2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry...
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Transcript of 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry...
![Page 1: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/1.jpg)
2D Symmetry(1.5 weeks)
![Page 2: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/2.jpg)
From previous lecture, we know that, in 2D,there are 3 basics symmetry elements:
Translation, mirror (reflection), and rotation.
What would happen to lattices that fulfill therequirement of more than one symmetryelement (i.e. when these symmetry elementsare combined!).
![Page 3: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/3.jpg)
Start with the translation T
Add a rotation A
A
lattice point
lattice point
latticepoint
A T
TT
T: scalar
Tp
T
A translation vector connecting twolattice points! It must be some integerof or we contradicted the basicAssumption of our construction.
T
p: integer
Therefore, is not arbitrary! The basic constrain has to be met!
Combination of translation with rotation:
![Page 4: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/4.jpg)
T
T T
tcos tcos
b
To be consistent with theoriginal translation t:
pTb p must be integer cos21cos2 ppTTTb
p1cos2 p cos n (= 2/) b
-1.5 -- -- -- -1 2 3T-0.5 2/3 3 2T 0 /2 4 T 0.5 /3 6 0 1 0 (1) -T 1.5 -- -- --
p > 4 orP < -2:no solution
T
T T
A A’
B’B
43210-1-2
Allowable rotationalsymmetries are 1, 2,3, 4 and 6.
![Page 5: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/5.jpg)
Look at the case of p = 2
= 120o
TTp
2
1T
2T
21 TT
o21 120 TT
angle
Look at the case of p = 1
n = 3; 3-fold
n = 4; 4-fold
= 90o
TTp
21 TT
o21 90 TT
1T
2T
3-fold lattice.
4-fold lattice.
![Page 6: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/6.jpg)
Look at the case of p = 0
= 60o
TTp
0
1T
2T 21 TT
o21 60 TT
n = 6; 6-fold
Look at the case of p = 3 n = 2; 2-fold
TTp
3
Look at the case of p = -1 n = 1; 1-fold
1 2
TTp
1
Exactly the same as 3-fold lattice.
![Page 7: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/7.jpg)
1-fold2-fold3-fold4-fold6-fold
Parallelogram21 TT
general21 TT
Hexagonal Net21 TT
o21 201 TT
Can accommodate1- and 2-foldrotational symmetries
Can accommodate3- and 6-foldrotational symmetriesSquare Net
21 TT
o21 90 TT
Can accommodate4-fold rotationalSymmetry!
![Page 8: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/8.jpg)
Combination of mirror line with translation:
m
Unless
0.5T
centered rectangular
constrain
Or21 TT
o21 09 TT
Primitive cell
Rectangular
1T
2T
m
![Page 9: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/9.jpg)
Lattice + symmetries of motif (point group) = plane group(5) (1, 2, 3, 4, 6, m, etc)
Parallelogram21 TT
general21 TT
Hexagonal Net21 TT
o21 201 TT
Square Net21 TT
o21 90 TT
(1)
(2)
(3)
21 TT
o21 09 TT
Double cell (2 lattice points)
Centered rectangular(4)
21 TT
o21 09 TT
Primitive cell
Rectangular(5)
![Page 10: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/10.jpg)
Oblique
Rectangular
Centered rectangular
Square
Hexagonal
1, 2
m
m
4
3,6
Five kinds of latticeThe symmetry that thelattice point can accommodate
+
+
+
+
+
Plane group
3D: space group.
Group theory
We will show the concept of group!
![Page 11: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/11.jpg)
Group theory: set of elements (things) for a law of combination is defined and satisfies 3 postulates. (1) the combination of any two elements is also a member of the group; (2) “Identity” (doing nothing) is also a member of the group. “I” aI=Ia=a (a : an element) (3) for element, an inverse exists. a; a-1
a . a-1 = I a-1. a = I
Example: Group {1, -1}; rank 2rank (order) of the group = number of elements contained in a set.
1 -1
1 -11-1 1-1
Another Example: Group {1, -1, i, -i}; rank 4
http://en.wikipedia.org/wiki/Group_(mathematics)
We will show examples for point groups later!
![Page 12: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/12.jpg)
n12346
m[ ][ ][ ][ ][ ]
In a point, there is no translation symmetry!
Therefore, consider 2D point group, we only considerrotation and mirror!
Put rotation symmetry and mirror together ?
![Page 13: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/13.jpg)
Example:
m1
m2
R
R L
L
{1, 1, 2, A} group of rank 4
1 1 2 A
1
1
2
A
1 1 2 A
11 2A
1 12 A
112A
Abelian group: a.b=b.a
2mm: point group2 + m
(1)
(2)(3)
(4) 1 1: 11 2: 2
1 3: A
1 4: 1
![Page 14: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/14.jpg)
3/53/43/23/ 1 AAAAA6-fold 21 A
A 1 is a subset of
2-fold axis
3/43/2 1 AA subgroup
3-fold axis
![Page 15: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/15.jpg)
1 2
?12 LL
RChirality not changed:
T
Rotation is the right choice!
12 ||
?12 A
212 A Combination theorem
2
1 A 12
(1) (2)
(3)(4)
if
2mm
![Page 16: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/16.jpg)
Show it is a group
1 1 2 A
1
1
2
A
1 1 2 A
11 2A
1 12 A
112A
Satisfy 3 postulates?
Rank 4
The number of motif in the pattern is exactly the same as therank (order) of the group!
![Page 17: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/17.jpg)
Hermann and MauguinInternational notation
Rotation axis
n 1, 2, 3, 4, 6
Schonllies notation CnC1, C2, C3, C4, C6
C: cyclic group – all elements are “powers” of some basicOperation e.g. 4
2/23
2/2/32
2/2/ 1 AAAAAAA
http://en.wikipedia.org/wiki/Group_(mathematics)#Cyclic_groups
Notation:
![Page 18: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/18.jpg)
Hermann and MauguinInternational notation
Mirror plane
m
Schonllies notation CS
Cnv : Rotational symmetry with mirror plane vertical to the rotation axis. E.g. 2mm – C2v .
![Page 19: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/19.jpg)
2/A
1
?12/ A
(1) (2)
(3)
2
212/ A
4
S:C4v
HM: 4mmmm
m m
m
m
Only independent symmetry elements.
The rank of this group is ?
R
L
L
4 + m
![Page 20: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/20.jpg)
1
213/ A
2
/6S:C6v
HM: 6mm
The rank of this group is 12!
1
(1) (2)
(3)
3/2A
(1)L
(2)R
(3)R
213/2 A
2
S:C3v
HM: 3mm (correct?)
The rank of this group is 6!
2 is not independent of 1.HM (international notation): 3m
![Page 21: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/21.jpg)
So far we have shown 10 point group or specifically 10 2-D crystallographic point group.HM notation , , , , , , , , , ; Schonllies notation , , , , , , , , , .
10 2-D crystallographic point group
5 2-D lattices
2-D crystallographicspace group
1 2 3 4 6 m 2mm 3m 4mm 6mm
C1 C2 C3 C4 C6 Cs C2v C3v C4v C6v
![Page 22: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/22.jpg)
Oblique
Primitive Rectangular
Centered rectangular
Square
Hexagonal
1, 2
m
m
4
3,6
Compatible with
Compatibility: 2mm, 3m, 4mm, 6mm
![Page 23: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/23.jpg)
2mm m
m
Put mirror planes along the edgeof the cell.
m
m
Primitive RectangularCentered rectangular
m, 2mmCompatible with
Square 4, 4mmCompatible with
![Page 24: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/24.jpg)
30o
T
m with m3T
||m with m3Hexagonal Compatible with
![Page 25: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/25.jpg)
Red ones Blue ones
T
m with m3T
||m with m3
Hexagonal Compatible with 6mm
![Page 26: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/26.jpg)
Oblique
Primitive Rectangular
Centered rectangular
Square
Hexagonal
,
,
,
,
, , ,
Compatible with
1 2
m 2mm
m 2mm
4 4mm
3 6 3m 6mm
![Page 27: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/27.jpg)
General oblique net.
2T
1T
atoms
Type of lattice
Point group
Symbol used to describe the space group
P (for primitive) 1
Space group: p1
Upper case P for 3Dlower case p for 2D
![Page 28: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/28.jpg)
Primitive oblique net + 2 = p2
2T
1T
A
A TB
(1)
(2) (3)
2T
1T
A
plane group: p2
![Page 29: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/29.jpg)
p2
positions with symmetry the lattice point!
![Page 30: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/30.jpg)
p2
![Page 31: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/31.jpg)
General relation between new symmetry position generated bycombining rotation with translation
)2/tan(2/
x
T)2/cot()2/()2/tan(
2/ Txx
T
BAT
)2/cot()2/( Tx
at along the -bisector of T
/2/2
T
A A
(1)(2)
(3)
A2
T
x
/2
B
Question: what kind of symmetryoperation is required in order formotif (1) get to motif (3)?
A : (1) (2);
T
: (2) (3);
![Page 32: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/32.jpg)
/2/2
A
12
(1)(2)
Could we always rotate /2 respect to thedashed line T!
You can always define it that way!
/2
![Page 33: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/33.jpg)
4 + lattice
2T
1T
4 } { 22/32/ AAAA
2/A 1
|| ||
Correct?
Combination of A/2 with T
p4
2T
1T
21 TT
o21 90 TT
2/2/ BAT
2/)4/cot()2/( TTx at
![Page 34: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/34.jpg)
![Page 35: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/35.jpg)
p + 3 = p3
2T
1T
120o
3 }1 { 23/23/43/2 AAAA
Combination of A2 /3 with
3/23/2 BAT
32)3/cot()2/(
TTx
along the -bisector of
at
T
T
X2/3)2/( 22 TTTX
30o
3/23/2 BA
3/2B
mass center
X/3323
1
2
3
3
TTX
![Page 36: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/36.jpg)
2T
1T
60o 60o
(1)
(2) (3)
(1)(2): A2/3;(2)(3): Translation T
(1)(3): B2/3;
2T
1T
60o 60o
21 TT
o21 201 TT
![Page 37: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/37.jpg)
p + 6 = p6 p has to be hexagonal net as well!
2T
1T
6 }1 { 23/3/53/23/43/23/ AAAAAAAA
3-fold
2-fold
From 2-fold rotation
From 3-fold rotation
Combination of A /3 and A- /3 with T
3/3/ BAT
2/3)6/cot()2/( TTx at
2/3T
2/T
T
![Page 38: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/38.jpg)
Combination of mirror symmetry with the translation!
m + p + c
p + m = pm
? T
(1) R
(2) L
(3) L T
@ 2/T
Independent mirror plane
^ is defined with respectto mirror line (plane)
![Page 39: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/39.jpg)
c + m = cm
not an independent mirror plane!(lattice point!)
(1) (3)
(1) R
(2) L
(3) L
T ||T
T ||T
|||| )( TTTT
Glide plane with glide component
Two-step operation
m m m
cm
g g
![Page 40: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/40.jpg)
p + g = pg possible?
g g
? T
(1) R
(2) L (3) L (1) (3)?
2/@ TT
2/@ )(||
|| TTTTT
General form:
)(||
|| TTTT
Remind: 2/@ T
![Page 41: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/41.jpg)
c + g = cg possible?
g gm
2/)2/(@ )2/2/( 121 TTT
2T
1T
2222 2/2/ ;2/ TTTT
g gm
4/@ )2/2/( 12/2122
TTTTT
4/@ 1T
cg = cmrectangular net:
pm, pg, cm!
![Page 42: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/42.jpg)
p + 2mm = p2mm
c + 2mm = c2mm
![Page 43: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/43.jpg)
p (square) + 4mm
Red: p4.Blue: pm.
m
p4mm
Special case of a rectangular.
![Page 44: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/44.jpg)
p (Hexagonal net) + 3m
p360o 60o
60o 60o
two ways centered rectangular net
m edge m || edge
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p3
Cell edge|| Cell edge
p31mp3m1
![Page 46: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/46.jpg)
p31mp3m1
3m3m
3m
3
Not yet done! Glide plane (or line).
![Page 47: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/47.jpg)
p (Hexagonal net) + 6mm = p6 + p3m1 + p31m
Red BlueMirror line
Glide linep6mm
![Page 48: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/48.jpg)
2mm compatible with Rectangular!
mirror plane?
p2mm
What if the mirror line is not passing through the rotation axis?
![Page 49: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/49.jpg)
For example this way? Why not?
How about this way? Why not?
Leave all the two fold rotationaxes maintain undisturbed!
OK
![Page 50: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/50.jpg)
Center rectangular net (c2mm)?
(m ok? ) (g ok? )
p2mg
X XTwo fold rotation symmetries+ offset mirror line
![Page 51: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/51.jpg)
p2gg
Two fold rotation symmetries + offset glide line
![Page 52: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/52.jpg)
Three different ways:
OK? X
![Page 53: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/53.jpg)
p4gm
The same results
This is not C4gm! Because center position is not a lattice!
![Page 54: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/54.jpg)
System (4) Lattice (5) Point group (10) Plane group (17)
Obliquea b
general
Rectangulara b, = 90o
Squarea = b, = 90o
Hexagonala = b,
= 120o
Primitiveparallelogram
Primitiveor centeredrectangular
Square
Hexagonalequilateral
2
pm pg cm
3
3m6
6mm
p3
p3m1 p31mp6
p6mm
4
4mm
p4
p4mm p4gm
m
2mmp2mm p2mg p2gg
c2mm
1
p2
p1
![Page 55: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/55.jpg)
Hermann-Mauguin Notation: pnab or cnab (1) First letter: p for primitive cell, c for centered cell (2) n: highest order of of rotational symmetry (1, 2, 3, 4, 6) (3) Next two symbols indicate symmetries relative to one translation axis. The first letter (a) is m (mirror), g (glide), or 1 (none). The axis of the mirror or glide reflection main axis. The second letter (b) is m (mirror), g (glide), or 1 (none). The axis of the mirror or glide reflection is either || or tilted 180o/n (when n>2) from the main axis.
a
b
1, 2
a
b
3
60o
b
445o
a b
630o
a
Old notes
![Page 56: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/56.jpg)
The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group. E.g. p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections. p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°. cmm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis. p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.
Shortfull
pmp1m1
pgp1g1
cmc1m1
pmmp2mm
pmgp2mg
pggp2gg
p4mp4mm
p6mp6mm
p1: p111p3: p311p4: p411p6: p611
p3m1
Old notes
![Page 57: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/57.jpg)
Symbol forthe plane group
# of the particular planegroup in the set
Symbol for the group in 3D
Point group
Crystal system
p2 No. 2 p211 2 oblique
The information of the international X-ray table
Diagram
symmetry elementsin the net.
First line
![Page 58: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/58.jpg)
origin at 2
(0 0)
x
y(x y)
(1-x 1-y)
) ( yx
) ( yx) ( yx General position(Unique for every
Plane or space group))-1 1( yx
=
Number/cell(rank of position)
2 e 1
Site symmetryAlways 1 for general position
) ( yx) ( yx
Special position(on a symmetry
Element)
1 d 2
1 c 2
1 b 2
1 a 2
)/21 2/1(
)1/2 0(
)0 /21(
)0 0(
x
Wyckoff symbol
byconvention
![Page 59: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/59.jpg)
Notation for asymmetric used to represent point group symmetry: (a) : Asymmetric unit in the plane of the page (b) : Asymmetric unit above the plane of the page (c) : Asymmetric unit below the plane of the page (d) : Apostrophe indicating a left-handed asymmetric unit. Clear circle indicating right-handedness. (e) : Two asymmetric units on top of each other (f) : Two asymmetric units on top of one another, one left-handed and the other right-handed.
+
,
+,
and are mirror images of each other.,
Old notes
![Page 60: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/60.jpg)
Another example
pmm No. 6 p2mm mm Rectangular
,
,
,
,
,
,
,
,
Origin at 2mm
(x y)
) ( yx ) ( yx
) ( yx
2 m ) 2/1( y) 2/1( y2 m ) 0( y) 0( y2 m )/21 (x)/21 (x2 m )0 (x1 2mm )/21 2/1(1 2mm )0 2/1(1 2mm )/21 0(1 2mm )0 0(
)0 (x
4 1 ) ( yx) ( yx ) ( yx) ( yxihgfedcba
![Page 61: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/61.jpg)
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![Page 62: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/62.jpg)
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![Page 63: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/63.jpg)
![Page 64: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/64.jpg)
pmg No. 7 p2mg mm Rectangular
,
,
(x y)
) ( yx
) 2
1( yx
) 2
1( yx
Origin at 2
,
,
4 1 ) ( yx) ( yx ) 2
1( yx)
2
1( yx
2 2 )0 0( )0 2
1(
Not an independentspecial position
(mirror)
2 m ) 4
1( y )
4
3( y
2 2 )2
1 0( )
2
1
2
1(
An independentspecial position
a
b
c
d
How about glide plane? Atoms do not coincide!Glide is never a candidate for a special position!
![Page 65: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/65.jpg)
rank
Symmetryof the
equipoints
designation
yy
xx
1
1
Condition limitingpossible reflection(structure factor)
Old notes
![Page 66: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/66.jpg)
0, 0
1, 0
0, 1
x, y
1-y, x1-x, 1-y
y, 1-x
4 d 1
0, 0
1, 0
0, 1
1/2, 0
2 c 2
0, 1/2
1, 1/2
01/2, 1
= 41/2
0, 0
1, 0
0, 1
1/2, 1/2
1 b 4
0, 0
1, 0
0, 1
1, 1
1 a 4
Old notes
41/4 = 1
![Page 67: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/67.jpg)
Supplement
![Page 68: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/68.jpg)
Does the crystallographic group abelian?Some yes, some no!
m
m
mm
m
m
m
(1)1
12/ A
2/1 A(2)
(3) (3)
(1)
(2)(3)
(3)
2
1(1) (2)
(3)
1A
A1
(3)
(3)(1)
(2)(3)
Commutative: a.b=b.a
Noncommutative group a.bb.a
![Page 69: 2D Symmetry (1.5 weeks). From previous lecture, we know that, in 2D, there are 3 basics symmetry elements: Translation,mirror (reflection),and rotation.](https://reader030.fdocuments.net/reader030/viewer/2022032607/56649ec95503460f94bd6211/html5/thumbnails/69.jpg)
1 1 2 A/2
1
1
2
A
Group: 4mm
A A3/2
A/2
A3/2
1
2
3
4
3 4
3
4
1
1
2
A
A/2
A3/2
3
4
1 2 A/2 A A3/23 4
1
24
A/2 A A3/2 2 3 4
A3/2 1 A/2 A 3 4 1
A A3/2 1 A/2 1
A/2 A A3/2 1 1 2 3
4 1 2 3 1A A3/2
3 4 1 2 A3/2 1 A/2
2 3 4 1 1 A/2 A
(1)
Ask yourself how to get (1) to the rest of position?