Materials Science for Chemical EngineersK. - K. Koo
3. ATOMIC AND IONIC ARRANGEMENTS
Arrangements of atoms and ions play an important role in determining the microstructure and properties of materials.
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3-1.Terminologies
No order/Short-Range Order/Long Range Order:
- Short range order: The special arrangement of the atoms extends only to the
atom’s nearest neighbors.
- Long-range order: the special atomic arrangement extends over much larger
length scales ∼ >100nm.
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Crystal:
Material in which the atoms are situated in a repeating or periodic array over
large atomic distance (three dimensional long-range order).
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Figure 3.3 (a) Photograph of a silicon single crystal. (b) Micrograph of a
polycrystalline stainless steel showing grains and grain boundaries
Crystal Lattice ( Space Lattice or Point Lattice ):
the repeating geometric pattern of arrangement of atoms, ions, groups of
atoms, or molecules in a crystal.
Unit Cell:
a small part of the crystal lattice that exhibits all of the essential features of
the lattice - geometry and atom positions
Crystal
- 7 crystal systems and the 14 crystal lattices ( Bravais lattices )
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Lattice Parameters (Lattice constants):
The lattice parameter, which described the size and shape of the unit cell,
include the lengths of the unit cell edges and angles between crystallographic
axes.
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Simple Cubic Body-centered Cubic Face-centered Cubic
SC: a = 2r
FCC: Face diagonal: f 2 = (4r) 2 = 2a
2
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Atomic Sizes: Average Distance between nuclei in solid can be measured by
x-ray diffraction.
atomic radius ( Rat = 1/2 dat ) = 1/2 interatomic distance between neighboring
atoms in solid
covalent radius ( Rcov = 1/2dcov ) = 1/2 intermolecular distance in molecule
Ionically Bonded Crystals ( NaCl, etc. )
dion = R + r = interatomic distance between neighboring ions
R = ionic radius of larger ion ( usually anion )
r = ionic radius of smaller ion ( usually cation )
Cations (r < Ratom): Na Ratom = 0.186 nm, Na+ r = 0.098 nm
Anions (R > Ratom): Cl Rcov = 0.098nm, Cl- R = 0.181nm
Anions R(nm) Cations r(nm)
F- 0.133 Li+ 0.078
Cl- 0.181 Na+ 0.098
Br- 0.196 K+ 0.133
Ca+2 0.106
Fe+2 0.087
Al+3 0.057
Position Fraction inside Cube
Atomic Densities
Linear Density of atoms in a crystal: (No. of atoms along line) /(length of
line) - linear densities are different for different family of directions.
Planar Density : (No. of atoms on a plane) / (area of plane)
- planar densities are different for different families of planes
Volumetric Density : (No. of atoms in a volume) / (volume)
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Ex) Simple cubic
Lattice Positions:
coordinates of a point in a system: x y z, a b c, h k l
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Lattice Directions
- a direction [a b c](square bracket) is indexed as a ray extending from the
origin through the location with the lowest integer index a, b, c.
- family of directions: set of crystal directions which are structurally
equivalent
[¯100], [0 ¯10], [00 ¯1]
tetragonal crystal : < 100 > = [100], [010], [¯100], [0¯10]
< 001 > = [001], [00¯1]
representing by the inverse of axial intercepts
- assigning Miller Indices
1. Draw plane inside unit cell
2. Take origin on corner(s) closest to but not in plane.
3. Imagine parallel plane passes through origin
4. Find distances along unit cell edges from origin to plane
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Direction in HCP cells are denoted with either the three-axis or four-axis.
hkil : h+k = -i
the three axis notation(h’k’l’) to the four-axis notation(hkil) for directions
h =(1/3)(2h’-k’)
k=(1/3)(2k’-h’)
i=(-1/3)(h’+k’)
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Coordination Number ( CN ):
number of nearest neighbor atoms or ions of a given atom or ion
1. Covalently bonded atoms: CN set mostly by directional character of bonds
and by valence requirements.
Diamond CNcarbon = 4
2. Non-directional bonded atoms (ionic, metallic, Van der Waals ):
CN set by PACKING considerations and CN depends on r/R ratio.
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Interstitial Sites
In any of the crystal structures, there are small holes between the usual atoms
into which smaller atoms may be placed. These locations are called interstitial
site.
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A. simple cubic ( SC )
C. face centered cubic ( FCC ) or cubic close packed ( CCP )
D. hexagonal close packed ( HCP )
E. diamond cubic ( DC )
(2) Coordination Number = 6
(3) Atomic Packing Factor ( APF ) = 0.52
(4) Interstitial(Coordination) Sites in cell : one of the 8 fold(cubic) site
(5) Polonium( 84
Po ), -Mn
Po: radioactive metallic element that is similar to tellurium and bismuth;
occurs in uranium ores but can be produced by bombarding bismuth with
neutrons in a nuclear reactor
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(1) atoms /unit cell = 8 (1/8)[corner] + 1(1)[center] = 2
(2) Atoms in contact along body diagonal
(3) CN = 8
(4) APF = 0.680
(5) No. of 6fold sites per cell = 6(1/2) + 12(1/4) = 6
No. of 4fold sites per cell = 24(1/2) = 12
(6) alkali metals (Li, . . ., Cs), Ba, α- Fe (ferrite), W, Mo, V, Cr
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C. Face Center Cubic (FCC) Structure ( Cubic Close Packed ( CCP)
Structure )
(1) atoms /unit cell = 8 (1/8) [corners] + 6(1/2) [faces] = 4
(2) Atoms in contact along face diagonal
(3) Coordination Number = 12
(4) APF = 0.74
(5) No. of 6 fold sites per cell = 1(1) center + (12)(1/4) edges = 4
No. of 4 fold sites per cell = 8(1) = 8
(6) inert gases (when solidified) : Ne, Ar, Kr, Xe
metals: Al, g - Fe ( austenite ), Cu, Ag, Au, etc.
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D. Hexagonal Close Packed (HCP) Structure
(1) atoms in cell = 12(1/6) [corners] + 2(1/2)[face] + 3(1) [interior] = 6
atoms /unit cell = 2
metal : Be, Mg, Ti, Zn, Cd, etc.
(5) theoretically, c/a = 1.63: In many HCP crystals, c/a differs slightly from
ideal value of 1.63 predicted for spheres.
element c/a ratio
the diamond cubic unit cell - FCC
(1) no. of atoms in cell = 8 (1/8) [corners] + 6(1/2)[faces] + 4(1) [inside] = 8
(2) APF = 0.34
(3) CN = 4 (to satisfy covalent bonding for valance of 4 for C)
(4) C (diamond), Si, Ge, Sn(gray), etc.
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DC 4 0.340
SC 6 0.524
BCC 8 0.680
*Molecular crystals:
many small, discrete, covalently bonded molecules by van der Vaals or
hydrogen bonding
CO2: FCC
CO2 H2O
( (), (), (), () )
-Ability of any element or compound to crystallize into at least two different
crystalline structures, but they should be identical in the liquid or gaseous
state.
- if the material is an element, polymorphs are called allotropes.
-Differences in morphological appearance do not necessary reflect
polymorphism.
-Molecular conformation and packing caused by various intermolecular
forces, hydrogen bonding, van der Waals forces, interactions with solvents and
additives, etc..
-Which polymorphic form of a compound is formed depends on the
preparation and crystallization conditions: method of synthesis, temperature,
pressure, solvent, cooling and heating rate, seed crystals, etc.
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brookite(, orthorhombic)
specific heat(cal/g K) 0.12 0.18 at 300K
structure DC hexagonal FCC
Fullerene(C60)
1985, R. Smalley, R. Curl, J. Heath, S. O’Brien, H. Kroto (Rice Univ.)
The C60 fullerene in crystalline form
Carbon nanotube/1991, S. Iijima, (NEC, Nippon Electric Company)
2010.3.8( ) . 5.16
. .
7 . 300-390
( 51-66 )
(2010.10.10) 24.78 16
. ‘ ’ , 3800 ( 427
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Example 3. SiO2
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–> calcium carbonate(CaCO3)
Calcite Aragonite
1822 Mitscherlich arsenates, phosphates, sulfur polymorphism

(SO2) . S8
30 . 95.3 α Sα()
.
, ·· . α .
.
Arsenates are salts or esters of arsenic acid.
A phosphate, an inorganic chemical, is a salt of phosphoric acid.
1912 Von Laue discovery of x-ray diffraction
1965 McCrone “Every compound has different polymorphic forms, and
that, in general, the number of forms known for a given compound is
proportional to the time and money spent in research on that compound.”
solvent(water). Stoichiometric
guest). Nonstoichiometric
CH3OH·3C6H4(OH)2 OH 3
,
kleithron . ·
··· .
3
, M·6H2O(M=Ar, CH4, CO2, H2S, Cl2 ), M·17H2O(M=C3H8, CHCl3 )
.
A clathrate hydrate, in particular, is a special type of gas hydrate in which a
lattice of water molecules encloses molecules of a trapped gas. Large amounts
of methane naturally frozen in this form have been discovered both in
permafrost formations and under the ocean sea-bed. Researchers have begun
to investigate silicon and germanium clathrates for possible semiconducting,
superconducting, and thermoelectric properties.
- A clear distinction between polymorph and pseudopolymorphs has to be
made, for example, SA, SB,… for solvates, HA, HB,… for hydrates
The importance of study on polymorphism/psudopolymorphism
All physicochemical properties of solid state with the polymorphic/
psudopolyporphic form vary:
hardness, crystal shape, refractive index, viscosity,
optical and electrical properties,
solubility, dissolution rate, hygroscopicity,
Therefore understanding polymorphism/psudopolymorphism is very important
in material design of organic crystals with specific functions in many fields.
1) Bioavailability of Pharmaceuticals
2) non-linear optical materials
3) intercalation in organic crystals
4) melting behavior and morphology in disperse systems: cream, butter and
chocolate etc.
S/l separation, crystal morphology, purity, handling efficiency, etc..
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One of the essential amino acid, Pharmaceutical intermediate
Food intermediate-Aspartame (l-aspartyl-phenylalanine methyl ester)
(L-aspartic acid)
Monohydrate
Anhydrate
P2211
P21
Orthorhombic
Monoclinic
Needle
Flake
Monohydrate
Anhydrate
P2211
P21
Orthorhombic
Monoclinic
Needle
Flake
Ind.Eng.Chem.Res. 49, 12632-12637, 2010
I: Cubic, II: Tetragonal, III: Orthorhombic, IV: Orthorhombic, V: Tetragonal
Phase stabilized AN Caked AN
Isomers: difference compounds having the same formula.
a) constitutional isomers: the constituent atoms are connected in a different
order (ex. C2H6O: CH3CH2OH(ethanol) and CH3OCH3(dimethyl ether))
b) stereo isomers: differ only in the special arrangement of their constituent
atoms. – Enantiomers and diastereomers
- Enantiomers: Organic compounds that contain a chiral carbon usually have
two non-superposable structures. These two structures are mirror images of
each other and are, thus, commonly called enantiomorphs (enantio =
opposite ; morph = form), hence this structural property is now commonly
referred to as enantiomerism.
- Diastereomers have different physicochemical properties.
Ex) unsaturated dicarboxilic acid (HO2-C-CH=CH-C-O2H)
1) Maleic acid 2) Fumaric acid
(Cis-butenedioic acid) (Trans-butenedioic acid)
A chiral molecule is a type of molecule that has a non-superposable mirror image. The
presence of an asymmetric carbon atom is often the feature that causes chirality in
molecules. The term chirality is derived from the Greek word for hand, χειρ (kheir).
In chemistry, chirality usually refers to molecules. Two mirror images of a chiral molecule
are called enantiomers or optical isomers. Pairs of enantiomers are often designated as
"right-" and "left-handed".
General rule
- filling of interstitial sites in anion sublattice by cations
- Ionic Radii: sizes of anion and cation are determined by radius ratio
- Electrical Neutrality: filled to preserve electrical nutrality
A. MX type (Structure of Ionic Crystals with Two Atoms per Lattice Site)
B. MX2 type (Structure of Ionic Crystals with Three Atoms per Lattice Site)
C. M2X3 type
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A. MX type ( Ionic Crystals with two Atoms per Lattice Site )
A-1. Cs + Cl
- Structure - Simple Cubic
(2) CNCs+ = CNCl- = 8 ( r/R = 0.167 nm/0.181 nm = 0.92 )
(3) ions in contact along unit cell body diagonal
2r + 2R = √3 a
halides with large r/R - CsCl, CsBr, CsI, TlBr, NH4Cl
intermetallics - CuZn ( b'-brass ), AgMg, NaBi, AlNi, etc.
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A-2. Na+Cl- (Rock Salt) Structure - FCC
(1) FCC Cl- sublattice, 4 of the octahedral(6f) sites occupied by Na+
(2) no. of Na+ in cell = 8(1/8) corners + 6(1/2) faces = 4
no. of Cl- in cell = 12(1/4) edges + 1(1) center = 4 or vice versa!
(3) CN Na+ = CN Cl- = 6 ( r /R = 0.097nm/0.081nm = 0.536 )
(4) ions in contact along unit cell : a = 2R + 2r
(5) Compounds with Na+Cl- Structure:
most alkali halides - NaCl, LiF, KBr
Metal Oxide - MgO, CaO, FeO, NiO, etc./Misc. - LiH, AgBr, PbS, etc.
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A-3. Zinc Blende ( ZnS, Sphalerite ) and Wurzite ( ZnS ) Structure – FCC
(1) FCC S -2
sublattice
half of the tetrahedral (4 fold) sites occupied by Zn +2
(2) no. of S -2
in cell = 8(1/8) corners + 6(1/2) faces = 4
no. of Zn +2
in cell = 4(1) = 4 : interior
(3) CN Zn+2 = 4, CN S-2 = 4 ( r/R = 0.074nm/0.184nm = 0.402 )
(4) b = 4 R + 4 r = √3a
(5) compounds with Zinc Blende structure: ZnS(zinc blende), BeO, SiC ( Carborundum )
GaAs, AlP, InSb ( III-V Compounds)
*Wurzite ( ZnS) Structure : HCP
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B. MX2 type (Ionic Crystals with Three Atoms per Lattice Site )
B-1. Fluorite ( CaF2 ) and Antifluorite Structure – FCC
FCC
All the tetrahedral (4 fold) sites occupied by F-
(2) no. of Ca+ in cell = 8(1/8) corners + 6(1/2) faces = 4
no. of F- in cell = 8(1) = 8: inside
(3) CN Ca+2 = 8, CN F- = 4
(4) b = 4 R + 4 r = √3a
(5) Compounds with Fluorite Structure
CaF2, SrCl2, PbF2, ThO2, UO2, ZrO2
*Antifluorite structure : Li2O, Na2O, K2O
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1) Si +4
2) - Crystobalite is one of 7 atmospheric pressure SiO2 polymorphs
3) All contain SiO4 -4
coordination polyhedra.
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Perovskite ( CaTiO3 ) Structure - SC
CaTiO3 , BaTiO3 : Ferroelectric, Piezelectric
Figure The perovskite unit cell showing the A and B site cations and oxygen ions
occupying the face-center positions of the unit cell.
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Inverse spinel structure - FeMgFeO4, FeNi FeO4 (ferrimagnetic ceramics)
3.4 crystalline polymers
(translation )
Schoenflies symbol: used in spectroscopy and the symmetry of molecules
Hermann-Mauguin symbol: symmetry in crystals. international symbol
symmetry operation: axis, point, plane

symmetry element:
_______________________________________________________________ symmetry symmetry symbol
2. reflection mirror plane m σ
3. inversion inversion center 1 i
4. rotoinversion (roto)inversion axis n
rotoreflection rotoreflection axis Sp
_______________________________________________________________
3.6 Liquid Crystal
molecules less symmetry than in a crystal, but more symmetry than in a liquid
Crystal Liquid Crystal (mesomorphic phase) Isotropic Liquid
Types of liquid crystals: nematics, smectics A, B, C,...
discotics, cholesterics
low molecular weight materials: rod-like(or lath-like), disc-like molecules
polymeric materials: main chain, side chain
Applications: electro-optics ( LCD, ..), spatial light modulators, etc.
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