The Seventeen Plane Groups (Two-dimensional Space Groups)

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1 Korean J. Crystallography Vol. 16, o. 1, pp.11~20, 2005 The Seventeen Plane Groups (Two-dimensional Space Groups) ƒá Á a Á Ÿ a Á~ a Áªœ a Á ž a ano-œ ª, ªƒ ª œ œ a ˆ ª Ÿ ª ( campus) The Seventeen Plane Groups (Two-dimensional Space Groups) Jin-Gyu Kim, Youn-Joong Kim, Young-Sang Kim a, Jaejung Ko a, Sang Ook Kang a, Won-Sik Han a and Il-Hwan Suh a Division of ano-material & Environment Science, Korea Basic Science Institute 52 Yeoeun-Dong, Yusung-Gu, Daejeon , Korea a Department of Material Chemistry, Korea University, 208 Seochang, Chochiwon, Chungnam , Korea Two-dimensional lattice w 6ƒ 5 Bravais lattice w š 10 two-dimensional point groups 5 ~ Bravais lattice w. 10 point groups w 17 two-dimensional space groups š w. Abstract Six basic symmetries and five Bravais lattices existing in the two-dimensional lattice are derived and then ten two-dimensional point groups are classified by each of five Bravais lattices. Finally seventeen two-dimensional space groups belonging to the ten point groups are studied. 1. Two-dimensional Point Group Œ w w 2π/n ~ n- k Œ Ÿ ž (self-coincidence) ª (congruence) Fig. 1. Lattice points in a plane normal to the n- fold rotation axis passing through O. n- w. Fig. 1 two-dimensional lattice point w n-fold rotation axis Ž š w. Lattice point lattice point same environment lattice point n-fold rotation axis. Vector T n-fold rotation axis w O w w period vector w. T, T. T ', T '' lattice point, T ' T '' T «w w lattice vector w. m integer w w. T ' T '' mt 2 T cos(2π/n) mt 11

2 12 ƒá Á Á Ÿ Á~ Áªœ Á ž w wz Table 1. Rotation symmetries existing in two-dimensional lattice m cos 2π/n 2π/n n /2 0 1/ Fig. 3. General oblique lattice in two dimensions, showing a choice of the fundamental translation vector a, b and a unit cell. translation vectors j» a, b ~ ~ γ w w ww latticeƒ. w lattice oblique lattice w. oblique lattice lattice point w 1-2-fold rotation symmetry wù 3-, 4-, 6-, mirror symmetry x w w. 3. Three Special Lattices Fig. 2. The graphical symbols for symmetry operations. w, 1ß cos(2π/n)m/2ß 1 n Table 1 1, 2, 3, 4, 6 ª. two-dimensional lattice 1-, 2-, 3-, 4-, 6-fold rotation symmetryƒ mirror reflection (m) }. graphical symbol Fig Oblique Lattice w two-dimensional lattice ƒ 1-, 2-, 3-, 4-, 6-fold rotation mirror symmetries operationª invariant lattice invariantw w symmetry š w. oblique lattice Fig. 3 1-, 2-fold rotation symmetryw invariantw. Fig. 3 oblique lattice lattice 1-, 2-, 3-, 4-, 6-fold rotation mirror symmetries w operationw invariant lattice» œw a b w w. w w w Fig. 4 3ƒ» special lattice type. Fig. 4(a) aþb, γ 90 o rectangular lattice 2-fold rotation m-symmetries w. Fig. 4(b) a b, γ 90 o square lattice 4-fold rotation m-symmetries w. Fig. 4(c) a b, γ 120 o hexagonal lattice 3-, 6-fold rotation m-symmetries w. ¾ two dimensional lattice w oblique lattice 3 special latticesƒ Ÿ} w. 4. Bravais Lattice (1) Two-dimensional lattice Ž lattice point. Fig. 5 two-dimensional lattice Ž ž X 1 AX 1 Þ X 1 B X 1 lattice pointƒ.

3 16«1y, 2005 The Seventeen Plane Groups (Two-dimensional Space Groups) 13 Fig. 6. Face-centered lattice. Fig. 7. C-entered rectangular lattice, a b, γ 90 o. jù unit cell ABED 1/2 AX m X m 'D latticeƒ. Fig. 4. Special two-dimensional lattices. These three, together with the oblique lattice (Fig. 3) form the four two-dimensional Bravais lattices. Fig. 5. Two dimensional lattice. w Ž lattice point X m ž w X m ', X m " w AX m X m B lattice point (2) Two-dimensional lattice face center lattice pointƒ. Fig. 6 ž lattice point w ù ~ X 1, X 2, X 3, X 4 ž ª w face center lattice pointƒ face-centered unit cell ABED primitive cell BX 1 EX 2 ƒ unit cell w. Fig. 4(a) rectangular lattice face center lattice pointƒ 2-fold rotation mirror symmetries w Fig. 7 rectangular lattice C-centered latticeƒ ƒ w. Fig. 4(b) square lattice face center lattice point square latticeƒ» square lattice centered lattice ƒ. Fig. 4(c) hexagonal lattice face center lattice point 3-fold 6- fold symmetry w w face-center lattice pointƒ.

4 14 ƒá Á Á Ÿ Á~ Áªœ Á ž w wz Table 2. Lattice symmetry direction for two dimensions Lattice Symmetry direction (position in Hermann-Mauguin symbol) Primary Secondary Tertiary Two dimensions Oblique Rectangular [10] [01] Square Hexagonal Rotation point in plane 10 [ ] [ 11] [ 01] [ 11] 10 [ ] [ 11] [ 01] [ 12] [ 11] [ 21] two-dimensional lattice oblique, square hexagonal lattice primitive š rectangular lattice primitive C-centered latticeƒ two-dimensional lattice 5 Bravais latticeƒ w. 5. Ten two-dimensional Point Groups Two-dimensions lattice symmetry direction Table 2. 1) Two-dimensional lattice 5 Bravais lattices Table 2 symmetry direction ˆw rotation reflection operation x ª 10 point groups» w, point group symbols Fig. 8 ùkü. 2) Table 3 five two-dimensional lattice types w point groups Šw ùkü. 1, 2, 1 m, 2 mm, 4, 4 mm, 3, 3 m, 6, 6 mm. Fig. 8. The ten two-dimensional crystallographic point groups. Equivalent points are shown. 6. The 17 Plane Groups (Two-dimensional Space Groups) ¾ w two-dimensional lattice symmetry w. (a) Two dimensional lattice 5 rotation symmetries mirror symmetry ww Table 3. The five two-dimensional lattice types and point groups Lattice cell parameters acentric point group centric point group o. of point groups oblique a b, γ 90 o primitive rectangular a b, γ 90 o m 2 mm centered rectangular a b, γ 90 o m 2 mm 2 square a b, γ 90 o 4, 4 mm 2 hexagonal a b, γ 120 o 3, 3 m 6, 6 mm 4 o. of point groups

5 16«1y, 2005 The Seventeen Plane Groups (Two-dimensional Space Groups) 15 6 basic symmetry w. (b) Translational element žw wr 5 } w Bravais lattice»š wr w glide line ¼. Ÿ ª ~ 2Ü2 matrix 2 œ column vector ùký. (c) Two-dimensional lattice ª basic symmetry 10ƒ ª } w 10 two-dimensional point groups. (d)» 10 two-dimensional point groups 5 Bravais lattices, š w glide line ww 17 two-dimensional space groupsƒ. ~ point group ž space group Ÿ ~ space group w point group w š w. point group centric noncentric space group, reflection condition translation symmetry ªw space group ùkù. two-dimensional space groups symmetry elements general position w diagram International Tables for Crystallography». 1) (1) oncentric point group 1 space group (1.1) P1(1) Oblique Origin arbitrary Coordinates: (1) x, y F( hk) f j exp2πi( hx j + ky j ) f j { cos2π( hx j + ky j ) + isin2π( hx j + ky j )} 2 I( xy) f j cos2π( hx + ky) + f j sin2π( hx + ky) 2 I ( x y ) F( hk) F( hk) F( hk) ƒ w Patterson symmetry P2. (2) Centric point group 2 space group (2.1) P2(2) oblique Origin at 2 Coordinates: (1) x, y (2) x, y 2 F( hk) 2 f j cos2π( hx j + ky j ) I( xy) f j cos2π( hx + ky) 2 I ( x y ) w Patter- F( hk) F( hk) F( hk) son symmetry P2. (3) Acentric point group m space group (3.1) Pm(3) P1m1 Rectangular Origin on m m//[10] w 2 tƒ. Coordinates: (1) x, y (2) x, y F( hk) f j { exp2πi( hx j + ky j ) + exp2πi( hx j + ky j )} f j exp2πiky j { exp2πhx j + exp2πi( hx j )} I(hk) I( h k ) I( h k) I(h k )ƒ w Patterson function f j 2cos2πhx j exp2πiky j f j 2cos2πhx j ( cos2πky i + isin2πky j ) 2 I( hk) f j 4cos 2 2πhx j ( cos2πky j + isin2πky j ) ( cos2πky j sin2πky i j ) 2 f j 4cos 2 2πhx j ( cos 2 2πky j + sin 2 2πky j )

6 16 ƒá Á Á Ÿ Á~ Áªœ Á ž w wz 1 P( xy) -- F( hk) 2 cos2π( hx + ky) A hk Patterson symmetry P2mm. (3.2) Pg(4) P1g1 Rectangular Origin on g m glide line ã g//[10] 2 tƒ. Coordinates: (1) x, y (2) x, y + 1/2 (4) Centric point group 2mm space group (4.1) P2mm(6) Rectangular Origin at 2mm 2//rotation point in plane 2 tƒ m//[10] tƒ ƒ. F( hk) f j { exp2πi( hx j + ky j ) I(hk) I( h k ) I ( h k) I(h k )ƒ w Patterson symmetry P2mm. Reflection condition. when k 2n. + exp2πi( hx j + ky j )expkπi } f j 2cos2πhx j exp2πiky j coskπ I( hk) 2 f j 4cos 2 2πhx j ( cos 2 2πky j + sin 2 2πky j )cos 2 kπ F( 0k) f j exp2πiky j ( 1 + expkπi) 0 only (3.3) Cm(5) C1m1 Rectangular Origin on m General positions Pm + (1/2, 1/2). Coordinates: (1) x, y (2) x, y (3) x + 1/2, y + 1/2 (4) x + 1/2, y + 1/2 F( hk) f j [ exp2πi( hx j +ky j ){ 1+expπi( h + k) } + exp2πi( hx j + ky j ){ 1 + expπi( h + k) }] f j [{ exp2πi( hx j + ky j ) +exp2πi( hx j + ky j )}{ 1 + expπi( h + k) }] 0 only when h + k 2n. Coordinates: (1) x, y (2) x, y (3) x, y (4) x, y Patterson symmetry: P2mm (4.2) P2mg(7) Rectangular Origin at 21g w space group symmetry operation «w space group symmetryƒ w rule w symmetry œ w. 2//rotation point in plane tƒ m//[10] m ù (4.1) origin ù g//[01] tƒ ù (4.2) Eq. (4.1) t Eq. (4.2) w

7 16«1y, 2005 The Seventeen Plane Groups (Two-dimensional Space Groups) 17» w mirror plane x 1/4 w. m//[10] at x 1/4 t ƒ. g//[10] at x 1/4 g//[01] at y 1/4 Coordinates: (1) x, y (2) x, y (3) x + 1/2, y (4) x + 1/2, y 2 F( hk) 2 f j { cos2π( hx j + ky j ) + cos2πi( hx + h 2 ky)} 2 Reflection F( h0) 2 f j cos2πhx j (1 + coshπi) 0 only when h 2n. õ P2mg a- b- ã P2gm. (4.3) P2gg(8) Rectangular Origin at 2 ù 2//rotation point in plane tƒ g//[10]ƒ ù š g//[01]ƒ ù (4.3) 4 coordinates. (1) x, y (2) x, y (3) x + 1/2, y + 1/2 (4) x + 1/2, y + 1/2 2 F( hk) 2 f j { cos2π( hx j + ky j ) + cos2πi( hx j + h 2 ky j + k 2)} Reflection conditions: 2 F( h0) 2 f j { cos2πhx j ( 1 + coshπi) } 0 only when h 2n. 2 F( 0k) 2 f j cos2πky j ( 1 + coskπi) 0 only when k 2n. (4.4) C2mm(9) Rectangular Origin at 2mm Coordinates P2mm + (1/2, 1/2) C-centered lattice reflection conditions hk: h + k 2n. Coordinates: (1) x, y (2) x, y (3) x, y (4) x, y Reflection conditions: hk: h + k 2n Patterson symmetry: C2mm (4.4) Eq. (4.3) Eq. (4.4)ƒ ew» w g// [10] glide line x 1/4 š g//[01] glide line y 1/4 w. (5) Centric point group 4 space group (5.1) P4(10) Square Origin at 4 Coordinates 4//rotation point in plane matrix

8 18 ƒá Á Á Ÿ Á~ Áªœ Á ž w wz l ù. g//[10] at x 1/4 4 tƒ ƒ. (1) (6) Coordinates: (1) x, y (2) x, y (3) y, x (4) y, x Structure factor j F( hk) 2f j { cos2π( hx j + ky j ) + cos2π( hy j kx j )} F( h k) F( k h) F( k h) š Patterson symmetry P4. (6) Centric point group 4mm space group (6.1) P4mm(11) Square Origin at 4mm Coordinates 4//rotation point in plane l 4 ƒ ù (2) (5) (3) (8) (4) (7) Space-group diagram 1) ùkù m//[11] at x 1/2 and y 1/2 tƒ (1) (3) (2) (4) m//[10]: ƒ ƒ š 4 coordinates (5) x, y (7) y, x (6) x, y (8) y, x w space-group diagram 1) ùkù g//[11] tƒ. m//[11]:. coordinatesƒ 8 coordinatesƒ ù : Coordinates: (1) x, y (2) x, y (3) y, x (4) y, x Coordinates: (5) x, y (6) x, y (7) y, x (8) y, x (6.2) P4gm(12) Point group: 4mm Square Origin at 41g Coordinates 4//rotation point in plane l 4 ƒ ù (1) (3) (2) (4) (1) x, y (2) x, y (3) y, x (4) y, x (5) x + 1/2, y + 1/2 (6) x + 1/2, y + 1/2 (7) y + 1/2, x + 1/2 (8) y + 1/2, x + 1/2 (7) Acentric point group 3 space group (7.1) P3(13) Hexagonal Origin at 3 Coordinates 3//rotation point in plane l 3 ƒ ù

9 16«1y, 2005 The Seventeen Plane Groups (Two-dimensional Space Groups) 19 (1) (2) (3) (1) (2) (6) Coordinates: (1) x, y (2) y, x y (3) x + y, x Coordinate transformation matrix transpose inverse matrixƒ Miller index transformation matrix. 3) Coordinate transformation matrixƒ A A T š A T cofactor [A T ] 1. matrix l equivalent reflection : (3) (4) Coordinates: (1) x, y (2) y, x y (3) x + y, x Coordinates: (4) y, x (5) x + y, y (6) x, x y (8.2) P31m(15) Hexagonal Coordinates 3//rotation point in plane l 3 ƒ ù (1) (2) (3) (1) m//[12] l 3 ƒ ƒ. hk (hx + ky) even function cosine 6 reflections w P(xy). Patterson function P( xy) 1 cos2π Ā - F ( hk ) 2 (1) (5) (2) (6) (3) (4) Patterson symmetry P6. (9.1) P6(16) žw. (8) Acentric point group 3m space group (8.1) P3m1(14) Hexagonal Origin at 3m1 Coordinates 3//rotation point in plane l 3 ƒ ù (1) (2) (3) (1) m//[10] l 3 ƒ ƒ. (1) (5) Coordinates: (1) x, y (2) y, x y (3) x + y, x Coordinates: (4) y, x (5) x y, y (6) x, x + y (9) Centric point group 6 space group (9.1) P6(16) Hexagonal Origin at 6 Coordinates 6//rotation point in plane l 6 ƒ. (1) (6) (2) (4) (3) (5)

10 20 ƒá Á Á Ÿ Á~ Áªœ Á ž w wz Coordinates: (1) x, y (2) y, x y (3) x + y, x Coordinates: (4) x, y (5) y, x + y (6) x y, x Coordinate transformation matrix l equivalent reflection w. Coordinate transformation matrixƒ A A T š A T cofactor [A T ] 1. matrix l equivalent reflection : (3) (7) (4) (11) (5) (12) (6) (10) m//[12] l tƒ ù. (1) (11) Patterson symmetry P6. (10) Centric point group 6mm space group (10.1) P6mm(17) Hexagonal Origin at 6mm Coordinates 6//rotation point in plane l 6 ƒ ù (1) (6) (2) (12) (3) (10) Coordinates: (1) x, y (2) y, x y (3) x + y, x Coordinates: (4) x, y (5) y, x + y (6) x y, x Coordinates: (7) y, x (8) x + y, y (9) x, x y Coordinates: (10) y, x (11) x y, y (12) x, x + y (2) (4) (3) (5) m//[10] l 6 ƒ ƒ. (1) (8) References 1) International Tables for Crystallography, Vol. A. Edited by Theo Hahn, Fourth, revised edition, Kluwer Academic Publishers, p. 15 (1995). 2) Charles Kittel, Introduction to Solid state Physics. Second Edition, John Wiley & Sons, Inc. p. 7 (1956). 3) Suh, I.-H., Oh, I.-K., Yoon, Y. K. and Kim, M.- J., J. of the Korean Physical Society, 19(4), (1986). (2) (9)

THE FIVE TYPES OF PLANAR 2-D LATTICES. (d) (e)

THE FIVE TYPES OF PLANAR 2-D LATTICES (a) (d) (b) (d) and (e) are the same (e) (c) (f) (a) OBLIQUE LATTICE - NO RESTRICTIONS ON ANGLES BETWEEN THE UNIT CELL EDGES (b) RECTANGULAR LATTICE - ANGLE BETWEEN