symmetry and properties of crystals (mse 638) 2d space...

Symmetry and Properties of Crystals(MSE 638)

2D Space Group

Somnath Bhowmick

Materials Science and Engineering, IIT Kanpur

February 16, 2018

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

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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)

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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).

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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

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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

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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◦

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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

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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

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p2

Motif has 2-fold symmetry as well

Identify relation (rotation/translation) among all L

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α = 90◦

T ? Aπ/2 = Bπ/2

x = T2 cot(π/4) = T

2

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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

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p4

Identify relation (rotation/translation) among all L

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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)

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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)

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International table for crystallography: p1

Multiplicity Wyckoff letter Site symmetry Coordinates

1 a 1 (x, y)

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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)

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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)

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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

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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)

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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)

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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

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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

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pm

p(primitive lattice) m(point group)

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cm

c(centered lattice)m(point group)

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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

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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

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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)

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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)

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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

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)

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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)

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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)

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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?

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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.

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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?

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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.

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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.

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International table for crystallography: pg

Multiplicity Wyckoff Site Coordinatesletter symmetry

2 a 1 (x, y), (x, y + 12)

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Other possibilities

pg - possible

cg - same as cm

p2gg - same as c2mm

c2gg - same as c2mm

p4gg - impossible

p3g1 - impossible

p31g - impossible

p6gg - impossible

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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?

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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)

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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)

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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)

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Summary

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p4mm

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