Symmetry and Properties of Crystals (MSE 638) 2D Space Group

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1 Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick Materials Science and Engineering, IIT Kanpur February 16, 2018

2 Symmetry and crystallography Symmetry operations in 2D Translation (T ) Rotation (1, 2, 3, 4, 6) Reflection (m) Symmetry of a lattice Number of 2D lattice 5 Derived by combining T with 2, 3, 4, 6 fold rotation and m Symmetry of a point group Number of 2D point groups 10 1, 2, 3, 4, 6 fold rotation and m form cyclic point group Four other point groups obtained by combining 2, 3, 4 and 6 fold rotation with m Symmetry of a crystal symmetry of the lattice, in general Example: graphene vs hexagonal boron nitride (h-bn) Hexagonal lattice, but h-bn has 3-fold rotation, while graphene has 6-fold rotation symmetry 2 / 46

3 List of symmetry operations in 2D and notation Symmetry element Chirality change Analytical symbol Geometrical symbol Translation No T Reflection Yes m 6-fold rotation No 6 4-fold rotation No 4 3-fold rotation No 3 2-fold rotation No 2 () 1-fold rotation No 1. How to derive the 2D lattices? For a lattice to have certain symmetry, what should be the magnitude and angle between two translations? Example: if the lattice must have 4-fold symmetry, then T 1 = T 2 and T 1 T 2 = 90 (square lattice) Example: if the lattice must have 3-fold symmetry, then T 1 = T 2 and T 1 T 2 = 120 (hexagonal lattice) 3 / 46

4 2D lattices Lattice Translations a 1 a 2 Symmetry Oblique (P) a 1 a 2 α 2 Rectangle (P) a 1 a , m Diamond (P) a 1 = a 2 α 2, m Centered rectangle (NP) a 1 a , m Square (P) a 1 = a , m, 4 Hexagonal (P) a 1 = a 2 60, 120 2, m, 3, 6 Triangular (P) a 1 = a 2 60, 120 2, m, 3, 6 P = Primitive (one lattice point per unit cell). NP = Non-primitive (multiple lattice points per unit cell). 4 / 46

5 List of 2D point groups Hermann-Mauguin Schönflies 1 1 C C 2 3 m C s 4 2mm C 2v 5 4 C 4 6 4mm C 4v 7 3 C 3 8 3m C 3v 9 6 C mm C 6v Goal: derive 2D space group (plane group) by taking each 2D lattice and dropping as many point groups as possible in each lattice 5 / 46

6 Possible plane groups Lattice type Point group Plane group Oblique 1, 2 p1, p2 Primitive rectangle m, 2mm pm, p2mm Centered rectangle m, 2mm cm, c2mm Square 4, 4mm p4, p4mm Hexagonal 3, 3m, 6, 6mm p3, p3m, p6, p6mm Looks like there are only 12 plane groups!! Let s derive them in a systematic way Need to learn a few more rules for combining two symmetry elements 6 / 46

7 Combine translation and rotation T A α = B α, translation rotation(a) = rotation(b) Location of the second rotation axis B is along the perpendicular bisector of the translation T, at a distance given by x = T 2 cot(α/2) Next: apply it for α = 180, 120, 90, 60 7 / 46

8 α = 180 T A π = B π x = T 2 cot(π/2) = 0 The second 2-fold axis (B π ) is going to be located in the middle of the translation vector Next: use this concept to get all 2-fold rotation axes present in an oblique unit cell 8 / 46

9 p2 2 = {1, 2} a 1 a 2, a 1 a 2 = α Three translations: a 1, a 2, a 1 + a 2 Combine each translation with each rotation axis p(primitive lattice)2(point group) Identify relation (rotation/translation) among all symmetry elements 9 / 46

10 p2 Motif has 2-fold symmetry as well Identify relation (rotation/translation) among all L 10 / 46

11 α = 90 T A π/2 = B π/2 x = T 2 cot(π/4) = T 2 11 / 46

12 p4 4 = {1, 2, 4, 4 1 } a 1 = a 2, a 1 a 2 = 90 Three translations: a 1, a 2, a 1 + a 2 Combine each translation with each rotation axis p(primitive lattice)4(point group) Identify relation (rotation/translation) among all symmetry elements 12 / 46

13 p4 Identify relation (rotation/translation) among all L 13 / 46

14 p3 3 = {1, 3, 3 1 } a 1 = a 2, a 1 a 2 = 120 Three translations: a 1, a 2, a 1 + a 2 Combine each translation with each rotation axis x = a 2 cot(π/3) p(primitive lattice)3(point group) 14 / 46

15 p6 6 = {1, 2, 3, 3 1, 6, 6 1 } a 1 = a 2, a 1 a 2 = 60 Three translations: a 1, a 2, a 1 + a 2 Combine each translation with each rotation axis x = a 2 cot(π/6) p(primitive lattice)6(point group) 15 / 46

16 International table for crystallography: p1 Multiplicity Wyckoff letter Site symmetry Coordinates 1 a 1 (x, y) 16 / 46

17 International table for crystallography: p2 Multiplicity Wyckoff Site Coordinates letter symmetry 2 e 1 (x, y), ( x, ȳ) 1 d 2 ( 1 2, 1 2 ) 1 c 2 ( 1 2, 0) 1 b 2 (0, 1 2 ) 1 a 2 (0,0) 17 / 46

18 International table for crystallography: p4 Multiplicity Wyckoff Site Coordinates letter symmetry 4 d 1 (x, y), ( x, ȳ), (ȳ, x), (y, x) 2 c 2 ( 1 2, 0), (0, 1 2 ) 1 b 4 ( 1 2, 1 2 ) 1 a 4 (0,0) 18 / 46

19 Hexagonal lattice translation vectors as reference axes (0,x) (y,x+y) (x,y) (x,x) x + x 2 = x 3 x 2 + x 3 = x y + y 2 = ȳ 3 y 2 + y 3 = ȳ (x,0) (x+y,x) 19 / 46

20 International table for crystallography: p3 Multiplicity Wyckoff Site Coordinates letter symmetry 3 d 1 (x, y), (ȳ, x y), ( x + y, x) 1 c 3 ( 2 3, 1 3 ) 1 b 3 ( 1 3, 2 3 ) 1 a 3 (0,0) 20 / 46

21 International table for crystallography: p6 Multiplicity Wyckoff Site Coordinates letter symmetry 6 d 1 (x, y), (ȳ, x y), ( x + y, x) ( x, ȳ), (y, x + y), (x y, x) 3 c 2 ( 1 2, 0), (0, 1 2 ), ( 1 2, 1 2 ) 2 b 3 ( 2 3, 1 3 ), ( 1 3, 2 3 ) 1 a 6 (0,0) 21 / 46

22 Possible plane groups Lattice type Point group Plane group Oblique 1, 2 p1, p2 Primitive rectangle m, 2mm pm, p2mm Centered rectangle m, 2mm cm, c2mm Square 4, 4mm p4, p4mm Hexagonal 3, 3m, 6, 6mm p3, p3m, p6, p6mm Obtained 5 out of 12 (anticipated) plane groups so far Next: consider mirror Need to learn a few more rules for combining two symmetry elements 22 / 46

23 L L L Combine translation and reflection L m I m' m g I I τ T T T T/2 T/2 L T τ= T L g τ g' +T L τ T/2 T T T τ= T τ T m = m Location of m : x = T 2 T m = g τ τ = T Location of g τ : x = T 2 x = T 2 Glide: 2-step symmetry operation, reflection + translation Reflection: a special case of glide with τ = 0 T g τ = g τ+t Glide by: τ + T Location of g τ+t : 23 / 46

24 pm p(primitive lattice) m(point group) 24 / 46

25 cm c(centered lattice)m(point group) 25 / 46

26 International table for crystallography: pm Multiplicity Wyckoff Site Coordinates letter symmetry 2 c 1 (x, y), ( x, y) 1 b.m. ( 1 2, y) 1 a.m. (0, y) Note that, m is horizontal 26 / 46

27 International table for crystallography: cm Multiplicity Wyckoff Site Coordinates letter symmetry (0, 0)+, ( 1 2, 1 2 )+ 4 b 1 (x, y), ( x, y) 2 a.m. (0, y) Note that, m and g τ horizontal 27 / 46

28 International table for crystallography: p2mm Multiplicity Wyckoff Site Coordinates letter symmetry 4 i 1 (x, y), ( x, y), ( x, ȳ), (x, ȳ) 2 h.m. ( 1 2, y), ( 1 2, ȳ) 2 g.m. (0, y), (0, ȳ) 2 f..m (x, 1 2 ), ( x, 1 2 ) 2 e..m (x, 0), ( x, 0) 1 d 2mm ( 1 2, 1 2 ) 1 c 2mm (0, 1 2 ) 1 b 2mm ( 1 2, 0) 1 a 2mm (0, 0) 28 / 46

29 International table for crystallography: c2mm Multiplicity Wyckoff Site Coordinates letter symmetry (0, 0)+, ( 1 2, 1 2 )+ 8 f 1 (x, y), ( x, y), ( x, ȳ), (x, ȳ) 4 e.m. (0, y), (0, ȳ) 4 d..m (x, 0), ( x, 0) 4 c 2.. ( 1 4, 1 4 ), ( 3 4, 1 4 ) 2 b 2mm (0, 1 2 ) 2 a 2mm (0, 0) 29 / 46

30 International table for crystallography: p4mm Multiplicity Wyckoff Site Coordinates letter symmetry 8 g 1 (x, y), ( x, ȳ), (ȳ, x), (y, x) ( x, y), (x, ȳ), (y, x), (ȳ, x) 4 f..m (x, x), ( x, x), ( x, x), (x, x) 4 e.m. (x, 1 2 ), ( x, 1 2 ), ( 1 2, x), ( 1 2, x) 2 d.m. (x, 0), ( x, 0), (0, x), (0, x) 2 c 2mm. ( 1 2, 0), (0, 1 2 ) 1 b 4mm ( 1 2, 1 2 ) 1 a 4mm (0,0) 30 / 46

31 International table for crystallography: p3m1 Multiplicity Wyckoff Site Coordinates letter symmetry 6 e 1 (x, y), (ȳ, x y), ( x + y, x) (ȳ, x), ( x + y, y), (x, x y) 3 d.m. (x, x), (x, 2x), (2 x, x) 1 c 3m. ( 2 3, 1 3 ) 1 c 3m. ( 1 3, 2 3 ) 1 a 3m. (0,0) 31 / 46

32 International table for crystallography: p31m Multiplicity Wyckoff Site Coordinates letter symmetry 6 d 1 (x, y), (ȳ, x y), ( x + y, x) (y, x), (x y, ȳ), ( x, x + y) 3 c..m (x, 0), (0, xx), ( x, x) 2 b 3.. ( 1 3, 2 3 ), ( 2 3, 1 3 ) 1 a 3.m (0,0) 32 / 46

33 International table for crystallography: p6mm Multiplicity Wyckoff Site Coordinates letter symmetry 6 d 1 (x, y), (ȳ, x y), ( x + y, x) ( x, ȳ), (y, x + y), (x y, x) 3 c 2 ( 1 2, 0), (0, 1 2 ), ( 1 2, 1 2 ) 2 b 3 ( 2 3, 1 3 ), ( 1 3, 2 3 ) 1 a 6 (0,0) 33 / 46

34 Possible plane groups Lattice type Point group Plane group Oblique 1, 2 p1, p2 Primitive rectangle m, 2mm pm, p2mm Centered rectangle m, 2mm cm, c2mm Square 4, 4mm p4, p4mm Hexagonal 3, 3m, 6, 6mm p3, p3m1, p31m, p6, p6mm There are 13 plane groups, one more than we anticipated!! They are generated by adding 10 point groups to 5 2D lattices. But we have discovered one more symmetry operation glide. Adding mirror to the lattice resulted glide symmetry in some cases. Can we add glide symmetry to lattice and generate new plane groups? 34 / 46

35 Some interesting facts about mirror and glide planes Logic: mirror or glide plane should not generate additional symmetries not congruent with the lattice. Both mirror and glide changes the chirality of the motif. Place a glide plane along a translation T. Then g 2 τ = T and τ = T 2. However, m 2 = E, and this is why m is and g τ is not a point group. If τ = T, then the glide plane is nothing but a mirror plane. 35 / 46

36 Updated list of symmetry operations in 2D and notation Symmetry element Chirality change Analytical symbol Geometrical symbol Translation No T Reflection Yes m Glide Yes g τ fold rotation No 6 4-fold rotation No 4 3-fold rotation No 3 2-fold rotation No 2 () 1-fold rotation No 1. Can we have something like pg (primitive rectangular lattice + glide) or cg (centered rectangular lattice + glide)? If the answer is yes, then can we replace one m by a g and generate new plane group? How many such possibilities are there? 36 / 46

37 Possible plane groups - adding glide like point group Lattice type Point group Plane group Oblique 1, 2 p1, p2 Primitive rectangle m, 2mm pm, p2mm pg, p2gg, p2mg Centered rectangle m, 2mm cm, c2mm cg, c2gg, c2mg Square 4, 4mm p4, p4mm p4gg, p4mg Hexagonal 3, 3m, 6, 6mm p3, p3m1, p31m, p6, p6mm p3g1, p31g, p6gg, p6mg 13 plane groups generated by adding 10 point groups to 2D lattices. Let s try point group like approach with glide as well. Point groups like 2mg, 4mg, 6mg not possible!! Point groups like g, 2gg, 4gg, 3g, 6gg - may be possible. 37 / 46

38 List of rules used to derive point and plane groups There are 6 cyclic point groups. Rotation and reflection combined to derive 4 point groups. 1 m 2 m 1 = A 2γ where m 1 m 2 = γ 2 m 2 A 2γ = m 1, where m 1 m 2 = γ Translation, rotation, reflection combined to derive 13 plane groups: 1 T A α = B α, B α on perpendicular bisector of T at x = T 2 cot( α 2 ) 2 T m = m, where m m distance x = T 2 3 T m = g τ, where m g τ distance x = T 2 & τ = T 4 T g τ = g τ+t, where g τ g τ distance x = T 2 The last rule has not been used so far. It states how a combination of a translation and a glide plane generates plane group. 38 / 46

39 International table for crystallography: pg Multiplicity Wyckoff Site Coordinates letter symmetry 2 a 1 (x, y), ( x, y ) 39 / 46

40 Other possibilities pg - possible cg - same as cm p2gg - same as c2mm c2gg - same as c2mm p4gg - impossible p3g1 - impossible p31g - impossible p6gg - impossible 40 / 46

41 Possible plane groups - adding glide like point group Lattice type Point group Plane group Oblique 1, 2 p1, p2 Primitive rectangle m, 2mm pm, p2mm pg, p2gg ( c2mm) Centered rectangle m, 2mm cm, c2mm cg ( cm), c2gg ( c2mm) Square 4, 4mm p4, p4mm Hexagonal 3, 3m, 6, 6mm p3, p3m1, p31m, p6, p6mm 13 plane groups generated by adding 10 point groups to 2D lattice. 4 plane groups generated by adding g and 2gg like point group in lattice. Out of 4, only 1 is new!! So far, we have derived 14 plane groups. Is there an alternate route, rather than adding point group in lattice? How about interleaved mirror/glide plane? 41 / 46

42 International table for crystallography: p2mg Multiplicity Wyckoff Site Coordinates letter symmetry 4 d 1 (x, y), ( x, ȳ), ( x + 1 2, y), (x + 1 2, ȳ) 2 c.m. ( 1 4, y), ( 3 4, ȳ) 2 b 2.. (0, 1 2 ), ( 1 2, 1 2 ) 2 a 2.. (0, 0), ( 1 2, 0) 42 / 46

43 International table for crystallography: p2gg Multiplicity Wyckoff Site Coordinates letter symmetry 4 c 1 (x, y), ( x, ȳ), ( x + 1 2, y ), (x + 1 2, ȳ ) 2 b 2.. ( 1 2, 0), (0, 1 2 ) 2 a 2.. (0, 0), ( 1 2, 1 2 ) 43 / 46

44 International table for crystallography: p4gm M WL SS Coordinates 8 d 1 (x, y), ( x, ȳ), (ȳ, x), (y, x), ( x + 1 2, y ), (x + 1 2, ȳ ), (y + 1 2, x ), (ȳ + 1 2, x ) 4 c..m (x, x ), ( x, x ), ( x + 1 2, x), (x + 1 2, x) 2 b 2.mm ( 1 2, 0), (0, 1 2 ) 2 a 4.. (0,0), ( 1 2, 1 2 ) 44 / 46

45 Summary 45 / 46

46 p4mm 46 / 46

Mathematics in Art and Architecture GEK1518

Mathematics in Art and Architecture GEK1518 Helmer Aslaksen Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/ Symmetry and Patterns Introduction