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--

Class 13

Crystallography and Crystal Structures continued

Suggested Reading M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007). Ch. 8, pp. 181-197;

. , pp. - Ch. 1 C. Kittel, Introduction to Solid State Physics, 3rdEdition, Wiley (1956).

Excerpt from ASM Metals Handbook.

Chs. 1 and 3 S.M. Allen and E.L. Thomas, The Structure of Materials, Wiley (1999).

102

s. an . ey, rysta s an rysta tructures, ey .

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Symmetry OperatorsSymmetry Operators

an initial position to a final position such that the initial

and final patterns are indistinguishable.

1. Translation* All crystals exhibittranslational symmetry.

.

3. Rotation

4. Inversion (center of symmetry)

Any other symmetry elementsmust be consistent withtranslational symmetry of the

5. Roto-inversion (inversion axis)

lattice

These are compound

6. Roto-reflection

7. Glide (translation + reflection)

(combinations of 1-4)

103

.

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Other Symmetry OperatorsOther Symmetry OperatorsTranslations interact with symmetry operators 1-6.

Results in the final two symmetry operators.

7. Glide Plane = Mirror Plane + Translation

a b c n d

8. Screw Axis = Rotation Axis + Translation

2

31, 32

1, 2, 3

61, 62, 63, 64, 65 104

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7. Glide Planes7. Glide Planes

Combine reflection and translation

Spacing btw. Lattice points

AA

A

Mirror plane(Glide plane)

Glide direction

Nomenclature for glide planes

Glide Direction Glide Magnitude Designation

, ,

face diagonal n

face diagonal d

When oin from a s ace rou to the arent oint rou all as bs

105

cs, ns, and ds are converted back into ms.

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8. Screw Axes8. Screw AxesP

Combine rotation

and translationm

nt mPn

(i.e., corresponds to the spacing btween lattice points)

, , ,

unit translation parallel to screw axisP P

pitch of screw axis /

#

t t m n P

m

of cells/steps back to starting position

P

n-fold rotation followed by a translation parallel to

the rotation axis Pby a vectort= mP/n.

P

106

Adapted from L.V. Azaroff, Introduction to Solids,

McGraw-Hill, New York, 1960, p. 22.

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2

2

m

mt P P

n 4

n

The operation of a 42 screw axis

parallel to thez direction; (a) atom A

=

P

t90

by 90; (b) the atom is translatedparallel to aby a distance oft= 2P/4,

i.e. P/2, to create atom B; (c) atom B

90

is rotated counter clockwise by 90

and translated parallel tozby a

distance oft= 2P/4, i.e. P/2, to give tatom ; atom s at z = , t e

lattice repeat, and thus is repeated atz

=0; (e) repeat of the symmetry

=t

90

.

Repeat of the symmetry operation on

atom D brings everything back into

coincidence, a requirement for

90

Final Configuration

107

. , , , , , , .symmetry; (f) a standard

crystallographic depiction of a 42screw axis viewed along the axis.

These can be difficult to visualize. A nice, but inverted

visualization is provided on page 133 of Allen & Thomas.Come see me if youd like a copy.

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Space GroupsSpace Groups

take a 3-dimensional periodic object into itself.

Describes the entire symmetry ofthe crystal (internal

in space.

There are 17 space groups in 2D

108

There are 230 space groups in 3D

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109R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, 2006

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Symmetry operators must leave every

22--D exampleD exampleType Lattice

with

symmetry elements .

Form for symmetry symbols

e.g., 2mm, 6mm, etc.

Consider the oblique lattice

2mm

180ab

Symmetry requires a 2-fold rotation axis

(i.e., diad) at the center of the cell.

Therefore, diads must be present on each

lattice point.

R. Tilley,

Crystals

and

Crystal

Structures

There will be implied diads centered on cell

sides.110

, o n

Wiley &

Sons,

Hoboken,NJ, 2006

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Point

Summary of what was presented on the previous page

symmetry

Implied

symmetry

111

M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge University Press (2007) p. 233

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Now consider the rectan ular

Symmetry of 2Symmetry of 2--D plane latticesD plane latticesSymmetry of 2Symmetry of 2--D plane latticesD plane latticesType Lattice

with

symmetry elements

primitive lattice

180b

2mm

2mm

As before, symmetry requirements

dictate that diads lie at the center of, ,

the center of the cell sides.

mirror symmetry along the cell edges

which necessitates the addition of

112

cell edge.

R. Tilley, Crystals and Crystal Structures, John Wiley &Sons, Hoboken, NJ, 2006

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Point

symmetry

See this site for high resolution diagrams and tables

http://img.chem.ucl.ac.uk/sgp/mainmenu.htm

Implied

symmetry

Implied

mirrors

113


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Point

symmetry

Implied mirrors

Implied

114


symmetry

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Form/Order of Symmetry SymbolsForm/Order of Symmetry Symbols

22--DDOrder of Hermann-Mauguin symbols for point groups:

Primary Secondary Tertiary

Lattice Primary Secondary Tertiary

Crystallographic position

Oblique, Rotation point --- ---

Retangular, , oc Rotation point [10] [01]

S uare Rotation oint 10 01

mp

op

t

Hexagonal, Rotation point [10]hp [01] [11] [11] [12] [21]

Example2 m m

115primary 2-fold rotation axis mirror to [10] mirror to [01]

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Ten plane point groups (2D) without an asymmetric objectTen plane point groups (2D) without an asymmetric object

1 2 3 4 6 m

2mm 3m 6mm4mm

116

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Ten plane point groups (2D) with an asymmetric objectTen plane point groups (2D) with an asymmetric object

1 2 3 4 6 m

m

2mm 3m 6mm4mmm mm

m

m

m

mmm

m

m

m

m

117

m

m

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Symmetry of 2Symmetry of 2--D plane latticesD plane latticesSymmetry of 2Symmetry of 2--D plane latticesD plane lattices

When all of the translations for the five plane lattices are

combined with the symmetry elements found in the 10

point groups plus glide lines, 17 plane groups are found.

e nes are -

equivalents of the 3-D

glide planes weintroduced earlier.

gure . e p ane groupspm an cm: a t e p ane att ce op; t e pattern orme y a ng t e mot o po nt

group m to the lattice in (a), representing the plane grouppm; (c) the plane lattice oc; (d) the pattern formed by adding

the motif of plane group m to the lattice in (c), representative of plane group cm. Mirror lines are heavy and glide lines in(d) are heavy dashed. From R. Tilley, Crystals and Crystal Structures, John Wiley & Sons, Hoboken, NJ, 2006.

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17 217 2--D plane groupsD plane groups17 217 2--D plane groupsD plane groups


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17 217 2--D plane groupsD plane groups17 217 2--D plane groupsD plane groups


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Higher-order?Higher-order?

The maximum degree of symmetry that a crystal

can ossess is defined b the lattice.

lattice:

Square Lattice 4mm (see next page)

Allowed motif symmetry 4 and 4mm

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Square + 4mmSquare + 4mm

Can we put a motif with 4mm symmetry on a square lattice?

4mm

Unit cell Primitive: p

o n symme ry o mo : mm

Space or Plane group: p4mm

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Symmetry in 3Symmetry in 3--DDSymmetry in 3Symmetry in 3--DD

-objects in terms of symmetry?

Hermann-Mauguin symbols for

point groups.


There is a specific order that we

use to assign symbols. This is

provided on the next slide.

Detailed descriptions of the point

123

octahedron are provided after

the table of symbols.R. Tilley, Crystals and Crystal Structures,

John Wiley & Sons, Hoboken, NJ, 2006

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33--DD


33--DD

Lattice Primary Secondary Tertiary

Crystallographic Position

Triclinic --- --- ---

Monoclinic [010], unique axis --- ---

[001], unique axis --- ---

b

c

Orthorhombic [100] [010] [001]

Tetagonal [001] [100],[010] [110],[110]

, , ,

Trigonal, Hexagonal axes [001] [100],[010],[110]

Hexagonal [001] [100],[010],[1 10] [1 10],[120],[2 10]

Cubic [100],[010],[001] [111],[111],[111],[111] [110],[110],[011],[011][101],[101]

124In 3In 3--D we formulate symbols just like 2D we formulate symbols just like 2--D!D!

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SPACE GROUPSSPACE GROUPS[We often use short notation. A few are shown below][We often use short notation. A few are shown below]

Triclinic space groups:

P1 and P1

Monoclinic space groups: (we consider only a few examples)

Short symbols: P2 P21 P2/m C2/c

Com lete s mbols: P 1 2 1 P 1 2 1 P 1 2/m 1 C1 2/c 1

Note from the last page

that we use one symbolfor monoclinic anyway

The same convention is used for

ymmetry a ong: aax s ax s cax s

r or om c space groups:Short symbols: P2221 Pma2 Pban Cmmm

Complete symbols: P2221 Pma2 P 2/b 2/a 2/n C2/m 2/m 2/m

More on this later. 125

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Synopsis of relationships between crystal systems, Bravaislattices, point groups, and space groups

7 Crystal Systems

Rotation

ReflectionInversion

Add points &centering

32 Point Groups 14 Bravais LatticesApplies to latticeApplies to motif on

lattice point

crew

Glide

126230 Space Groups

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Point group for a regular

tetrahedron

43mPoint group for a regular

octahedron


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R. Tille Cr stals and Cr stal

Structures, John Wiley & Sons,

Hoboken, NJ, 2006

128

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R. Tille Cr stals and Cr stal Structures John

129

Wiley & Sons, Hoboken, NJ, 2006

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Symmetry of the CubicSymmetry of the Cubic PP--LatticeLatticeSymmetry of the CubicSymmetry of the Cubic PP--LatticeLattice

4 3 2

Letter

symbol

P m m

100 111 110


130Adapted from W. Borchardt-Ott, Crystallography, 2

nd

Edition, Springer, Berlin, Germany, 1995, p. 99

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Galena (PbS)Galena (PbS)4 3 2

F 3

m m

m m

https://reader009.{domain}/reader009/html5/0419/5ad7c638a7a42/5ad7c64f2d5e3.jpg

htt ://www.minservice.com/cartsho 3/catalo /FERR0240 alena ._ _

http://en.wikipedia.org/wiki/File:Galena-unit-cell-3D-ionic.png

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Crystal FormsCrystal FormsCrystal FormsCrystal Forms

132

Adapted from W. Borchardt-Ott, Crystallography, 2nd

Edition, Springer, Berlin, Germany, 1995, p. 124

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230 Space Groups230 Space Groups

Describe complete symmetry of a crystal

internal and external .

.

,

Tables for Crystallography.

133

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Summary of CrystallographySummary of Crystallography

7 Crystal Systems

Based on the number of self consistent combinations of rotation axis in 3D-defines basic P

units cells.

14 Bravais Lattices

Arrangements of lattice points consistent with the above combinations of rotation axes i.e.some unit cells can also be F, I or C centered.

32 Point Grou s

Combinations of symmetry elements acting through a point - each belongs to a crystal class.

Describes macroscopic shape of ideal crystals.

Describes symmetry of properties such as thermal expansion, elastic modulus, refractive

index and conductivit .

230 Space Groups

Point group symmetry plus translational symmetry e.g. screw axes and glide planes each

belongs to a crystal structure.

Not all of the s ace rou s are of e ual im ortance and man of them have no exam les of

real crystals at all.About 70% of the elements belong to the space groups Fm3m, Im3m, Fd3m, F43m and

63/mmc. Over 60% of organic and inorganic crystals belong to space groups P21/c, C2/c, P21,

P1, Pbca, P2 2 2 .

134

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