atomic arrangement in solidsmaecourses.ucsd.edu/~jmckittr/mae251-wi11/251 lecture...atomic...
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1MATS 227 / MAE 251/CHEM 222Lecture notes 1
Atomic arrangement in solids
• Solids are classified by theway the atoms/ions arearranged
• In crystalline solids, there isa periodic, long range orderto the arrangement
• In amorphous solids (glass)there is no long range order
2MATS 227 / MAE 251/CHEM 222Lecture notes 1
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2-fold rotation leaves object unchanged
3-fold rotation leaves object unchanged
120˚ 120˚
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parallelogram
rectangle square
1 2
m,2mm
m,2mm
4,4mm
hexagonal
3,3m
6,6mm
centered rectangle
5 distinct 2D nets
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Determination of 2D point groups
C6vC6
C4vC4
C3vC3
C2vC2
C1v = C1h = CsC1
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Determination of 2D point groups
6mm6
4mm4
3m3
2mm2
m1
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The 17 plane groups
p2mg, p4mg, p31m(6) Rotation, mirror and glide
p2gg(5) Rotation and glide
p2mm, c2mm, p4mm, p3m1,p6mm
(4) Rotation and mirror
pg(3) Glide
pm, cm(2) Mirror
p1, p2, p3, p4, p6(1) Rotation
Plane groupSet of symmetriesTranslation +
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p1
cm
pgpm
p2 p2mm
c2mm p2gg p2mg
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p4 p4mm p4gm
p3 p3m1 p31m
p6p6mm
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p1 pm
p2gg
pg
cm
http://escher.epfl.ch/escher/
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Determine the plane group
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Three dimension point symmetry operations
• We need to add someadditional operations– Inversion (i or )• x, y, z → -x, -y, -z
– Horizontal mirror plane
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•+z-z
•
Ch or σh 6
Enantiomorphbelow the plane
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– Improper rotation (Sn = σhCn)• Rotation of 2π/n followed by reflection in a horizontal plane (plane ⊥
to rotation axis)– The only one we have to consider as a point group is S4
α
S4
S3 = C3h = 6_
S6 = 3 = C3i
_
S2 = i = 1_
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–Glide plane instead of glide axis•Glide is still translation + reflection
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– Rotoinversion (iCn or )• C2i = m C3i = S6 C4i = S4
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n
3 6
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Adding another 2-fold axis
• We can add an additional 2-fold axis to some groupsforming the D groups
• If we take C4 and add a perpendicular 2-fold axis
Above the planeBelow the plane
C4 (4) D4 (422)
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• Consider C6 and D6
2f axis
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= C3i
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23MATS 227 / MAE 251/CHEM 222Lecture notes 1