Analytical Methods for Materials
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1 Analytical Methods for Materials Lesson 13 Crystallography and Crystal Structures continued Chapter 6 in Waseda Excerpt from ASM Metals Handbook. Suggested Reading 337
2 Notation for crystal structures 1. Hermann-Mauguin (HM) aka the International system 2. Pearson 3. Strukturbericht 4. Schoenflies 338
3 The International (i.e., Hermann-Maugin) system is preferred for describing crystal structures in metals and ceramics. The other systems have their uses too. We use all of them to describe space groups! 339
4 Space group data allows us to define allowed and missing reflections in a diffraction pattern. Tells us everything we need to know about symmetry and motif placement! In this class, the first place you ll see this is on X-ray (i.e., ICDD, JCPDS, or PDF) cards. 340
5 ICDD (i.e., JCPDF)card
6 ICDD (i.e., JCPDF)card
7 ICDD (i.e., JCPDF)card for Al 343
8 Space group data allows us to define allowed and missing reflections in a diffraction pattern. Tells us everything we need to know about symmetry and motif placement! Another place you ll see this is in the International Tables for Crystallography. 344
9 International Tables for Crystallography From the data tabulated in the International Tables, we can get the following: Herman-Mauguin (HM) Symbol (Long, Short) Point Group (HM, Schoenflies) Location and identification of symmetry elements Wyckoff site multiplicity and symmetry Location of general and special positions Extinction conditions for X-rays Possible subgroups and supergroups You don t need to worry about some of these things yet.
10 RECALL Herman Mauguin (International) Symbol The HM symbol is derived from the type of Bravais lattice and the symmetry elements present in the crystal. (Letter) (primary) (secondary) (tertiary) Each letter describes centering present in the lattice: P Primitive C C-centered B B-centered A A-centered I Body centered F Face centered R Rhombohedral H Hexagonal See the next viewgraph.
11 Space Group Letter Symbols Letter Number of lattice "Centering" symbol Lattice type points per unit cell Coordinates of lattice points P A B P 1 0,0,0 A-face centered 2 0, 0, 0 ; 0,, B-face centered 2 0, 0, 0 ;, 0, C C-face centered 2 0,0,0 ; 1, 1,0 2 2 I F R H Body centered 2 0, 0, 0 ;,, Face centered (all) 4 0,0,0 ;,,0 ;,0, ; 0,, Primitive 1 0,0,0 (Rhombohedral axes) Centered 3 0,0,0 ;,, ;,, (obverse setting) (Hexagonal axes) ,0,0 ;,, ;,, (reverse setting) Centered Hexagonal 3 0,0,0 ;,,0 ;,, We use specific symbols to denote crystal systems. 347
12 The next three symbols denote symmetry elements present in certain directions: Crystal System Triclinic Symmetry Direction Primary Secondary Tertiary None Monoclinic* [010] Orthorhombic [100] [010] [001] Tetragonal [001] [100]/[010] [110] Hexagonal/ [001] [100]/[010] [120]/[1(-1)0] Trigonal Cubic [100]/[010]/ [001] [111] [110] * For this structure list primary in secondary slot. Place 1 in primary and tertiary slots for Long notation.
13 The bracketed numbers on the previous page represent directions originating from the crystal origin: [100] Axis parallel or plane perpendicular to the a-axis. [010] Axis parallel or plane perpendicular to the b-axis. [001] Axis parallel or plane perpendicular to the c-axis. [110] Axis parallel or plane perpendicular to the line running at 45 to the a-axis and b-axis. [110] Axis parallel or plane perpendicular to the long face diagonal of the ab face of a hexagonal cell. [111] Axis parallel or plane perpendicular to the body diagonal.
14 Symmetry elements in space group symbols Symbol Lattice type Comments m Mirror plane reflection a b c n Axial glide plane [010],[001] Glide vector a/ 2 Axial glide plane [001],[100] Glide vector b/ 2 Axial glide plane [100],[010] Glide vector c / 2 [1 10],[110] Glide vector c / 2 [100],[010],[1 10] Glide vector c / 2, hexagonal axes [1 10],[120],[210],[110] Glide vector c / 2, hexagonal axes Diagonal glide plane [001]; [100]; [010] Glide vector ( a b); ( b c); ( a c) Diagonal glide plane [1 10]; [01 1]; [101] Diagonal glide plane [110]; [011]; [101] Glide vector ( a b c); ( a c) Glide vector ( a b c); ( a b c); ( a b c) d Diamond glide plane [001]; [100]; [010] Glide vector ( a b); ( b c); ( a c) Diamond glide plane [1 10]; [01 1]; [101] Diamond glide plane [110]; [011]; [101] Glide vector ( a b c); ( a b c); ( a b c) Glide vector ( a b c); ( a b c); ( a b c) None 2, 3, 4, 6 n fold rotation axis A counter clockwise rotation of 360 / n 1 2 m, 3, 4, 6 Center of symmetry n fold rotoinversion axis A counter clockwise rotation of 360 / n followed by inverstion 2, 3, 3, 4, n fold screw axis, n A counter clockwise right-handed screw rotation of 360 / n followed ,4,6,6, ,6, p by translation by ( p/ n) T We use specific symbols to denote internal symmetry. 350
15 The full Herman-Mauguin symbol shows: Both rotation/screw axes running parallel to the specified direction. Any mirror/glide plane perpendicular to the same direction. or The two symbols separated by a forward slash (/) C 1 2/m 1 P Some examples: P 2/m 2/n 2 1 /a I 4 1 /a 2/m 2/d P 6 3 /m 2/m 2/c F 4/m 3 2/m
16 The short Herman-Mauguin symbol Short HM symbols contain only the mirror/glide plane components for primary, secondary and tertiary directions. If no mirror/glide plane, contains the rotation/screw axis. In certain cases (monoclinic, tetragonal and hexagonal), both the rotation/screw axis and the mirror/glide plane are retained for the primary direction only. Long C 1 2/m 1 P P 2/m 2/n 2 1 /a I 4 1 /a 2/m 2/d P 6 3 /m 2/m 2/c F 4/m 3 2/m Short C2/m P Pmna I4 1 /amd P6 3 /mmc Fm3m
17 Symmetry of a Cubic P-Lattice c P P 4 m 3 2 m m 3 m LONG SHORT a a a a b c c c a P 4 m 100 b a P b a P... 2/m 110 b Primary Secondary Tertiary Fig. (a) Cubic unit cell. (b) Space group P 4 m 3 2 m. In diagrams (c), (d) and (e), only the symetry elements corresponding to the symmetry directions <100>, <111>, <110> are shown. P 4 m 3 2 m Adapted from W. Borchardt-Ott, Crystallography, 2 nd Edition, Springer, Berlin, Germany, 1995, p
18 3 tetrad axes 4 triad axes 6 diad axes 9 mirror planes (hidden) Fig The location of 4-, 3-, and 2-fold symmetry axes with respect to a cubic outline for 432. Note that the axes connect symbols on the opposite sides of the crystal and run through the center. Adapted from C. Klein and B. Dutrow, Manual of Mineral Science, 23 rd Edition (John Wiley & Sons, 2007) 354
19 Galena (PbS) 4 m 3 2 Fmm 3 m
20 Crystal Forms Adapted from W. Borchardt-Ott, Crystallography, 2 nd Edition, Springer, Berlin, Germany, 1995, p
21 Fig (a) the three planes of symmetry parallel to the faces of a cube. (b) The six diagonal planes of symmetry of a cube. (c) One of the tetrad axes of a cube. (d) one of the triad axes of a cube. (e) one of the diad axes of a cube. (f) The thirteen axes of symmetry shown by a cube. From C. Kittel, Introduction to Solid State Physics, 2 nd Edition, (John Wiley & Sons, New York, 1959) p
22 How do we identify the crystal system from a HM symbol? Cubic - Secondary symmetry symbol 3 or 3 (i.e. Ia3, Pm3 m, Fd3 m) Tetragonal - Primary symmetry symbol 4, 4, 41, 42 or 43 (i.e. P41212, I4/ m, P4/ mcc) Hexagonal - Primary symmetry symbol 6, 6, 61, 62, 63 or 65 (i.e. P6 mm, P6 3 / m3 m) Trigonal - Primary symmetry symbol 3, 3, 31, or 32 (i.e. P31 m, R3, R3 c, P312) Orthorhombic - All three symbols following the lattice descriptor are mirror planes, glide planes, 2-fold rotation or screw axes (i.e. P nma, Cmc2 1, Pnc2) Monoclinic - Lattice descriptor followed by a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. C c, P2, P2 / ) 1 n Triclinic - Lattice descriptor followed by either a 1 or 1 358
23 Pearson Symbols W.B. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, Wiley-Interscience, New York, Classification of structures based on: Crystal system Bravais lattice Number of atoms per unit cell a B # Crystal system Bravais Lattice (centering) Number of atoms per unit cell 359
24 Notation for Crystal Structures cont d Symbols for the 14 Bravais Lattices Symbol System Lattice Symbol ap Triclinic P mp Simple monoclinic P mc Base-centered monoclinic C op Simple orthorhombic P oc Base-centered orthorhombic C of Face-centered orthorhombic F oi Body-centered orthorhombic I tp Simple tetragonal P ti Body-centered tetragonal I hp Hexagonal P hr Rhombohedral R cp Simple cubic P cf Face-centered cubic F ci Body-centered cubic I 360
25 Pearson Symbols for Some Typical Crystal Structures Crystal structure Pearson Symbol Simple cubic cp1 Body-centered cubic ci2 Face-centered cubic cf4 Hexagonal close-paced hp2 CsCl cp2 NaCl cf8 Diamond cubic cf8 Zinc blende cf8 Fluorite cf12 a a a cp1 1 atom/unit cell Cubic crystal system Primitive Bravais Lattice 361
26 What s the Pearson symbol for this one? AlNi 3 cp4 362
27 Strukturbericht Symbols designate the type of crystal structure. A elements; B AB type compounds (i.e., composition near at.%); C AB 2 compounds; D - A m B n compounds; E, F, G, H K More complex compounds ; L Alloys; O Organic compounds ; S Silicates. Many visual examples can be viewed at: This system is inconsistent. For example, TiAl is denoted as L1 0 even though it has AB stoichiometry (see the next viewgraph). 363
28 What s the Strukturbericht symbol for this one? What is the international symbol for this one? AlNi 3 L1 2 Pm3m 364
29 Excerpt from C.S. Barrett, Crystal Structure of Metals, in ASM Handbook, 8th Edition, vol. 8 (1973), ASM International, Metals Park, OH, pp
30 Excerpt from C.S. Barrett, Crystal Structure of Metals, in ASM Handbook, 8th Edition, vol. 8 (1973), ASM International, Metals Park, OH, pp A very good place to look for crystallography data is the Bilbao Crystallographic Server 366
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