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**Symbols of the 32 three dimensional point groups**

III Crystal Symmetry 3-3 Point group and space group Point group Symbols of the 32 three dimensional point groups Rotation axis X x X Rotation-Inversion axis X

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**X + centre (inversion): Include X for odd order**

X 2 or X m even: only for even rotation symmetry X2 + centre; Xm +centre: the same result (Include X m for odd order)

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**Ratation axis with mirror plane normal to it X/m**

Rotation axis with mirror plane (planes) parallel to it Xm x mirror x

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**Rotation axis with diad axis (axes) normal to it X2**

Rotation-inversion axis with diad axis (axes) normal to it X 2 X X

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**Rotation-inversion axis with mirror plane (planes) parallel to it X m**

Rotation axis with mirror plane (planes) normal to it and mirror plane (planes) parallel to it X/mm x mirror x mirror

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**Only one symbol which denotes all directions in the crystal.**

System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Triclinic 1, 1 Only one symbol which denotes all directions in the crystal. Monoclinic 2, m, 2/m The symbol gives the nature of the unique diad axis (rotation and/or inversion). 1st setting: z-axis unique 2nd setting: y-axis unique 1st setting 2nd setting

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System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Orthorhombic 222, mm2, mmm Diad (rotation and/or inversion) along x-axis Diad (rotation and/or inversion) along y-axis Diad (rotation and/or inversion) along z-axis Tetragonal 4, 4 , 4/m, 422, 4mm, 4 2m, 4/mmm Tetrad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x- and y-axes Diad (rotation and/or inversion) along [110] and [1 1 0] axis

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System and point group Position in point group symbol Stereographic representation Primary Secondary Tertiary Trigonal and Hexagonal 3, 3 , 32, 3m, 3 m, 6, 6 , 6/m, 622, 6mm, 6 m2, 6/mmm Triad or hexad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x-, y- and u-axes Diad (rotation and/or inversion) normal to x-, y-, u-axes in the plane (0001) Cubic 23, m3, 432, 4 3m, m3m Diads or tetrad (rotation and/or inversion) along <100> axes Triads (rotation and/or inversion) along <111> axes Diads (rotation and/or inversion) along <110> axes

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

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Triclinic Rotation axis X 1 Rotation-Inversion axis X 1 X + centre Include X (odd order) 1 For odd order includes X already!

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Monoclinic 1st setting X 2 X 2 = m X + centre Include X (odd order) 2 𝑚 2 mirror mirror 2 𝑚 2 = m 2

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**X2 + centre, Xm +centre Include X m (odd order) 2/m**

1 Monoclinic 2st setting 2 X2 12 ≡2 Xm 1m ≡m X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) 2/m

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**Rotation-Inversion axis X**

Orthorhombic 2/m Rotation axis X Already discussed Rotation-Inversion axis X 2/m X + centre Include X (odd order) 2/m X2 222 Xm 2mm (2D) = mm2 X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) mmm ≡ 2/m2/m2/m

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Tetragonal Rotation axis X 4 4 Rotation-Inversion axis X X + centre Include X (odd order) 4 𝑚 X2 422 Xm 4mm X 2 or X m even 4 2𝑚 X2 + centre, Xm +centre Include X m (odd order) 4/mmm ≡ 4/m 2/m 2/m

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**Rotation-Inversion axis X**

Trigonal Rotation axis X 3 Rotation-Inversion axis X 2/m X + centre Include X (odd order) 3 X2 32 Xm 3m X 2 or X m even 2/m X2 + centre, Xm +centre Include X m (odd order) 3 𝑚≡ 3 2 𝑚

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Hexagonal Rotation axis X 6 6 Rotation-Inversion axis X X + centre Include X (odd order) 6 𝑚 X2 622 Xm 6mm X 2 or X m even 6 𝑚2 X2 + centre, Xm +centre Include X m (odd order) 6/mmm ≡ 6/m 2/m 2/m

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**Rotation-Inversion axis X m3≡ 2/m 3 X + centre Include X (odd order)**

Cubic Rotation axis X 23 Rotation-Inversion axis X 2/m m3≡ 2/m 3 X + centre Include X (odd order) X2 432 Xm 2/m X 2 or X m even 4 3𝑚 X2 + centre, Xm +centre Include X m (odd order) m3m ≡ 4/m 3 2/m

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**Examples of point group operation**

At a general position [x y z], the symmetry is 1, Multiplicity = 4 The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. y x

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**(2)At a special position [100], the symmetry is**

2. Multiplicity = 2 At a special position [010], the symmetry is 2. Multiplicity = 2 At a special position [001], the symmetry is 2. Multiplicity = 2

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#2 Point group 4 At a general position [x y z], the symmetry is 1. Multiplicity = 4

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**(2) At a special position [001], the symmetry is**

4. Multiplicity = 1 #3 Point group 4

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**At a general position [x y z], the symmetry**

is 1. Multiplicity = 4 (2) At a special position [001], the symmetry is 𝑚. Multiplicity = 2

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**(3) At a special position [000], the symmetry is 4 . Multiplicity = 1**

xyz, -x-yz, y-x-z, -yx-z 2 g m 0 ½ z, ½ 0 -z f ½ ½ z, ½ ½ -z e 0 0 z, 0 0 -z d ½ ½ ½ c ½ ½ 0 b 0 0 ½ a 000

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**Transformation of vector components**

Original vector is P =[ 𝑝 1 , 𝑝 2 , 𝑝 3 ]= 𝑥, 𝑦, 𝑧 P =𝑥 x +𝑦 y +𝑧 z i.e. When symmetry operation transform the original axes x , y , z to the new axes x′ , y′ , z′ New vector after transformation of axes becomes P ′ =[ 𝑝 ′ 1, 𝑝′ 2, 𝑝′ 3 ]= 𝑢, 𝑣, 𝑤 i.e. P′ =𝑢 x ′ +𝑣 y′ +𝑤 z′

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**The angular relations between the axes may be specified by drawing up a table of direction cosines.**

Old axes x y z New axes x′ a11 = cos x′ x a12 = cos x′ y a13 = cos x′ z y′ a21 = cos y′ x a22 = cos y′ y a23 = cos y′ z z′ a31 = cos z′ x a32 = cos z′ y a33 = cos z′ z

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Then 𝑢=𝑥∗cos x′ x +𝑦∗cos x′ y + 𝑧∗cos x′ z i.e. 𝑝′ 1 = a 11 ∗ 𝑝 1 + a 12 ∗ 𝑝 2 + a 13 ∗ 𝑝 3 In a dummy notation 𝑝′ 1 = a 1j ∗ 𝑝 j Similarly 𝑝′ 2 = a 2j ∗ 𝑝 j 𝑝′ 3 = a 3j ∗ 𝑝 j i.e. 𝑝′ i = a ij ∗ 𝑝 j

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**Moreover, by repeating the argument for the reverse transformation and we have**

𝑥=𝑢∗cos x ′ x +𝑣∗cos y ′ x + 𝑤∗cos z ′ x 𝑝 1 = a j1 ∗ 𝑝′ j Similarly, 𝑝 2 = a j2 ∗ 𝑝′ j 𝑝 3 = a j3 ∗ 𝑝′ j i.e. “old” in terms of “new” 𝑝 i = a ji ∗ 𝑝′ j

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**For example: #1 Point group 4**

The direction cosines for the first operation is Old axes x y z New axes x′ = - y a11 = 0 a12 =−1 a13 = 0 y′ = x a21 = 1 a22 = 0 a23 = 0 z′ = z a31 = 0 a32 = 0 a33 =1

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**After symmetry operation, the new position is [x y z] in new axes.**

We can express it in old axes by 𝑝 i = a ji ∗ 𝑝′ j = 𝑝′ j ∗a ji i.e. [ 𝑝 1 𝑝 2 𝑝 3 ]=[ 𝑥 𝑦 𝑧 ] 0 − =[ 𝑦 𝑥 𝑧 ] 𝑝 1 𝑝 2 𝑝 3 = − x y z = y x z or

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**14 plane lattices + 32 point groups**

230 Space groups Crystal Class Bravais Lattices Point Groups Triclinic P 1, 1 Monoclinic P, C 2, m, 2/m Orthorhombic P, C, F, I 222, mm2, 2/m 2/m 2/m Trigonal P, R 3, 3 , 32, 3m, 3 2/m Hexagonal 6, 6 , 6/m, 622, 6mm, 6 m2, 6/m 2/m 2/m Tetragonal P, I 4, 4 , 4/m, 422, 4mm, 4 2m, 4/m 2/m 2/m Isometric P, F, I 23, 2/m 3 , 432, 4 3m, 4/m 3 2/m

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**Table for all space groups Look at the notes!**

B. Space group Table for all space groups Look at the notes! Good web site to read about space group

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**Symmetry elements in space group**

Point group Translation symmetry + point group Translational symmetry operations The first character: P: primitive A, B, C: A, B, C-base centered F: Face centered I: Body centered R: Romohedral

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**Glide plane also exists for 3D space group with more possibility**

Symmetry planes normal to the plane of projection Symmetry plane 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 e Diagonal glide plane n Diamond glide plane 1/4 along line & 1/4 normal to plane d

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

Projection plane

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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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Glide planes ---- translation plus reflection across the glide plane * axial glide plane (glide plane along axis) ---- translation by half lattice repeat plus reflection ---- three types of axial glide plane i. a glide, b glide, c glide (a, b, c) 1 2 along line in plane ≡ along line parallel to projection plane

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e.g. b glide , b --- graphic symbol for the axial glide plane along y axis c.f. mirror (m) graphic symbol for mirror

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**If b glide plane is ⊥ 𝑧 axis**

If the axial glide plane is normal to projection plane, the graphic symbol change to c glide glide plane⊥ 𝑧 axis If b glide plane is ⊥ 𝑧 axis glide plane symbol

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b , underneath the glide plane c glide: 𝑐 2 along z axis or 𝑎+𝑏+𝑐 2 along [111] on rhombohedral axis

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ii. Diagonal glide (n) 𝑎+𝑏 2 , 𝑏+𝑐 2 , 𝑎+𝑐 2 or 𝑎+𝑏+𝑐 2 (tetragonal, cubic system) If glide plane is perpendicular to the drawing plane (xy plane), the graphic symbol is If glide plane is parallel to the drawing plane, the graphic symbol is

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iii. Diamond glide (d) 𝑎+𝑏 4 , 𝑎+𝑏+𝑐 4 (tetragonal, cubic system)

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**Symbols of symmetry axes**

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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**i. All possible screw operations**

screw axis --- translation τ plus rotation screw Rn along c axis = counterclockwise rotation 360/R o + translation n/R c

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2 21 3 31 32 4 41 42 43

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6 61 62 63 64 65

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62

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Symmorphic space group is defined as a space group that may be specified entirely by symmetry operation acting at a common point (the operations need not involve τ) as well as the unit cell translation. (73 space groups) Nonsymmorphic space group is defined as a space group involving at least a translation τ.

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**Cubic – The secondary symmetry symbol will always be either 3 or –3 (i**

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) 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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Examples Space group P1

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Space group P 1

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Space group P112

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Space group P121

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Space group P1121

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**Condition limiting possible reflections**

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Space group P1211

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Space group B112 +( )

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( 𝑥 , 𝑦 ,z) (1/2+ 𝑥 , 𝑦 ,1/2+z) (x,y,z) y x (1/2+x,y,1/2+z)

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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) (000) (½½0) (½0½) (0½½) Sr 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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**Another example from the note**

The usage of space group for crystal structure identification Space group P 4/m 3 2/m Reference to note chapter 3-2 page 26

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**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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#1 Simple cubic

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**Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5 (can interchange if desired)**

#2 CsCl structure CsCl Vital Statistics Formula CsCl Crystal System Cubic Lattice Type Primitive Space Group Pm3m, No. 221 Cell Parameters a = Å, Z=1 Atomic Positions Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5 (can interchange if desired) Density 3.99

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**#3 BaTiO3 structure Temperature 183 K 278 K 393 K rhombohedral (R3m)**

Orthorhombic (Amm2) Tetragonal (P4mm). Cubic (Pm 3 m)

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