22-1

States of Matter

SM VIII (post)

Crystallography

Experimental Basis Crystal Systems Closed Packing Ionic Structures skip in 2008 Ref 12: 8 Prob in-text: 12: 9, 10

end: 12: 59, 61 - 63, 65a,b, 70

Adv Rdg 16: 1,2, 4,5, 8,9

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Experimental Basis is X-ray diffraction;

see HT Fig. 21.1, Pet. Fig. 12.43 & 12.44.

Recall: diffraction & interference occur

if slits (openings) in a barrier

and wavelength are of similar size.

This requirement is satisfied when considering

distances between planes in crystals & (similar to distances between atoms)

wavelength of X-rays:

λ of X-rays atomic distances

~ 10 –10 m ~ 100 pm

0.1 nm 0.1 nm

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HT Fig. 22.1 Block Diagram of X-Ray Crystallography

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Pet. Fig. 12.43 Experimental Basis of Crystallography

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Pet. Fig. 12.44 Diffraction by crystal planes

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General Comments

• crystals have 3D repeating pattern of

molecular arrangements

• “unit cell” is smallest repeating unit • whole crystal can be built

by stacking unit cells in all directions,

without gaps/voids

Illustration:

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Crystal Systems 7 systems exist

see HT Fig. 22.2

all unit cells are parallelepipeds, (six faces, opposite faces parallel)

but differ in length/angle relationships

in addition, different “lattice types” may exist

for each system,

e.g., simple,

body centered,

face centered

altogether: 14 lattice types

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HT Fig. 22.2 Crystal Systems

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Crystal systems …

In CHEM 101/3: deal mostly w/ cubic system;

(once w/ hexagonal system)

See Pet. Fig. 12.38

Distinguish

• simple cubic

• body centered cubic, bcc

• face centered cubic, fcc

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Pet. Fig. 12.38 Cubic Crystal Systems

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Counting Particles in a Unit Cell (see Pet. Fig. 12.42)

• basic assumption: atoms/molecules are hard spheres in touching contact

• counting particles in unit cell:

body center: 1 (not shared)

face center: 12 (shared by 2 cells)

edge center: 14 ( shared by 4 cells)

corner : 18 (shared by 8 cells)

Total count of particles per unit cell :

Simple cubic: 8 x 1/8 = 1

bcc : (8 x 1/8) + 1 = 2

fcc : (8 x 1/8) + (6 x 1/2) = 4

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Pet. Fig. 12.42 Counting in Unit Cells

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Coordination Number = # of nearest neighbors, in touching contact

Practice:

Analyze Pet. Fig. 12.38 &

12.42

simple cubic: 6

bcc: 8

fcc 12

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Close Packing

= max. occupation of space by spherical objects, atoms in particular.

(see Pet. Fig. 12.39)

1st layer (“a”) of spheres:

6 spheres surround a central sphere

in hexagonal fashion

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Pet. Fig. 12.39 Close Packing

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close packing …

2nd layer (b)

spheres center on dips (dimples, indentations)

of first layer

notice: only 1/2 of dips are filled

leaving 2 types of dips in the 2nd layer:

type “c”, can see through layer a (octahedral in Pet.)

type “h”, can see tops of layer a (tetrahedral in Pet.)

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close packing …

3rd layer

• if spheres center on type “c” dips:

layers repeat abcabcabc…

= cubic closest packing (ccp),

has face centered cubic (fcc) unit cell

see Pet. Fig. 12.40

coordination # = 12 , # per unit cell = 4

• if spheres center on type “h” dips:

layers repeat ababab…

= hexagonal closest packing (hcp),

has “body centered” hexagonal (“bch”) unit cell (this statement is not quite true; the “central yellow” atom is somewhat off center) see Pet. Fig. 12.41

coordination # = 12 , # per unit cell = 2

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blank, deliberately

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Pet. Fig. 12.40 Cubic Closest Packing (ccp)

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Pet. Fig. 12.41 Hexagonal Closest Packing (hcp)

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Packing Efficiency (PE) • unit cell is only partially occupied by atoms

(“hard spheres”)

• the rest is empty space (voids)

• PE = space occupiedvol. of unit cell x 100%

• can be determined by simple geometric

calculations; see HT Fig. 22.3, 22.4, 22.5

Summary:

system PE (%)

simple cubic 52.4

bcc 68.0

closest packing

ccp (= fcc) 74.0

hcp (= bch)

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HT Fig. 22.3 PE in Simple Cubic Unit Cell

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HT Fig. 22.4 PE in bcc Unit Cell

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HT Fig. 22.5 PE in fcc Unit Cell

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Applied Problems Need the following as background

• vol. of unit cell → PE vol. of particles

• crystal type # of particles per unit cell

• mass of unit cell

= (# of particles) x (mass of 1 particle) (m = MM/NA)

• density = mass of unit cell

volume of unit cell

Practice: Sample Final, #11

Pet. 12: 9, 10, 62 - 66

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Practice Germanium, Ge, crystallizes in a (complex) cubic system; Density, D, of Ge = 5.36 g/cm3 ; Length of unit cell edge = 565 pm. Q. What is the # of Ge atoms per unit cell? Approach a.) Determine volume of unit cell, V. Use cubic formula, V = a3 b.) Determine mass of unit cell, from density. c.) Determine mass of a Ge atom. d.) Relate mass of 1 Ge atom to mass of unit cell → # of atoms per unit cell

a.) V = (565 pm)3 = (565 pm x 1 x 10-12 m

1 pm )3 = 1.804 x 10-28 m3

b.) mass of unit cell: D = m/V; m = D V

m = 5.36g

cm3 x (100cm

m )3 x 1.804 x 10-28 m3= 9.67 x 10-22 g

c.) mass of Ge atom:

m (Ge) = MM NA

= 72.6 g/mol

6.02 x 1023 atoms/mol = 1.206 x 10-22 g/atom

d.) # of Ge atoms per unit cell:

mass of unit cellmass of Ge atom =

9.67 x 10-22 g 1.206 x 10-22 g = 8.02, close to 8.

∴there are 8 Ge atoms in a unit cell,

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Ionic Crystal Structures

Skip in 2008

• type of structure depends, to large extent, on

ratio of ionic radii (usually cation/anion)

• 2 important cases:

1.) if ratio between 0.4 - 0.7,

get “rock- salt” (=NaCl) structure,

e.g., NaCl, RbI, CaO, AgCl

2.) if ratio between 0.7 – 1.0,

get CsCl structure,

e.g., CsCl, CsI

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NaCl Structure • see Pet. Fig. 12.48

• anions occupy ccp (fcc; abcabc ...) positions;

• somewhat expanded (puffed up);

no longer touching each other

• “octahedral holes” become large enough

to accommodate cations for a tight fit (the center of the fcc cell is such an “octahedral hole”)

• can also be seen as 2 interpenetrating fcc systems

• unit cell has 4 Cl– & 4 Na+ ions

• coordination # (# of cations touching anions & vv.):

= 6

i.e., Na+ has 6 Cl– neighbors;

Cl– has 6 Na+ neighbors

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Pet.Fig.12.48 NaCl Unit Cell

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CsCl Structure

• see Pet. Fig. 12.49

• anions occupy simple cubic (“primitive”)

positions;

• somewhat expanded

• cations fit into body center positions

• can also be perceived as 2 inter-penetrating

simple cubic networks

• unit cell has 1 Cl– & 1 Cs+ ion (mistake in pre notes)

• coordination # = 8

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Pet.Fig. 12.49 CsCl Unit Cell