# Symbolism Symmetry Operation Schnflies Notation International Notation Proper

Symbolism Symmetry Operation Schnflies Notation International Notation Proper Rotation (by 2//n) Cn (C2/, C3, C4, ) n (2/, 3, 4, ) Identity Improper Rotation E = C1 Sn = h Cn (S3, S4, S5, ) 1 n 1 n (3, 4, 5,) Inversion (x,y,z) (x,y,z) i = S2/ 1 Mirror plane Principal Axis h = S1 Mirror plane Principal Axis v , d (= S1) /m (n/m is the designator: 4/m) 2/ m= y In Schnflies notation, what does the symbol S2 mean? (x,y,z) x In International notation, what does the symbol 2/ mean? y (x,y,z) x 1 Symbolism: Crystal Systems What rotational symmetries are consistent with a lattice (translational symmetry)? Crystal System Minimum Symmetry Primitive Unit Cell Triclinic None a b c; Monoclinic

One 2/-fold axis (b-axis) a b c; = = 90, 90 Orthorhombic Three orthogonal 2/-fold axes a b c; = = = 90 Tetragonal One 4-fold axis (c-axis) a = b c; = = = 90 Cubic Four 3-fold axes a = b = c; = = = 90 Trigonal One 3-fold axis a = b = c; = = a = b c; = = 90, = 12/0 Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 12/0 c a b Lattice Types = angle between b and c = angle between a and c = angle between a and b 2/ Symbolism: Bravais Lattices 7 Crystal Systems = 7 Primitive Lattices (Unit Cells): Including Centering: 14 Bravais Lattices Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types Triclinic None a b c; P Monoclinic

One 2/-fold axis (b-axis) a b c; = = 90, 90 P C Orthorhombic Three orthogonal 2/-fold axes a b c; = = = 90 P C (A) I F Tetragonal One 4-fold axis (c-axis) a = b c; = = = 90 P I Cubic Four 3-fold axes a = b = c; = = = 90 P I F Trigonal One 3-fold axis a = b = c; = = a = b c; = = 90, = 12/0 R (rhombohedral) P Hexagonal One 6-fold axis (c-axis) a = b c; = = 90, = 12/0 P Centering? c a b = angle between b and c = angle between a and c = angle between a and b 3 Symbolism: Point Groups Schnflies Notation Type Symbol Features Uniaxial

n Single rotation axis Cn Low Symmetry Dihedral Polyhedral nh + mirror plane Cn axis nv + n mirror planes || Cn axis 1 Asymmetric (NO symmetry) s Mirror plane only i Inversion center only n Rotation axis Cn + n C2/ axes Cn axis nd + n mirror planes || Cn axis nh + mirror plane Cn axis T, Th , Td Tetrahedral; 4 C3 axes (cube body-diagonals) O, Oh Octahedral; 4 C3 axes + 3 C4 axes (cube faces) I, Ih Icosahedral; 6 C5 axes 4 Symbolism: Crystallographic Point Groups Allowed Rotations = C1 C2 C3 C4 C6 32 Point Groups International Symbol Crystal System

Schnflies Symbol Full Abbrev. Triclinic 1 1 1 1 1 s 1m1 or 11m m 2/ 12/1 or 112/ 2/ 2/ / No 2/h 12//m1 or 112//m 2//m 4 / Yes 2/v 2/mm 2/mm 2/ 2/2/2/ 2/2/2/ 4 / No 2/h 2//m 2//m 2//m 44 mmm 44 8 / Yes 42/m 4/m 42/m 4/m

(Holohedral) Monoclinic (Holohedral) Orthorhombic (Holohedral) Tetragonal i 4 4 4h Directions Order / Inversion? 1 / No 2/ / Yes [010] or [001] [100][010][001] [001]{100}{110} 2/ / No 4 / No 4 / No 4 / No 8 / Yes 8 / No 2/d 4v 4mm 4mm 8 / No 4 42/2/ 42/2/ 8 / No 5 Symbolism: Crystallographic Point Groups (cont.) International Symbol Crystal System Schnflies Symbol Full Abbrev. Directions

Trigonal 3 3 3 3 3 [001]{100}{2/10} Hexagonal 3v 3m or 3m1 3m or 3m1 6 / No 3 3m 3 m1 32/ or 32/1 332/ m or 3 m1 or 32/1 6 / No 3d 6 3h 6h 12/ / Yes 6 6 6 m2/ 6/m 6 6 [001]{100}{2/10} Cubic 6 / No 6 / No 6 m2/ 6/m 12/ / Yes 12/ / No 3h (Holohedral) 3 / No 6 / Yes

6 (Holohedral) Order / Inversion? 6v 6mm 6mm 12/ / No 6 62/2/ 62/ 12/ / No 6h / m 32//m 6/m2/ 2//m m3 6/mmm 2/4 / Yes T 43m 2/3 43m 2/3 12/ / No 2/4 / Yes Th Td {100}{111}{110} 4 / m 3 2/ / m m3m 2/4 / No 6 Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m) Schnflies International Coordinates (1) E 1

x, y, z (2/) C2/ = C42/ 2/ 0,0,z (3) C4 (4) Schnflies International Coordinates (9) i x, y, z x, y, z (10) h 4+ 0,0,z y, x, z (11) S 43 0,0,z; 0,0,0 y, x, z C43 4 0,0,z y, x, z (12/) S4 0,0,z; 0,0,0 y, x, z (5) C2/ 2/ 0,y,0 x, y, z (13) v

m x,0,z x, y, z (6) C2/ 2/ x,0,0 x, y, z (14) v m 0,y,z x, y, z (7) C2/ 2/ x,x,0 y, x, z (15) d m x,,z y, x, z (8) C2/ 2/ x,,0 y, x, z (16) d m x,x,z y, x, z m x,y,0 x, y, z 4/mmm y x + 7 Symmorphic Space Groups Point Group = Space Group = Symmorphic Space Groups (73): Nonsymmorphic Space Groups (157):

8 Symmorphic Space Groups Space Group: I4/mmm Asymmetric Unit: BaFe2As2 Ba (2a): Fe (4d): As (4e): 0 0 0 0 0 0 0.3544 9 Symmorphic Space Groups What is the empirical formula of this compound? How many formula units are in the unit cell? Asymmetric Unit: Dy Dy Ni B B (4e) (4d) (8j) (16l) (16m) 0 0 0.273 0.274 0.269 0 0.5 0.5 0.111 0.269 0.203 0.25 0 0 0.152 10 11

## Recently Viewed Presentations

• BEOL min. pitch = 2.λ. source.k 1 / NAλ. source =193nm / NA=1.35 / k 1. is the process coefficient =0.25 (difficult printability)Min. Pitch ~ 72nm with one mask Solution: Double Patterning Technology. DPT utilizes the existing lithography intelligently. Most...
• Structuring and Analyzing Arguments: The Classical, Toulmin, and Rogerian and Ad Herennium Models Key Terms: Deductive vs. Inductive Reasoning Deductive Reasoning = in traditional Aristotelian logic, the process of reasoning in which a conclusion follows necessarily from the stated premises;...
• A yellow legged rooster was crossed with a white legged hen. All fifteen of the offspring were yellow legged. What is the genotype of the rooster? Practice Test cross. Suppose you work for a company that sells plant seeds. You...
• Developmental Genetics Topics : I. Genetic Basis of Differentiation II. Control of Gene Expression in Differentiation III. Nature of Determination Development ของ organism การเจริญพัฒนา Single fertilized egg cell เป็น embryo และไปเป็น adult Adult ประกอบด้วย cells จำนวนหลาย trillions (1012 ...
• ‏ New version of Glide XP - not yet released Integration of WaterMap with Glide XP Results of integrating watermap with Glide - tests on PDB data sets (decoy data from docking of 1000 decoys, Schrodinger decoy set) PDE4 decoys...
• Following Napoleon's defeat in Russia the European powers allied themselves against France. The European powers marched all the way into Paris causing the French to surrender. One of the terms of the surrender was that Napoleon be removed from power...
• Jet Noise Reduction - Chevron Nozzle Low Stage Number Compressor Turbine Aerodynamics Higher Temperature Blade Material Improved Coatings Improved Disk Material Developing Better Materials for 4-to-1 Hot Section Life Increase Summary GEAE Remains the Industry Leader in Aircraft Engine Technology...
• IEEE Framework Group on Earth Observations www.earthobservations.org Global Earth Observation System of Systems IEEE-GEO Collaboration Opportunities Group on Earth Observations Global Earth Observation System of Systems GEOSS Global, Coordinated, Comprehensive Sustained System of Observing Systems GEOSS Global Earth Observation System...