Translational symmetry, point and space groups in solids

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1 Translational symmetry, point and space groups in solids Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy ASCS26 Spokane Michele Catti

2 a = b = Å; Å; c=2.959 Å TiO 2 Space group P4 P4 22 /mnm Aperiodic unit (unit-cell content): two Ti Ti (,3) and four O (2,4,5,6) Asymmetric unit: one Ti Ti () () and one O (2) (2) ASCS26 Spokane Michele Catti 2

3 Translational symmetry: set of of lattice points or or lattice vectors ll ASCS26 Spokane Michele Catti 3

4 ranslational symmetry direct lattice set of lattice vectors l 3 l = l a +l 2 a 2 +l 3 a 3 = i l i a i ; (l, l 2, l 3 : integer numbers) a, a 2, a 3 (a, b, c) : unit-cell basis vectors ASCS26 Spokane Michele Catti V = a (a 2 a 3 ) = a 2 (a 3 a ) = a 3 (a a 2 ) = abc ( - cos 2 α -cos 2 β -cos 2 γ + 2cosα cosβ cosγ) /2 x s = i x s,i a i x s,, x s,2, x s,3 (x s, y s, z s ): atomic fractional coordinates (real numbers) x s,l = x s + l x s : position vector of s-th atom in the reference unit-cell x s,l : position vector of s-th atom in the l-th unit-cell 4

5 x s,l = x s + l s=4, l = [ 2 ] x s,l = x s + l s=4, l = [ 2 ] 5 ASCS26 Spokane Michele Catti

6 Scalar product of vectors in the oblique lattice reference: l () l (2) = ( i l i () a i ) ( j l j (2) a j ) = ij l i () l j (2) a i a j a i a j = a i a j cos (a i,a j ) = G ij = G ji ; metric matrix component l 2 = l l = ij l i l j a i a j = = l 2 a 2 + l 22 b 2 + l 32 c 2 + 2l l 2 abcosγ + 2l l 3 accosβ + 2l 2 l 3 bccosα = = ij l i l j G ij = l T G l; squared length of a lattice vector ASCS26 Spokane Michele Catti 6

7 Reciprocal lattice 3 h = h a * +h 2 a 2* +h 3 a 3* = i h i a i* ; (h, h2, h3 : integer numbers) h : reciprocal lattice vector a *, a 2*, a 3* (a *, b *, c * ) : reciprocal unit-cell basis vectors a * = (/V) a 2 a 3 a 2* = (/V) a 3 a a i * a j = δ ij a 3* = (/V) a a 2 Wave vector space K = i K i (2π a i* ) = i K i b i ; K, K 2, K 3 : real numbers ASCS26 Spokane Michele Catti 7

8 ij * = a i* a j* = a i* a j* cos (a i*,a j* ); reciprocal metric matrix component G * = G - V*= /V h 2 = h h = ij h i h j a i * a j *= ij h i h j G ij * = = h 2 a* 2 + h 22 b* 2 + h 32 c* 2 + 2h h 2 a*b*cosγ *+ 2h h 3 a*c*cosβ* + 2h 2 h 3 b*c*cosα* The scalar product of a direct and a reciprocal lattice vector is an integer number l h = ( i l i a i ) ( j h j a j* ) = ij l i h j a i a j* = = ij l i h j δ ij = i l i h i = l h +l 2 h 2 +l 3 h 3 ASCS26 Spokane Michele Catti 8

9 Rotation axes consistent with translational symmetry A O B 2π/n -2π/n A B A B = m OA = m OB ; A B = 2OA cos (2π/n) = 2OA cos (2π/n) = m OA 2cos (2π/n) = m (integer number) As cos (2π/n), then -2 m 2, and m = -2, -,,, 2. By solving for n, the solutions are n =, 2, 3, 4, 6 ASCS26 Spokane Michele Catti 9

10 Triclinic Monoclini c 32 crystallographic point goups Orthorho mbic Trigonal Tetragonal Hexagonal Cubic = m 2/m /m 6/m m 3 mm2 3m 4mm 6mm m 42m 6 m2 43m mmm 4/mmm 6/mmm m 3 m ASCS26 Spokane Michele Catti

11 Cubic point groups: symmetry axes and mirror planes ASCS26 Spokane Michele Catti

12 2 = R = 2 / t x = -x y = -y z = z + ½ Symmetry operator S : rotational R and translational t components S = {R t} t z y x R z y x + = ' ' ' Matrix notation: = R = 2 / t x = -x y = y+½ z = -z Example: twofold screw axis along y (2 // []) ASCS26 Spokane Michele Catti

13 Glide planes: R = m t = l/2 a: t = a/2; b: t = b/2; c: t = c/2; n: t = (a i +a j )/2 ASCS26 Spokane Michele Catti 3

14 Screw axes Screw axes ASCS26 Spokane Michele Catti 4

15 5 = R = 2 / t x = -y y = x z = z+/2 Fourfold screw axis 4 2 along z (4 2 // []) ASCS26 Spokane Michele Catti Glide plane c parallel to xz (c // ()) = R = 2 / t x = x y = -y z = z+½

16 Space group P2 /m O O O + O + O + O + O O O + O + O + O + Shift of the origin by b/4 ASCS26 Spokane Michele Catti 6

17 Space group P2 /m: derived from point group 2/m by combination with the P monoclinic lattice, and replacement of the 2 axis by the 2 screw axis Origin on the mirror plane Origin on the symmetry centre. x,y,z x,y,z 2. x,-y,z x,½-y,z 3. -x,½+y,-z -x,½+y,-z 4. -x,½-y,-z -x,-y,-z ASCS26 Spokane Michele Catti 7

18 Monoclinic P and C Bravais lattices Monoclinic P and C Bravais lattices P primitive C c b centred (non primitive) a ASCS26 Spokane Michele Catti 8

19 Triclinic P Triclinic P 4 Bravais lattices 4 Bravais lattices Monoclinic P, C Monoclinic P, C Orthorhombic P, C, F, I Orthorhombic P, C, F, I Tetragonal P, I Tetragonal P, I Cubic P, F, I Cubic P, F, I Rhombohedral R Hexagonal P ASCS26 Spokane Michele Catti 9

20 ASCS26 Spokane Michele Catti 2

21 2 ASCS26 Spokane Michele Catti

22 General position: set of symmetry-related points not lying on any purely rotational symmetry element; their number (multiplicity) is equal to the order of the symmetry group All symmetry operators act on the general position Special position: set of symmetry-related points lying on one or more purely rotational symmetry elements; the multiplicity is equal to the order of the symmetry group divided by the product of multiplicities of all symmetry elements on which each point lies The symmetry operators corresponding to symmetry elements on which the points lie do not act on the special position ASCS26 Spokane Michele Catti 22

23 TiO 2 - space group P4 2 /mnm x y z - Ti() 2a 3 - Ti(2) ½ ½ ½ 2 - O() 4f O(2) ½ 5 - O(3) O(4) ½ ASCS26 Spokane Michele Catti 23

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures

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