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Short-‐course on symmetry and crystallography
Part 2:
La8ces, :lings
Michael Engel Ann Arbor, June 2011
Bravais la8ce
b a c
Every la8ce point can be wriFen as: with integers i, j, k.
x = ia + jb + kc
The group of transla:ons is isomorph to , the 3D integers. Z3
Ques%on: How many parameters are needed to specify a la8ce? Defini%on: Two Bravais la8ces are equivalent if they have isomorphic symmetry groups.
Auguste Bravais 1811-‐1863
Bravais also studied magne3sm, the northern lights, meteorology, geobotany, phyllotaxis, astronomy, and
hydrography.
Types of unit cells
Observa%ons: • The unit cell of a la8ce is not unique. • Primi:ve unit cells have the smallest possible volume
of all unit cells. • Any non-‐primi:ve unit cell has volume with an integer n. • A la;ce reduc3on minimizes the orthogonality defect:
La8ce reduc:on is a technique to find “nice” unit cells.
Λ = det(a,b, c)nΛ
δ =abc
Λ
Defini%on: A unit cell is an elementary building block of the la8ce.
Classifica:on of Bravais la8ces Verify the following steps: • Step 1a: Only rota:onal symmetries of order 2, 3, 4, 6 can
appear in two and three dimensions. • Step 1b (crystallographic restric%on): Point symmetries in
two and three dimensions have order 2, 3, 4, 6. • Step 2 (inversion): Inversion is always a point symmetry of a
la8ce. • Step 3a (mirrors, 2D): If a two-‐dimensional la8ce has a
mirror axis, then there is a perpendicular direc:on which is also a mirror axis.
• Step 3b (mirrors, 3D): If the la8ce has a two-‐fold rota:on axis, then there is a mirror symmetry perpendicular to this axis. If the la8ce has a mirror symmetry, then the axis perpendicular to the mirror plane is a two-‐fold rota:on axis.
The five two dimensional Bravais la8ces • Possible point symmetry groups are: D1, D2, D4, D6. • Only the following five possibili:es exist:
oblique rectangular centered rectangular
hexagonal square
D4 D6
D1 D2 D2
Crystallographic point groups Some point groups are not compa:ble with periodicity. They cannot by point groups of crystals.
ß Generaliza:on of the crystallographic restric:on to three dimensions.
There are 32 crystallographic point groups (27 “axial” PGs and 5 “Platonic” PGs).
Hermann-‐Mauguin nota:on
• An n-‐fold rota:onal symmetry is denoted by the number n. • A mirror is represented by the leFer “m”. • If there is an inversion present, then a bar is added over a
number. • A combina:on of an n-‐fold rota:onal symmetry and a
perpendicular mirror is wriFen as “n/m”. • A point group consists of three
symmetry symbols, one for each of the axis (x, y, z).
In prac%ce: Use tables to look up symmetry groups. Understand only the basic meaning.
The 32 crystallographic point groups in 7 crystal systems
Remove point groups without inversion symmetry Some point groups are always subgroups of others
Classifica:on of three-‐dimensional Bravais la8ces The seven possible point symmetry groups of la8ces are: Ci, C2h, D2h, D3d, D4h, D6h, Oh.
The 14 three-‐dimensional Bravais la8ces:
Ci Point
Symmetry
C2h
D2h
D4h
D3d
D6h
Oh
Pearson symbol
hFp://en.wikipedia.org/wiki/Pearson_symbol
Exercise 1: Rock salt (NaCl)
Exercise 2: Zincblende (ZnS)
Exercise 3: ???
Exercise 4: Wurtzite ZnS
Exercise 5: Beta-‐Tin
Exercise 6: alpha-‐Manganese
Exercise 7: Silicon
Exercise 8: Perovskite (CaTiO3)
Great online resource for crystal structures
hFp://cst-‐www.nrl.navy.mil/la8ce/
Triclinic, Ci
Monoclinic, C2h
Orthorhombic, D2h
Tetragonal, D4h
Cubic, Oh
Rhombohedral, D3d Hexagonal, D6h
Subgroup rela:onship (no centering)
α, β, γ ≠ 90° a ≠ b ≠ c
α = β = 90°, γ ≠ 90° a ≠ b ≠ c
α = β = γ = 90° a ≠ b ≠ c
α = β = γ = 90° a = b ≠ c
α = β = γ = 90° a = b = c
α = β = γ ≠ 90° a = b = c
Hierarchy of the 14 Bravais la8ces
M. Hosoya, Acta Crystallographica A56, 259 (2000).
Comparison of fcc and hcp • Fcc is a Bravais laQce.
Every atom can be mapped onto every other atom by a transla:on symmetry. Equivalent: The primi:ve unit cell contains one atom. There are three 3-‐fold axes.
• Hcp is not a Bravais laQce. The Bravais la8ce is hexagonal. The primi:ve unit cell contains two atoms. There is one 3-‐fold axis.
hcp fcc bcc other complex
Structure of the elements
J. Donohue, The structure of the elements, Wiley (1974) M. I. McMahon, R. J. Nelmes, Chem. Soc. Rev. 35, 943 (2006)
H 1.0079
hydrogen
1
Li 6.941
lithium
3
Fr [223]
francium
87
Be 9.0122
beryllium
4
Mg 24.305
magnesium
12
Ca 40.078
calcium
20
Sr 87.62
stron%um
38
Ba 137.33
barium
56
Ra [226]
radium
88
Sc 44.956
scandium
21
Y 88.906
y_rium
39
Ti 47.867
%tanium
22
Zr 91.224
zirconium
40
Hf 178.49
hafnium
72
Rf [261]
rutherfordium
104
V 50.942
vanadium
23
Nb 92.906
niobium
41
Ta 180.95
tantalium
73
Db [262]
dubnium
105
Cr 51.996
chromium
24
Mo 95.94
molybdenum
42
W 183.84
tungsten
74
Sg [266]
seaborgium
106
Mn 54.938
manganese
25
Tc [98]
techne%um
43
Re 186.21
rhenium
75
Bh [264]
bohrium
107
Fe 55.845
iron
26
Ru 101.07
ruthenium
44
Os 190.23
osmium
76
Hs [277]
hassium
108
Co 58.933
cobalt
27
Rh 102.91
rhodium
45
Ir 192.22
iridium
77
Mt [268]
meitnerium
109
Ni 58.693
nickel
28
Pd 106.42
palladium
46
Pt 195.08
pla%nium
78
Ds [271]
darmstad%um
110
Cu 63.546
copper
29
Ag 107.87
silver
47
Au 196.97
gold
79
Rg [272]
roentgenium
111
Zn 65.39
zinc
30
Cd 112.41
cadmium
48
Hg 200.59
mercury
80
Cn [285]
copernicum
112
B 10.811
boron
5
Al 26.982
aluminium
13
Ga 69.723
gallium
31
In 114.82
indium
49
Tl 204.38
thallium
81
C 12.011
carbon
6
Si 28.086
silicon
14
Ge 72.61
germanium
32
Sn 118.71
%n
50
Pb 207.2
lead
82
Uuq [289]
ununquadium
114
N 14.077
nitrogen
7
P 30.974
phosporus
15
As 74.922
arsenic
33
Sb 121.76
an%mony
51
Bi 208.98
bismuth
83
O 15.999
oxygon
8
S 32.065
sulfur
16
Se 78.96
selenium
34
Te 127.60
tellurium
52
Po [209]
polonium
84
F 18.998
fluorine
9
Cl 35.453
chlorine
17
Br 79.904
bromine
35
I 126.90
iodine
53
At [210]
asta%ne
85
He 4.0026
hydrogen
1
Ne 20.180
neon
10
Ar 39.948
argon
18
Kr 83.798
krypton
36
Xe 131.29
xeon
54
Rn [222]
radon
86
Ce 140.12
cerium
58
Th 232.04
thorium
90
Pr 140.91
praseodymium
59
Pa 231.04
protac%nium
91
Nd 144.24
neodynium
60
U 238.03
uranium
92
La 138.91
lanthanum
57
Ac [227]
ac%nium
89
Pm [145]
promethium
61
Np [237]
neptunium
93
Sm 150.36
samarium
62
Pu [244]
plutonium
94
Eu 151.96
europium
63
Am [243]
americium
95
Gd 157.25
gadolinium
64
Cm [247]
curium
96
Tb 158.93
terbium
65
Bk [247]
berkelium
97
Dy 162.50
dysprosium
66
Cf [251]
californium
98
Ho 164.93
holmium
67
Es [252]
einsteinium
99
Er 167.26
erbium
68
Fm [257]
fermium
100
Tm 168.93
thulium
69
Md [258]
mendelevium
101
Yb 173.04
y_erbium
70
No [259]
nobelium
102
Lu 175.97
Lute%um
71
Lr [262]
lawrencium
103
Uut [284]
ununtrium
113
Uup [288]
ununpen%um
115
Uuh [292]
ununhexium
116
Uus ununsep%um
117
Uuo [294]
ununoc%um
118
Na 22.990
sodium
11
K 39.098
potassium
19
Rb 85.468
rubidium
37
Cs 132.91
caesium
55
complex under high pressure
Tilings Defini%on: A :ling (or tessela:on) is a two-‐dimensional (or three-‐dimensional) paFern that fills space with no overlaps and no gaps.
A :ling is a generaliza:on of a Bravais la8ce in the sense that one unit cell in one orienta3on is replaced by one unit cell in several orienta3ons or several unit cells.
Wall pain:ng in the Alhambra (Spain, 14th century) Tessela:on of pavement
The three regular :lings
Square :ling 44
Triangle :ling 36
Honeycomb :ling 63
é Vertex-‐figure First number: # of polygon edges Second number: # of polygons around a vertex
Semiregular or Archimedean :lings
Snub hexagonal I, 34.6 Snub hexagonal II, 34.6 Trihexagonal (Kagome), 3.6.3.6
The vertex of a regular n-‐gon (polygon with n ver:ces) corresponds to a frac:on of of the full turn. For a space-‐filling :ling, the following equality has to hold: with integers >2: n1, n2, . . .
(n− 2)/(2n)
(n1 − 2)/(2n1) + (n2 − 2)/(2n2) + . . . + (nm − 2)/(2nm)
Truncated square, 4.28 Truncated hexagonal, 3.122 Truncated Trihexagonal, 4.6.12
Elongated triangular, 33.42 Snub square (sigma), 32.4.3.4 Rhombitrihexagonal, 3.4.6.4
Random rhomb :ling
Blunt et al., Random tiling and topological defects in a 2D molecular network, Science 322, 1077-1081 (2008)
A random %ling is a :ling of a finite number of different 3les (up to transla:on and rota:on) and without any long-‐range order.
Three-‐dimensional embedding of the rhomb :ling
The higher-‐dimensional embedding allows to study topological proper3es of 3lings
Three basis vectors are necessary to index the :ling ver:ces.
Disloca:ons in :lings
Embedding of quasiperiodic :lings
The minimal embedding dimension of an n-‐fold symmetry is given by the Euler to:ent func:on.
Islamic :lings
Zaouïa Moulay Idriss II in Fez, Morocco (14th century)
[Makavicky2, App. Cryst. June 2011]
Seljuk Mama Hatun Mausoleum in Tercan, Turkey (~1200 C.E.) [Lu, Steinhardt, Science 2007]
Darb-‐i Imam Shrine Isfahan, Iran (1453 AD).
Persian carpet
Exam ques:ons
• How many parameters are needed to specify a Bravais la8ce? • What symmetries are allowed by the crystallographic
restric:on? • How can one proof the crystallographic restric:on? • How many point groups are there in 2D and 3D? • What Bravais la8ces exist in 3D? • Determine the Bravais la8ce of XXX.