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3D Symmetry _2 (Two weeks)

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**3D lattice: Reading crystal7.pdf**

Building the 3D lattices by adding another translation vector to existing 2D lattices Oblique (symmetry 1) + triclinic General Triclinic Primitive

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Oblique (symmetry 2) + projection 4 choices:

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Double cell side centered Double cell side centered

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Double cell body centered Some people use based centered, some use body centered. monoclinic

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**Rectangular (symmetry m) +**

90o 90o already exist! Rectangular (symmetry g) + : the same. cm + ?

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**Rectangular (symmetry 2mm) +**

P2mm P2mg p2gg Orthorhombic primitive

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Double cell side centered Orthorhombic base-centered Orthorhombic base-centered Double cell side centered

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Orthorhombic body-centered rectangular

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**Centered Rectangular (symmetry 2mm) +**

C2mm the same

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Face centered orthorhombic

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Square (symmetry 4, 4mm) + P4 P4mm p4gm Tetragonal primitive

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Tetragonal Body centered Tetragonal

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**p3 p3m1 p31m Hexagonal (symmetry 3, 3m) + not in this category**

Hexagonal primitive not in this category Why? Rhombohedral

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Hexagonal primitive Rhombohedral triple cell

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**p6 p6mm p31m Hexagonal (symmetry 3m, 6, 6mm) +**

can only located at positions: Hexagonal primitive p31m Hexagonal & 6 related can only fit 3P!

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11 lattice types already cubic (isometric) Special case of orthorhombic (222) with a = b = c Primitive (P) Body centered (I) Face centered (F) Base center (C) Tetragonal (I)? Cubic a = b c [100]/[010]/[001] [111] Tetragonal (P)

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**Another way to look as cubic:**

Consider an orthorhombic and requesting the diagonal direction to be 3 fold rotation symmetry P222 P23 Primitive I222 I23 Body centered F222 F23 Face centered C222 I23

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Bingo! 14 Bravais lattices!

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**Lattice type - compatibility with - point group**

reading crystal9.pdf. Crystal Class Bravais Lattices Point Groups Triclinic P (1P) 1, 1 Monoclinic P (2P), C(2I) 2, m, 2/m Orthorhombic P(222P), C(222C) F(222F), I(222I) 222, mm2, 2/m 2/m 2/m Rhombohedral P (3P), 3R 3, 3 , 32, 3m, 3 2/m Hexagonal P (3P) 6, 6 , 6/m, 622, 6mm, 6 m2, 6/m 2/m 2/m Tetragonal P (4P), I (4I) 4, 4 , 4/m, 422, 4mm, 4 2m, 4/m 2/m 2/m Isometric (Cubic) P (23P), F(23F), I (23I) 23, 2/m 3 , 432, 4 3m, 4/m 3 2/m

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**http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg = P = I**

= T P

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= P = I = B = T P

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**Next, we can put the point groups to the compatible lattices, just like the cases in 2D space group.**

3D Lattices (14) + 3D point groups 3D Space group There are also new type of symmetry shows up in 3D space group, like glide appears in 2D space (plane) group!

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The naming (Herman-Mauguin space group symbol) is the same as previously mentioned in 2D plane group! The first letter identifies the type of lattice: P: Primitive; I: Body centered; F: Face centered C: C-centered; B: B-centered, A: A-centered The next three symbols denote symmetry elements in certain directions depending on the crystal system. (See next page)

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**Monoclinic a b = 90o; c b = 90o.**

b axis is chosen to correspond to a 2-fold axis of rotational symmetry axis or to be perpendicular to a mirror symmetry plane. Convention for assigning the other axes is c < a. a c is obtuse (between 90º and 180º). Orthorhombic The standard convention is that c < a < b. Once you define the cell following the convention A, B, C centered

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**Hexagonal/ Rhombohedral**

Crystal System Symmetry Direction Primary Secondary Tertiary Triclinic None Monoclinic [010] Orthorhombic [100] [001] Tetragonal [100]/[010] [110] Hexagonal/ Rhombohedral [120]/[1 0] Cubic [100]/[010]/ [001] [111]

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**Consider 2P Monoclinic + 2**

/2 P2 /2

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**How about 2I Monoclinic + 2**

There is a lattice point in the cell centered!

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z (1) (3) (2) z z +1/2 (3) (1) (2) New type of operation In general Screw axis 21 2

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Specifying

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For a 3-fold screw axis: 3 31 32 4-fold screw axis: 43 41 41 42 43

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42 n1 n2 ……... nm-2 nm-1 No chirality

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3 31 32 2 21 4 41 42 43

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6 61 62 63 64 65

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62

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**Example to combine lattice with screw symmetry**

D A A: 2-fold + translation (to arise at B, C, or D) B C Rotation symmetry of B, C, and D is the same as A. A: 2 P + 2 = P2

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A: 21 21 P + 21 = P21 21 21 I + 2 = I2 or I + 21 = I21 A A: 2 E: 21 Same, only shifted E A: 21 E: 2 I2 = I21

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**Hexagonal lattice (P and R) with 3, 31, 32. Case P first!**

All translations in P have component on c of 0 or unity! A B C B C B and C: same point; B and C: equivalent point; Having

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P3 P31 P32

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**All translations of R has component on c of 1/3 or 2/3!**

Case R! A E D 2/3 All translations of R has component on c of 1/3 or 2/3! E D 1/3 Screw at Designation of Space group A D’ E’ 3 31 32 c/3 2c/3 2/3 c/3 2c/3 c 2/3 2c/3 c 4c/3 31 32 3 32 3 31 R3 R31 R32 R3 = = Hexagonal lattice (P, R) + 3, 31, 32 P3, P31, P32, R3.

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Square lattice P with 4, 41, 42, 43. The translation of P have component on c of 0 or unity! A C B B B C C A 4 41 42 43 c/4 c/2 3c/4 B /2 0 /2 c/4 /2 c/2 /2 3c/4 B 0 c/2 c 3c/2 B 4 41 42 43 B 2 21 P4 P41 P42 P43

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P4 P41 P42 P43

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Homework: Discuss the cases of I4, I41, I42, I43.

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**How to obtain Herman-Mauguin space group symbol by reading the diagram of symmetry elements?**

First, know the Graphical symbols used for symmetry elements in one, two and three dimensions! International Tables for Crystallography (2006). Vol. A, Chapter 1.4, pp. 7–11.

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**Symmetry planes normal to the plane of projection**

Graphical symbol Translation Symbol Reflection plane None m Glide plane 1/2 along line a, b, or c 1/2 normal to plane Double glide plane 1/2 along line & 1/2 normal to plane (2 glide vectors) e Diagonal glide plane 1/2 along line, 1/2 normal to plane (1 glide vector) n Diamond glide plane 1/4 along line & 1/4 normal to plane d

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**Symmetry planes parallel to plane of projection**

Graphical symbol Translation Symbol Reflection plane None m Glide plane 1/2 along arrow a, b, or c Double glide plane 1/2 along either arrow e Diagonal glide plane 1/2 along the arrow n Diamond glide plane 1/8 or 3/8 along the arrows d 3/8 1/8 The presence of a d-glide plane automatically implies a centered lattice!

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Symmetry Element Graphical Symbol Translation Symbol Identity None 1 2-fold ⊥ page 2 2-fold in page 2 sub 1 ⊥ page 1/2 21 2 sub 1 in page 3-fold 3 3 sub 1 1/3 31 3 sub 2 2/3 32 4-fold 4 4 sub 1 1/4 41 4 sub 2 42 4 sub 3 3/4 43 6-fold 6 6 sub 1 1/6 61 6 sub 2 62 6 sub 3 63

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Symmetry Element Graphical Symbol Translation Symbol 6 sub 4 2/3 64 6 sub 5 5/6 65 Inversion None 1 3 bar 3 4 bar 4 6 bar 6 = 3/m 2-fold and inversion 2/m 2 sub 1 and inversion 21/m 4-fold and inversion 4/m 4 sub 2 and inversion 42/m 6-fold and inversion 6/m 6 sub 3 and inversion 63/m

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c-glide b n-glide || c 21 c || a n 2 2 21 b-glide m m c || b a

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**From the point group mmm orthorhombic**

For orthorhombic: primary direction is (100), secondary direction is (010), and tertiary is (001). lattice for orthorhombic: C Short symbol No. 17 orthorhombic that can be derived

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**Principles for judging crystal system by space group**

Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m) Tetragonal – The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc) Hexagonal – The primary symmetry symbol will always be a 6, (-6), 61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm) Trigonal – The primary symmetry symbol will always be a 3, (-3) 31 or 32 (i.e P31m, R3, R3c, P312)

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Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc21, Pnc2) Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P21/n) Triclinic – The lattice descriptor will be followed by either a 1 or a (-1).

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**What can we do with the space group information**

contained in the International Tables? 1. Generating a Crystal Structure from its Crystallographic Description 2. Determining a Crystal Structure from Symmetry & Composition

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**Example: Generating a Crystal Structure**

Description of crystal structure of Sr2AlTaO6 Space Group = Fm 3 m; a= 7.80 Å Atomic Positions Atom x y z Sr 0.25 Al 0.0 Ta 0.5 O

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**From the space group tables**

32 f 3m xxx, -x-xx, -xx-x, x-x-x, xx-x, -x-x-x, x-xx, -xxx 24 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x d mmm 0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼, ¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0 8 c 4 3m ¼ ¼ ¼ , ¼ ¼ ¾ 4 b m 3 m ½ ½ ½ a 000

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Sr 8c; Al 4a; Ta 4b; O 24e 40 atoms in the unit cell stoichiometry Sr8Al4Ta4O24 Sr2AlTaO6 F: face centered (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) Sr (000) (½½0) (½0½) (0½½) 8c: ¼ ¼ ¼ (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾ (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼) Al ¾ + ½ = 5/4 =¼ 4a: (000) (½ ½ 0) (½ 0 ½) (0 ½ ½)

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Ta (000) (½½0) (½0½) (0½½) 4b: ½ ½ ½ (½½½) (00½) (0½0) (½00) (000) (½½0) (½0½) (0½½) O x00 24e: ¼ 0 0 (¼00) (¾½0) (¾0½) (¼½½) -x00 ¾ 0 0 (¾00) (¼½0) (¼0½) (¾½½) 0x0 0 ¼ 0 (0¼0) (½¾0) (½¼½) (½¾½) 0-x0 0 ¾ 0 (0¾0) (½¼0) (½¾½) (0¼½) 00x 0 0 ¼ (00¼) (½½¼) (½0¾) (0½¾) 00-x 0 0 ¾ (00¾) (½½¾) (½0¼) (0½0¼)

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Bond distances: Al ion is octahedrally coordinated by six O Al-O distance d = 7.80 Å − − − = 1.95 Å Ta ion is octahedrally coordinated by six O Ta-O distance d = 7.80 Å − − − = 1.95 Å Sr ion is surrounded by 12 O Sr-O distance: d = 2.76 Å

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**Determining a Crystal Structure from**

Symmetry & Composition Example: Consider the following information: Stoichiometry = SrTiO3 Space Group = Pm 3 m a = 3.90 Å Density = 5.1 g/cm3

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First step: calculate the number of formula units per unit cell : Formula Weight SrTiO3 = (16.00) = g/mol (M) Unit Cell Volume = (3.9010-8 cm)3 = 5.93 cm3 (V) (5.1 g/cm3)(5.93 cm3) : weight in a unit cell ( g/mole) / (6.022 1023/mol) : weight of one molecule of SrTiO3

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** number of molecules per unit cell : 1 SrTiO3. **

(5.1 g/cm3)(5.93 cm3)/ ( g/mole/6.022 1023/mol) = 0.99 number of molecules per unit cell : 1 SrTiO3. From the space group tables (only part of it) 6 e 4mm x00, -x00, 0x0, 0-x0,00x, 00-x 3 d 4/mmm ½ 0 0, 0 ½ 0, 0 0 ½ c 0 ½ ½ , ½ 0 ½ , ½ ½ 0 1 b m 3 m ½ ½ ½ a 000

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**Calculate the Ti-O bond distances: **

Sr: 1a or 1b; Ti: 1a or 1b Sr 1a Ti 1b or vice verse O: 3c or 3d Evaluation of 3c or 3d: Calculate the Ti-O bond distances: d 3c) = 2.76 Å (0 ½ ½) d 3d) = 1.95 Å (½ 0 0, Better) Atom x y z Sr 0.5 Ti O

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Crystal Systems and Space Groups Paul D

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