Explanation of Lattices with symmetry labeling. Using symmetries to explain physical & chemical properties.

1. 3D CRYSTALS
ABHISHEKRAWAT
1001875407
Solid State physics 1

2. Reason for choosing the topic
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3. Why it is important?
Importance:
Have varied employability
Knowing the structure can open more
possibilities
Structural knowledge for various purposes will
be enhanced
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4. Outline
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1. Solids: Lattices
2. Symmetry: Point group and Space group
3. Centrosymmetric & Non-Centrosymmetric Crystals
4. Analytical method of Deconvoluting Crystals
5. Conclusion

5. Introduction
Solids
Crystalline Solids Amorphous Solids
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 Have periodic arrangement
of atoms/ molecules.
 Are anisotropic in nature.
 Have definite Melting point.
 Have irregular arrangement of
atoms/ molecules.
 Are isotropic in nature.
 Have a range of Melting point.
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6. How crystalline solids are formed?
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For 2D lattice:
1 1 2 2
R n a n a
 
For 3D lattice:
1 1 2 2 2 3
R n a n a n a
  
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7. 2D lattice & symmetry
As,
ο
1
mt
360 m
2
cos = cos = =
n t 2

 
 
 
where,
m = integer
n = fold symmetry
t = trans. vector
n 1 2 3 4 5 6
(theta) 360o 180o 120o 90o 72o 60o
Cos (theta) 1 -1 -1/2 0 0.3 1/2
m 2 -2 -1 0 0.6 1 7

8. 3D lattices – Packing & Unit cells
Simple
Cubic
(P)
G.P. = (0,0,0)
Face
Centered
(F)
G.P. = (0,0,0); (0.5,0.5,0);
(0.5,0,0.5); (0,0.5,0.5)
Body
Centered
(I)
G.P. = (0,0,0); (0.5,0.5,0.5)
End
Centered
(A, B, C)
G.P. = (0,0,0); (0.5,0.5,0)
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9. Crystal Class & Bravais Lattice
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10. Crystal Class & Bravais Lattice
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11. Point group Vs Space Group
Symmetry
- Point group is defined
for a single basis/
moiety
- There are 32-point
groups
Molecular Sulphur
Point group = D4d
- Space group is
defined for 3D
occupation of basis/
moieties.
- There are 230 space
groups.
Monoclinic Sulphur
Space Group = P2/c
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12. Point group
Point group explains the symmetry of a molecule keeping one point fixed.
Point group consists of following elements:
• Identity element: E
• Principal axis of rotation: Cn & Dn
• Mirror Planes : σv , σh & σd
• Improper axis of rotation: Sn = Cn . σ
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13. Space Group
Space group explains the symmetry of the lattice consists of basis points.
Space group consists of following elements:
• Crystal type: P, F, I, C, R.
• Rotation/ mirror symmetry: n/ m
• Screw axis: nm
• Glide plane: a, b, c, n & d
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14. Space group
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15. Schoenflies Vs International notation
Non- Centrosymmetric Space Group:
- Can be Chiral.
- Can be Polar.
- Forms dipole upon mechanical stress.
Centrosymmetric space group:
- Are neither Chiral nor Polar.
- Forms highly symmetrical structure.
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16. Non-centrosymmetric Crystal
Lead Zirconate titanate (PZT):
Space Group: P4mm
Crystal Symmetry: Non- centrosymmetric
- Upon mechanical stress, forms dipole and
hence develops voltage.
- Can be used as spark ignitor
- Exhibits pyroelectric properties, upon heating
polar dielectric PZT produces electric responses.
- Can be used as heat sensor.
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17. Techniques to find crystal structure
https://chanl.unc.edu/instrument/xrd/
X-Rays Diffractometer Transmission Electron Microscope
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18. Analytical Method to deconvolute Crystal
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Fig: PXRD pattern of cubic Aluminum sample.

19. Analytical Method
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 
2 2 2
2
2 2
2 2 2
2 sin
4 sin
4 sin
d
d
a
h k l
 
 
 



 
Bragg’s equation: From 1st order diffraction

20. Analytical Method
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Fig: Indexing PXRD peaks of cubic Aluminum sample.

21. Applications of crystals
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Advanced Materials 30(14) DOI:10.1002/adma.201870094

22. Conclusion
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1. Crystals were always an important part of our lives.
2. Knowing the crystals can open many possibilities,.
3. Employing past knowledge, can help to solve many problems.
4. As it is one of the most systematic and organized field of science, hence
the crystallographic database will be valid forever.

23. 23
Dr. Dorothy Crowfoot Hodgkin
1964 Chemistry Noble Prize winner
Thank you