Symmetry and Properties of Crystals (MSE 638) 2D Space...
Transcript of Symmetry and Properties of Crystals (MSE 638) 2D Space...
![Page 1: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/1.jpg)
Symmetry and Properties of Crystals(MSE 638)
2D Space Group
Somnath Bhowmick
Materials Science and Engineering, IIT Kanpur
February 16, 2018
![Page 2: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/2.jpg)
Symmetry and crystallography
Symmetry operations in 2DI Translation (T )I Rotation (1, 2, 3, 4, 6)I Reflection (m)
Symmetry of a latticeI Number of 2D lattice – 5I Derived by combining T with 2, 3, 4, 6 fold rotation and m
Symmetry of a point groupI Number of 2D point groups – 10I 1, 2, 3, 4, 6 fold rotation and m – form cyclic point groupI Four other point groups obtained by combining 2, 3, 4 and 6 fold
rotation with m
Symmetry of a crystal 6= 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 has6-fold rotation symmetry
2 / 46
![Page 3: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/3.jpg)
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 74-fold rotation No 4 �3-fold rotation No 3 42-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 magnitudeand angle between two translations?Example: if the lattice must have 4-fold symmetry, then T1 = T2 andT1∠T2 = 90◦ (square lattice)Example: if the lattice must have 3-fold symmetry, then T1 = T2 andT1∠T2 = 120◦ (hexagonal lattice)
3 / 46
![Page 4: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/4.jpg)
2D lattices
Lattice Translations a1∠a2 Symmetry
Oblique (P) a1 6= a2 α 2
Rectangle (P) a1 6= a2 90◦ 2, m
Diamond (P) a1 = a2 α 2, mCentered rectangle (NP) a1 6= a2 90◦ 2, m
Square (P) a1 = a2 90◦ 2, m, 4
Hexagonal (P) a1 = a2 60◦, 120◦ 2, m, 3, 6Triangular (P) a1 = a2 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
![Page 5: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/5.jpg)
List of 2D point groups
Hermann-Mauguin Schonflies
1 1 C1
2 2 C2
3 m Cs4 2mm C2v
5 4 C4
6 4mm C4v
7 3 C3
8 3m C3v
9 6 C6
10 6mm C6v
Goal: derive 2D space group (plane group) by taking each 2D latticeand dropping as many point groups as possible in each lattice
5 / 46
![Page 6: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/6.jpg)
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
![Page 7: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/7.jpg)
Combine translation and rotation
T ? Aα = Bα, translation ? rotation(A) = rotation(B)
Location of the second rotation axis B is along the perpendicularbisector 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
![Page 8: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/8.jpg)
α = 180◦
T ? Aπ = Bπ
x = T2 cot(π/2) = 0
The second 2-fold axis (Bπ) is going to be located in the middle ofthe translation vector
Next: use this concept to get all 2-fold rotation axes present in anoblique unit cell
8 / 46
![Page 9: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/9.jpg)
p2
2 = {1, 2}a1 6= a2, a1∠a2 = α
Three translations: ~a1,~a2,~a1 + ~a2
Combine each translation with each rotation axis
p(primitive lattice)2(point group)
Identify relation (rotation/translation) among all symmetry elements
9 / 46
![Page 10: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/10.jpg)
p2
Motif has 2-fold symmetry as well
Identify relation (rotation/translation) among all L
10 / 46
![Page 11: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/11.jpg)
α = 90◦
T ? Aπ/2 = Bπ/2
x = T2 cot(π/4) = T
2
11 / 46
![Page 12: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/12.jpg)
p4
4 = {1, 2, 4, 4−1}a1 = a2, a1∠a2 = 90◦
Three translations: ~a1,~a2,~a1 + ~a2
Combine each translation with each rotation axis
p(primitive lattice)4(point group)
Identify relation (rotation/translation) among all symmetry elements
12 / 46
![Page 13: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/13.jpg)
p4
Identify relation (rotation/translation) among all L
13 / 46
![Page 14: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/14.jpg)
p3
3 = {1, 3, 3−1}a1 = a2, a1∠a2 = 120◦
Three translations: ~a1,~a2,~a1 + ~a2Combine each translation with each rotation axis
x = a2 cot(π/3)
p(primitive lattice)3(point group)
14 / 46
![Page 15: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/15.jpg)
p6
6 = {1, 2, 3, 3−1, 6, 6−1}a1 = a2, a1∠a2 = 60◦
Three translations: ~a1,~a2,~a1 + ~a2
Combine each translation with each rotation axis
x = a2 cot(π/6)
p(primitive lattice)6(point group)
15 / 46
![Page 16: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/16.jpg)
International table for crystallography: p1
Multiplicity Wyckoff letter Site symmetry Coordinates
1 a 1 (x, y)
16 / 46
![Page 17: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/17.jpg)
International table for crystallography: p2
Multiplicity Wyckoff Site Coordinatesletter symmetry
2 e 1 (x, y), (x, y)1 d 2 (12 ,
12)
1 c 2 (12 , 0)1 b 2 (0, 12)1 a 2 (0,0)
17 / 46
![Page 18: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/18.jpg)
International table for crystallography: p4
Multiplicity Wyckoff Site Coordinatesletter symmetry
4 d 1 (x, y), (x, y), (y, x), (y, x)2 c 2 (12 , 0), (0, 12)1 b 4 (12 ,
12)
1 a 4 (0,0)
18 / 46
![Page 19: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/19.jpg)
Hexagonal lattice translation vectors as reference axes
(x,0)
(0,x)
(x,x)
(x,y)
(y,x+y)
(x+y,x)
x+ x2 = x3 ⇒ x2 + x3 = x
y + y2 = y3 ⇒ y2 + y3 = y
19 / 46
![Page 20: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/20.jpg)
International table for crystallography: p3
Multiplicity Wyckoff Site Coordinatesletter symmetry
3 d 1 (x, y), (y, x− y), (x+ y, x)1 c 3 (23 ,
13)
1 b 3 (13 ,23)
1 a 3 (0,0)
20 / 46
![Page 21: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/21.jpg)
International table for crystallography: p6
Multiplicity Wyckoff Site Coordinatesletter symmetry
6 d 1 (x, y), (y, x− y), (x+ y, x)(x, y), (y, x+ y), (x− y, x)
3 c 2 (12 , 0), (0, 12), (12 ,12)
2 b 3 (23 ,13), (13 ,
23)
1 a 6 (0,0)
21 / 46
![Page 22: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/22.jpg)
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
![Page 23: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/23.jpg)
Combine translation and reflection
I Im m'
T⟂T/2⟂ I
Tm
T∥T⟂ τ T∥=
T/2⟂
τg τg'+T∥ TT∥
T⟂ τ T∥=
T/2⟂
τg
τ
T⊥ ? m = m′
Location of m′:x = T⊥
2
T ? m = gτ
τ = T‖
Location of gτ :x = T⊥
2
T ? gτ = g′τ+T‖Glide by: τ + T‖
Location of g′τ+T‖ :
x = T⊥2
Glide: 2-step symmetry operation, reflection + translationReflection: a special case of glide with τ = 0
23 / 46
![Page 24: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/24.jpg)
pm
p(primitive lattice) m(point group)
24 / 46
![Page 25: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/25.jpg)
cm
c(centered lattice)m(point group)
25 / 46
![Page 26: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/26.jpg)
International table for crystallography: pm
Multiplicity Wyckoff Site Coordinatesletter symmetry
2 c 1 (x, y), (x, y)1 b .m. (12 , y)1 a .m. (0, y)
Note that, m is horizontal
26 / 46
![Page 27: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/27.jpg)
International table for crystallography: cm
Multiplicity Wyckoff Site Coordinatesletter symmetry (0, 0)+, (12 ,
12)+
4 b 1 (x, y), (x, y)2 a .m. (0, y)
Note that, m and gτ horizontal
27 / 46
![Page 28: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/28.jpg)
International table for crystallography: p2mm
Multiplicity Wyckoff Site Coordinatesletter symmetry
4 i 1 (x, y), (x, y), (x, y), (x, y)
2 h .m. (12 , y), (12 , y)
2 g .m. (0, y), (0, y)
2 f ..m (x, 12), (x, 12)
2 e ..m (x, 0), (x, 0)
1 d 2mm (12 ,12)
1 c 2mm (0, 12)
1 b 2mm (12 , 0)
1 a 2mm (0, 0)
28 / 46
![Page 29: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/29.jpg)
International table for crystallography: c2mm
Multiplicity Wyckoff Site Coordinates
letter symmetry (0, 0)+, (12 ,12)+
8 f 1 (x, y), (x, y), (x, y), (x, y)
4 e .m. (0, y), (0, y)
4 d ..m (x, 0), (x, 0)
4 c 2.. (14 ,14), (34 ,
14)
2 b 2mm (0, 12)
2 a 2mm (0, 0)
29 / 46
![Page 30: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/30.jpg)
International table for crystallography: p4mm
Multiplicity Wyckoff Site Coordinatesletter symmetry
8 g 1 (x, y), (x, y), (y, x), (y, x)(x, y), (x, y), (y, x), (y, x)
4 f ..m (x, x), (x, x), (x, x), (x, x)
4 e .m. (x, 12), (x, 12), (12 , x), (12 , x)
2 d .m. (x, 0), (x, 0), (0, x), (0, x)
2 c 2mm. (12 , 0), (0, 12)
1 b 4mm (12 ,12)
1 a 4mm (0,0)30 / 46
![Page 31: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/31.jpg)
International table for crystallography: p3m1
Multiplicity Wyckoff Site Coordinatesletter symmetry
6 e 1 (x, y), (y, x− y), (x+ y, x)(y, x), (x+ y, y), (x, x− y)
3 d .m. (x, x), (x, 2x), (2x, x)
1 c 3m. (23 ,13)
1 c 3m. (13 ,23)
1 a 3m. (0,0)
31 / 46
![Page 32: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/32.jpg)
International table for crystallography: p31m
Multiplicity Wyckoff Site Coordinatesletter symmetry
6 d 1 (x, y), (y, x− y), (x+ y, x)(y, x), (x− y, y), (x, x+ y)
3 c ..m (x, 0), (0, xx), (x, x)
2 b 3.. (13 ,23), (23 ,
13)
1 a 3.m (0,0)
32 / 46
![Page 33: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/33.jpg)
International table for crystallography: p6mm
Multiplicity Wyckoff Site Coordinatesletter symmetry
6 d 1 (x, y), (y, x− y), (x+ y, x)(x, y), (y, x+ y), (x− y, x)
3 c 2 (12 , 0), (0, 12), (12 ,12)
2 b 3 (23 ,13), (13 ,
23)
1 a 6 (0,0)
33 / 46
![Page 34: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/34.jpg)
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
![Page 35: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/35.jpg)
Some interesting facts about mirror and glide planes
Logic: mirror or glide plane should not generate additional symmetriesnot congruent with the lattice.
Both mirror and glide changes the chirality of the motif.
Place a glide plane along a translation T . Then g2τ = T and τ = T2 .
However, m2 = E, and this is why m is and gτ is not a point group.
If τ = T , then the glide plane is nothing but a mirror plane.
35 / 46
![Page 36: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/36.jpg)
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 gτ - - - -
6-fold rotation No 6 74-fold rotation No 4 �3-fold rotation No 3 42-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 generatenew plane group?
How many such possibilities are there?
36 / 46
![Page 37: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/37.jpg)
Possible plane groups - adding glide like “point group”
Lattice type Point group Plane group
Oblique 1, 2 p1, p2
Primitive rectangle m, 2mm pm, p2mmpg, p2gg,���XXXp2mg
Centered rectangle m, 2mm cm, c2mmcg, c2gg,���XXXc2mg
Square 4, 4mm p4, p4mmp4gg,���XXXp4mg
Hexagonal 3, 3m, 6, 6mm p3, p3m1, p31m, p6, p6mmp3g1, p31g, p6gg,���XXXp6mg
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
![Page 38: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/38.jpg)
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 m2 ? m1 = A2γ where m1∠m2 = γ2 m2 ? A2γ = m1, where m1∠m2 = γ
Translation, rotation, reflection combined to derive 13 plane groups:1 T ? Aα = Bα, Bα on perpendicular bisector of T at x = T
2 cot(α2 )2 T⊥ ? m = m′, where m−m′ distance x = T⊥
23 T ? m = gτ , where m− gτ distance x = T⊥
2 & τ = T‖4 T ? gτ = g′τ+T‖
, where gτ − 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 planegenerates plane group.
38 / 46
![Page 39: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/39.jpg)
International table for crystallography: pg
Multiplicity Wyckoff Site Coordinatesletter symmetry
2 a 1 (x, y), (x, y + 12)
39 / 46
![Page 40: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/40.jpg)
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
![Page 41: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/41.jpg)
Possible plane groups - adding glide like “point group”
Lattice type Point group Plane group
Oblique 1, 2 p1, p2
Primitive rectangle m, 2mm pm, p2mmpg, p2gg (≡ c2mm)
Centered rectangle m, 2mm cm, c2mmcg (≡ 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” inlattice. 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
![Page 42: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/42.jpg)
International table for crystallography: p2mg
Multiplicity Wyckoff Site Coordinatesletter symmetry
4 d 1 (x, y), (x, y), (x+ 12 , y), (x+ 1
2 , y)
2 c .m. (14 , y), (34 , y)
2 b 2.. (0, 12), (12 ,12)
2 a 2.. (0, 0), (12 , 0)
42 / 46
![Page 43: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/43.jpg)
International table for crystallography: p2gg
Multiplicity Wyckoff Site Coordinates
letter symmetry
4 c 1 (x, y), (x, y), (x+ 12 , y + 1
2), (x+ 12 , y + 1
2)
2 b 2.. (12 , 0), (0, 12)
2 a 2.. (0, 0), (12 ,12)
43 / 46
![Page 44: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/44.jpg)
International table for crystallography: p4gm
M WL SS Coordinates
8 d 1 (x, y), (x, y), (y, x), (y, x), (x+ 12 , y + 1
2),(x+ 1
2 , y + 12), (y + 1
2 , x+ 12), (y + 1
2 , x+ 12)
4 c ..m (x, x+ 12), (x, x+ 1
2), (x+ 12 , x), (x+ 1
2 , x)
2 b 2.mm (12 , 0), (0, 12)
2 a 4.. (0,0), (12 ,12)
44 / 46
![Page 45: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/45.jpg)
Summary
45 / 46
![Page 46: Symmetry and Properties of Crystals (MSE 638) 2D Space Grouphome.iitk.ac.in/~bsomnath/mse638/WWW/2dsg.pdf · Symmetry and Properties of Crystals (MSE 638) 2D Space Group Somnath Bhowmick](https://reader035.fdocuments.us/reader035/viewer/2022071102/5fdbd988fe569a6a6c290603/html5/thumbnails/46.jpg)
p4mm
46 / 46