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Analytical Methods for Materials

Lesson 14Crystallography and Crystal Structures – continued

Suggested Reading• Ch. 6 in Waseda• Ch. 1 – C. Kittel, Introduction to Solid State Physics, 3rd Edition, Wiley (1956).• Excerpt from ASM Metals Handbook.• Chs. 1 and 3 – M. DeGraef and M.E. McHenry, Structure of Materials, Cambridge (2007).• Chs. 1 and 3 – S.M. Allen and E.L. Thomas, The Structure of Materials, Wiley (1999).• Chs. 3 and 4 – R. Tilley, Crystals and Crystal Structures, Wiley (2006).

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Graphical representation of space groups

• International Tables for Crystallography– Expensive eight volume book series

published by Springer. Volume A contains Space-group symmetry. You can access some of this information on line through the University of Alabama Libraries web site.

– An inexpensive Brief Teaching Edition of Volume A (ISBN 978-0-7923-6591-4) is available from the International Union of Crystallography (http://www.iucr.org).

• Information In Tables– Diagrams of symmetry elements

• One shows effect of operation of symmetry elements

• One shows location of various symmetry elements

– Origin– Asymmetric unit– Symmetry operations– Generators selected– Positions, multiplicities, site

symmetries, coordinates, reflection conditions

– Symmetry of special projections– Maximal non-isomorphic subgroups– Maximal isomorphic subgroups of

lowest index– Minimal non-isomorphic supergroups

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SG number →Full H-M notation →

Short H-M notation

← Lattice type

SG diagrams

Sitesymmetryof origin

(m3m = center)

Assymmetric unit

Location of motif← Symmetry operations

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GeneratorsDetermine

sequence of coordinate triplets

and reflection conditions

Wykoffpositions

with multiplicity, Wykoff letter and

site symmetry

Orthographic projections along symmetry

directions

Wykoff positions for Cu

If atoms are occupying position 8c, there are two

equivalent atoms at ¼, ¼, ¼ and ¼, ¼, ¾

We add these positions to our base

coordinates!

5

371

Refer to relations

between space groups

1

372

← Symmetry operations

Continued from first page

Transform the initial point x, y, zinto point under consideration

Other sources for information

• Pearson’s Handbook of Crystallographic Data for Intermetallic Phases– Very inclusive

• Smithells Metals Reference Book– Information for many crystal structures

• Various other handbooks

• The information provided will allow you to build up crystal structures.

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Excerpt from Pearson’s Handbook

• A prototype (i.e., structure type) is provided for each phase.

• Pearson symbol and space group tell us Bravais lattice, basis, and symmetry.

• Atoms, Point Set, (x,y,z) and Occ are the Wykoff generating sites.

These are sites of specific atoms in the crystal structure that the space group symmetry operators act on.

We also find these sites in the International Tables for Crystallography, Volume A, Space Group Symmetry.

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PearsonSymbol

Structure and , , , , PointPhase Type Space Group Atoms Setnm

Cu 0.36148 4 000 000 000 1004 3( 3 )

a b cx y z Occ

Cu acF Fm mFm m

Cu

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Wyckoff sites and Point Setsare used to determine site

occupancy.

See this site for high resolution diagrams and tableshttp://img.chem.ucl.ac.uk/sgp/mainmenu.htm

Cu

1 12 2

1 12 2

1 12 2

( , , ) (0, 0, 0)

(0, 0, 0)(0, , )( ,0, )( , ,0)

x y z

WyckoffCoords.

Example for Copper

Figure from Brandon & Kaplan, p. 35

0, 0, 01 1

2 2, ,0

1 12 20, ,

1 12 2,0,

FCC basis

Point set

Place Cu atoms on each of these sites and stack

cells 3DYou end up with FCC

Copper

Cu

1 12 2

1 12 2

1 12 2

( , , ) (0, 0, 0)

(0, 0, 0)(0, , )( ,0, )( , ,0)

x y z

WyckoffCoords.

Example for Copper - continued

Normal FCC unit cell

Point set

Place Cu atoms on each of these sites and stack

cells 3DYou end up with FCC

Copper

What about Tungsten (W, atomic number = 74)

• It is part of space group #229. I’ve presented relevant information on next 3 pages. You can also find this information online at:

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

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PearsonSymbol

Structure and , , , , PointPhase Type Space Group Atoms Setnm

W 0.3165 2 000 000 000 1002 3( 3 )

a b cx y z Occ

W acI Im mIm m

W

379

380

381

382

383

384

cI2: cubic, body-centeredOrigin: m m = centerPoint set 2a: atoms at 0,0,0 about BCC Wykoff sites

PearsonSymbol

Structure and , , , , PointPhase Type Space Group Atoms Setnm

W 0.3165 2 000 000 000 1002 3( 3 )

a b cx y z Occ

W acI Im mIm m

W

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230 Space Groups

• Describe complete symmetry of a crystal (internal and external).

• 230 possible combinations.

• Thus, there are 230 possible crystal structures!

• All possibilities can be found in the International Tables for Crystallography.