecular Solids
• Ionic Solids
• Change of State
• Vapor Pressure
• Heat of Vaporization
• Phase Diagrams
• Triple Point
• Critical Point

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Intermolecular Forces

• Every gas liquifies.
• Long-range attractive forces overcome thermal
  dispersion at low temperature. ( Tboil )
• At lower T still, intermolecular potentials are
  lowered further by solidification. ( Tfusion )
• Since pressure influences gas density, it also
  influences the T at which these condensations
  occur.
• What are the natures of the attractive forces?

4
London Dispersion Forces

• AKA induced-dipole-induced-dipole forces
• Electrons in atoms and molecules can be polarized
  by electric fields to varying extents.
• Natural electronic motion in neighboring atoms or
  molecules set up instantaneous dipole fields.
• Target molecules electrons anticorrelate with
  those in neighbors, giving an opposite dipole.
• Those quickly-reversing dipoles still attract.

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Induced Dipolar Attraction

• Strengths of dipolar interaction proportional to
  charge and distance separated.
• So weakly-held electrons are vulnerable to
  induced dipoles. He tight but Kr loose.
• Also l o n g molecules permit charge to
  separate larger distances, which promotes
  stronger dipoles. Size matters.

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Permanent Dipoles

• Non-polar molecules bind exclusively by London
  potential ? R6 (short-range)
• True dipolar molecules have permanently shifted
  electron distributions which attract one another
  strongly ? R4 (longer range).
• Gaseous ions have strongest, longest range
  attraction (and repulsion) potentials ? R2.
• Size being equal, boiling Tpolar gt Tnon-polar

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Strongest Dipoles

• Hydrogen bonding potential occurs when H is
  bound to the very electronegative atoms of N, O,
  or F.
• So H2O ought to boil at about 50C save for the
  hydrogen bonds between neighbor water molecules.
• Its normal boiling point is 150 higher!

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The Liquid State (Hawaii?)

• The most complex of all phases.
• Characterized by
• Fluidity (flow, viscosity, turbulence)
• Only short-range ordering (solvation shells)
• Surface tension (beading, meniscus, bubbles)
• Bulk molecules bind in all directions but
  unfortunate surface ones bind only
  hemispherically.
• Missing attractions makes surface creation costly.

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Type of Solids

• While solids are often highly ordered structures,
  glass is more of a frozen fluid.
• Glass is an amorphous solid. without shape
• In crystalline solids, atoms occupy regular array
  positions save for occasional defects.
• Array composed by stacking of the smallest unit
  cell capable of reproducing full lattice.

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Types of Lattices

• While there are quite a few Point Groups and
  hundreds of 2D wallpaper arrangments, there are
  only SEVEN 3D lattice types.
• Isometric (cubic), Tetragonal, Orthorhombic,
  Monoclinic, Triclinic, Hexagonal, and
  Rhombohedral.
• They differ in the size and angles of the axes of
  the unit cell. Only these 7 will fill in 3D
  space.

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Isometric (cubic)

• Cubic unit cell axes are all
• THE SAME LENGTH
• MUTUALLY PERPENDICULAR
• E.g.,Fools Gold is iron pyrite, FeS2, an
  unusual 4 valence.

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Tetragonal

• Tetragonal cell axes
• MUTUALLY PERPENDICULAR
• 2 SAME LENGTH
• E.g., Zircon, ZrSiO4. This white zircon is a
  Matura Diamond, but only 7.5 hardness.
• Real diamond is 10.

Diamonds are not tetragonal but
rather face-centered cubic.
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Orthorhombic

• Orthorhombic axes
• MUTUALLY PERPENDICULAR
• NO 2 THE SAME LENGTH
• E.g., Aragonite, whose gem form comes from the
  secretion of oysters its CaCO3.

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Monoclinic

• Monoclinic cell axes
• UNEQUAL LENGTH
• 2 SKEWED but PERPENDICULAR TO THE THIRD
• E.g., Selenite (trans. the Moon) a fully
  transparent form of gypsum, CaSO42H2O

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Triclinic

• Triclinic cell axes
• ALL UNEQUAL
• ALL OBLIQUE
• E.g., Albite, colorless, glassy component of this
  feldspar, has a formula NaAlSi3O8.
• Silicates are the most common minerals.

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Hexagonal

• Hexagonal cell axes
• 3 EQUAL C2
• PERPENDICULAR TO A C6
• E.g., Beryl, with gem form Emerald and formula
  Be3Al2(SiO3)6
• Diamonds are cheaper than perfect emeralds.

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Rhombohedral
_ 3

• Rhombohedral axis
• CUBE stretched (or squashed) along its diagonal.
  (abc)
• DIAGONAL is bar 3
• rotary inversion
• E.g., Quartz, SiO2, the base for amethyst with it
  purple color due to an Fe impurity.

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Identification (Point Symmetry Symbols)

• Lattice Type
• Isometric
• Tetragonal
• Orthorhombic
• Monoclinic
• Triclinic
• Hexagonal
• Rhombohedral

• Essential Symmetry
• Four C3
• C4
• Three perpendicular C2
• C2
• None (or rather i all share)
• C6
• C3

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Classes

• Although theres only 7 crystal systems, there
  are 14 lattices, 32 classes which can span 3D
  space, and 230 crystal symmetries.
• Only 12 are routinely observed.
• Classes within a system differ in the symmetrical
  arrangement of points inside the unit cube.
• Since it is the atoms that scatter X-rays, not
  the unit cells, classes yield different X-ray
  patterns.

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Common Cubic Classes

• Simple cubic
• Primitive P
• Body-centered cubic
• Interior I
• Face-centered cubic
• Faces F
• Capped C if only on 2 opposing faces.

• BCC
• FCC

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Materials Density

• Density of materials is mass per unit volume.
• Unit cells have dimensions and volumes.
• Their contents, atoms, have mass.
• So density of a lattice packing is easily
  obtained from just those dimensions and the
  masses of THE PORTIONS OF atoms actually WITHIN
  the unit cell.

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Counting Atoms in Unit Cells

• INTERIOR atoms count in their entirety.
• FACE atoms count for only the ½ inside.
• EDGE atoms count for only the ¼ inside.
• CORNER atoms are only 1/8 inside.

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Golds Density from Unit Cell

• Gold is FCC.
• a b c 4.07 Å
• Au atoms in cell
• 1/8 (8) ½ (6) 4
• M 4(197 g) 788 g
• Volume NAv cells
• (4.07?1010 m)3 ? Nav
• 3.90?105 m3 39.0 cc
• ? M / V 20.2 g/cc

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Bravais Lattices

• 7 lattice systems P, I, F, C options
• P atoms only at the corners.
• I additional atom in center.
• C pair of atoms capping opposite faces.
• F atoms centered in all faces.
• Totals 14 types of unit cells from which to
  tile a crystal in 3d, the Bravais Lattices.
• Adding point symmetries yields 230 space groups.

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New Names for Symmetry Elements

• What we learned as Cn (rotation by 360/n), is
  now called merely n. ? 3s a 3-fold axis.
• Reflections used to be ? but now theyre m (for
  mirror). So mmm means 3 ? mirrors.
• In point symmetry, Sn was 360/n and then ? but
  now it is just n, still a 360/n but now followed
  by an inversion (which is now 1).

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Triclinic Lattice Designation

• Triclinic ? ? ? ? ?
• All 7 lattice systems have centrosymmetry, e.g.,
  corner, edge, face, center inversion pts!
• Designation 1

• These are inversion points only because the
  crystal is infinite!
• While all 7 have these, triclinic hasnt other
  symmetry operations.
• Its 1 means inversion.

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Cubic (isometric) Designation

• The principal rotation axes are 4, but it is
  the four 3 axes that are identifying for cubes.
• The 4fold axes have an m ? to each.
• Each 3fold axis has a trio of m in which it
  lies. All 3 to be shown.

• The cube is m 3 m
• All its other symmetries are implied by these.

3
m
m
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The Three Cubic Lattices

• Where before we called them simple,
  body-centered, and face-centered cubics, the are
  now P m3m, I m3m, and F m3m, resp.
• The cubic has the highest and the triclinic the
  lowest symmetry. The rest of the Bravais
  Lattices fall in between.
• We will designate only their primitive cells.
• It will help when we get to a real crystal.

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Ortho vs. Merely Rhombic

• Orthorhombic all 90 but a ? b ? c. Trivial.
• Its mmm because

• Rhombohedral all ?s but ? 90 a b c
• Its 3m because

m
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Last of the Great Rectangles

• Tetragonal all 90 and a b ? c
• Principle axis is 4 which is ? m
• But it is also to mm
• So it is designated as 4/m mm
• Abbreviated 4/mmm

4
m
m
m
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Natures Favorite for Organics

• Monoclinic
• a ? b ? c
• ? ? 90 lt ?
• Then b is a 2-fold axis and ? to m
• So it is 2/m
• b is a 2 because the crystal is infinite.

m
2
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(finally) Hexagonal
So it is 6/m mm

• Hexagonal refers to the outlined rhomboid (
  ?120 ) of which there are six around the
  hexagon! So a 6
• That 6 has a ? m and two mm.
• m is a mirror because the crystals infinite.

6
m
m
m
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Lattice Notation Summary

• Lattice Type
• Isometric Cubic
• Tetragonal
• Orthorhombic
• Monoclinic
• Triclinic
• Hexagonal
• Rhombohedral

• Crystal Symmetries
• m 3 m ( m4 3 )
• 4 / m mm (4 ? m )
• mmm (m ? m ? m)
• 2 / m ( 2 ? m)
• 1 (invert only)
• 6 / m mm (6 ? m )
• 3 m ( 3 )

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X-ray Crystal Determination

• Since crystals are so regular, planes with atoms
  (electrons) to scatter radiation can be found at
  many angles and many separations.
• Those separations, d, comparable to ?, the
  wavelength of incident radiation, diffract it
  most effectively.
• The patterns of diffraction are characteristic of
  the crystal under investigation!

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Diffractions Source

• X-rays have ? ? d.
• X-rays mirror reflect from adjacent planes in the
  crystal.
• If the longer reflection exceeds the shorter by
  n?, they reinforce.
• If by (n½)?, cancel!
• 2d sin? n? , Bragg

reinforced
d sin?
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Relating Cell Contents to ?

• Atomic positions replicate from cell to cell.
• Reflection planes through them can be drawn once
  symmetries are known.
• Directions of the planes are determined by
  replication distances in (inverse) cell units.
• Interplane distance, d, is a function of the
  direction indices (Miller indices).

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Inverse Distances

• The index for a full cell move along axis b is 1.
  Its inverse is 1.
• That for ½ a cell on b is ½. Its inverse is 2.
• Intersect on a parallel axis is ?! Its inverse
  makes more sense, 0.
• Shown is (3,2,0)

c
a
b
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Interplane Spacings (cubic lattice)

• Set of 320 planes at right (looking down c).
• Their normal is yellow.
• (h,k,l) (3,2,0)
• Shifts are a/h, b/k, c/l
• Inverses h/a, k/b, l/c
• Pythagoras in inverse!
• d2hkl (h/a)2 (k/b)2 (l/c)2 for use in
  Bragg

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Bragg Formula
c
b
a

• 2 sin? / ? 1 / d (conveniently inverted)
• Let the angles opposite a, b, and c be ?, ?, and
  ?. (All 90 if cubic, etc.)
• Then Bragg for cubic, orthorhombic, monoclinic,
  and triclinic becomes
• 2 sin? / ? (h/a)2 (k/b)2 (l/c)2
  2hkcos?/ab 2hlcos?/ac 2klcos?/bc ½

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Unit Cell Parameters from X-ray

• Triclinic
• Monoclinic
• Orthorhombic
• Tetragonal
• Rhombohedral
• Hexagonal
• Cubic

• a ? b ? c ? ? ? ? ?
• a ? b ? c ? ? 90 lt ?
• a ? b ? c ? ? ? 90
• a b ? c ? ? ? 90
• a b c ? ? ? ? 90
• a b c ? ? 90 ? 120
• a b c ? ? ? 90

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New Space Symmetry Elements

• Glide Plane
• Simultaneous mirror with translation to it.
• a, b, or c if glide is ½ along those axes.
• n if by ½ along a face.
• d if by ¼ along a face.

• Screw axis, nm
• Simultaneous rotation by 360/n with a m/n
  translation along axis.

cell 2
32 screw
cell 1
a glide
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Systematic Extinctions

• Both space symmetries and Bravais lattice types
  kill off some Miller Index triples!
• Use missing triples to find P, F, C, I
• E.g., if odd sums hkl are missing, the unit
  cell is body-centered and must be I.
• Use them to find glide planes and screw axes.
• E.g., if all odd h is missing from (h,k,0)
  reflections, then there is an a glide (by ½) ? c.
• http//tetide.geo.uniroma1.it/ipercri/crix/struct.
  htm

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Natures Choice Symmetries

• 36.0 P 21 / c monoclinic
• 13.7 P 1 triclinic
• 11.6 P 21 21 21 orthorhombic
• 6.7 P 21 monoclinic
• 6.6 C 2 / c monoclinic
• 25.4 All (230 5 ) 225 others!
• 75 these 5 90 only 16 total for organics.
• Stout Jensen, Table 5.1

_
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Packing in Metals
A B A hexagonal close pack
A B C cubic close pack
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Relationship to Unit Cells
Is FCC
A B C cubic close pack
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ABA (hcp) Hexagonal
The white lines indicate an elongated hexagonal
unit cell with atoms at its equator and an offset
pair at ¼ ¾.
If we expand the cell to see its shape, we get a
diamond at both ends3 make a hexagon
whose planes are 90 to the sides of the
(expanded) cell.
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Alloys (vary properties of metals)

• Substitutional
• Heteroatoms swap originals, e.g., Cu/Sn (bronze)
• Intersticial
• Smaller interlopers fit in interstices (voids) of
  metal structure, e.g., Fe/C (steels)
• Mixed
• Substitutional and intersticial in same metal
  alloy, e.g., Fe/Cr/C (chrome steels)

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Phase Changes

• Phase changes mean
• Structure reorganization
• Enthalpy changes, ?H
• Volume changes, ?V
• Solid-to-Solid
• E.g., red to white P
• Solid-to-Liquid
• ?Hfusion significant
• ?Vfusion small

• Solid-to-Gas
• ?Hsublimation very large
• ?Vsublimation very large
• Liquid-to-Gas
• ?Hvaporization large
• ?Vvaporization very large
• All occur at sharply defined P,T, e.g., P 1 bar
  Tfusion normal FP

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Heating Curve (1 mol H2O to scale)
Csteam ?T
60
steam heats
water becomes steam
heat (kJ)
?Hvaporization
ice warms
water warms
?Hfusion
Cice ?T
Cwater ?T
ice becomes water
0
0C
100C
T
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Equilibrium Vapor Pressure, Peq

• At a given P,T, the partial pressure of vapor
  above a volatile condensed phase.
• If two condensed phases present, e.g., solid and
  liquid, the one with the lower Peq will be the
  more thermodynamically stable.
• The more volatile phase will lose matter by gas
  transfer to the less (more stable) one because
  such equilibrium are dynamic!

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Liquid Vapor Pressures

• Measure the binding potential in the liquids.
• Vary strongly with T since the fraction of
  molecules energetic enough at T to break free is
  e?Hvap / RT.
• Will be presumed ideal.
• Equal 1 bar at normal boiling point, Tboil.
• Decrease as liquid is diluted with another.

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Temperature Dependence of P

• The thermodynamic relationship between Gibbs Free
  energy, G, and gas pressure, P, can be shown to
  define P as a function of T.
• Well see this in Chapter 6.
• PT / PT e?Hvap / RT / e?Hvap / RT or
• Just the ratio of molecules capable of overcoming
  ?Hvap
• P P e ?Hvap / R ? (1/T ) (1/T )
• The infamous Clausius-Clapeyron equation.

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Raoults Law PA varies with XA

• Ideal solutions composed of molecules with AA
  binding energy the same as AB.
• Vapor pressures are consequence of the
  equilibrium between evaporation and condensation.
  If evaporation slows, P falls.
• But only XA of liquid at surface is A, then its
  evaporation rate varies directly with XA.
• PA P A XA and PB P B XB
• Where P means P of pure (X1) liquid.

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Consequences of Ideality

• Measured vapor pressures predict mole fractions
  (hence concentrations) of solutes.
• Pressure solution equilibria predict solute
  solution equilibria.
• While gases are adequately ideal, solutions
  almost never are ideal.
• Positive deviations of P from PX imply AB
  interactions are not as strong as AA ones.

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Pure Compound Phase Diagram

• Predicts the stable phase as a function of Ptotal
  and T.
• Characteristic shape punctuated by unique points.
• Phase equilibrium lines
• Triple Point
• Critical Point

P
Liquid
Solid
Gas
T
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Phase Diagram Landmarks

• Triple Point (PT,TT)
• where SLG coexist.
• Critical Point (PC,TC)
• beyond this exist no liquid/vapor property
  differences.
• P 1 bar
• Normal fusion TF and boiling TB points.

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Inducing Phase Changes

• Below PT or above PC
• Deposition of gas to solid induced by dropping T
  or raising P
• Sublimation is reverse.
• Between PT and PC
• Liquid condensation vs. vaporization.
• Normally, pressure on liquid solidifies it
  (unless ?solid lt ?liquid)

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Impure (solution) Phase Diagram

• Adding a solute to a pure liquid elevates its
  Tboil by lowering its vapor pressure.
• (Raoults Law)
• It also stabilizes liquid against solid (lowers
  Tfusion)
• Lower P wins, remember?
• Click to see the new liquid regions and
• 2 colligative properties in 1!

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ClausiusClapeyron Lab Fix

• dP/dT P?Hvap/RT 2
• from thermodynamics
• PPe?H/R(1/T)(1/T )
• But only if ?H ? f(T)
• If ?H a bT
• where b related to CP
• PP(T/T )b/R ea/R(1/T)(1/T )
• assumes only CP are fixed.
• A better approximation.

Clausius Clapeyron
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ClausiusClapeyron Parameters

• ?H a bT
• b (?HBP?H) / (BP298)
• a ?H 298 b

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End of Presentation

• Last modified 30 June 2001