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Symmetry and Properties of Crystals(MSE638)
Somnath Bhowmick
Materials Science and Engineering, IIT Kanpur
January 8, 2019
Contact details
Somnath BhowmickI Email: [email protected] Office: FB 410I Phone: 7161I Course web-page:http://home.iitk.ac.in/ bsomnath/mse638/WWW/index.html
Schedule: Tuesday 14:10-15.25, Friday 14:10-15.25
Venue: FB-413
Grading policy:I Quiz (2) – 30%I Midsem – 35%I Final – 35%I Surprise quiz – 5% (extra)
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Course content
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Course content
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Snowflake
Identify the symmetries
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Regular hexagon – same as snowflake
Total six lines of symmetry.
Lines are vertical (w.r.t plane) mirrors.
The 6-fold rotation axis is perpendicular to the plane (it has to be for2D).
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Lines of symmetry in 2D objects
Lines are vertical (w.r.t plane) mirrors.
How about rotational symmetry ?
The axis of rotation has to be perpendicular to the plane (for 2D).
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Reflection symmetry in a cube
Total 9 planes.
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Rotational symmetry in a cube
Total 13 axes.
3 tetrad (4-fold rotation)
4 triad (3-fold rotation)
6 diad (2-fold rotation)
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Inversion symmetry in a cube
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Story so far
We have learned about three types of symmetries in finite objectsI ReflectionI RotationI Inversion
Atoms are finite objects
Molecules are finite objects
Crystals are “infinite” objects
How to build a crystal starting from atoms/molecules?
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Ancient Architecture – Belur Temple in Karnataka
How can we get such nice patterns?
Take one object and repeat it.
What are the ways to repeat?
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Periodic Repetition
T T TT2
T3
Translation
Reflection
Rotation
Glide
gg2
g3
g g
Image taken from Buerger, Elementary Crystallography
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Frieze patterns
# Translation Glide Rotation (180◦) Mirror(v) Mirror(h)
1 Y N N N N
2 Y Y N N N
3 Y N N Y N
4 Y N Y N N
5 Y Y Y Y N
6 Y N N N Y
7 Y N Y Y Y
These are 2D objects repeated along a line.
Question: why do we consider only 2-fold roation symmetry?
Symmetry operates all over space, including other symmetry elements.
Other rotations generate planar patterns, instead of linear patterns.
Glide ≡ translate + reflect; translation vector ‖ reflection axis.
Reflection is a special case of glide.
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What is symmetry?
An object possesses a symmetry when there is an operation thatmaps the “new object” (after operation) exactly onto the “originalobject” (before operation).
Symmetries can be continious and discrete.
Example: Circle has continuous rotational symmetry
Example: Hexagon has discrete rotational symmetry
Discrete symmetries are only a subset of continuous symmetries
Crystallography – discrete or continuous symmetry ?
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Symmetry of gas, liquid and solid
What are the symmetries of empty space?
Continuous translational and rotational symmetry.
Empty space has maximum symmetry, it is completely homogeneousand isotropic.
Volume occupied by a gas molecule ∼ 33 A3 - lot of empty space.
Gases have continuous translational and rotational symmetry.
Liquids also have continuous translational and rotational symmetry.
Crystalline solids have discrete translational and rotational symmetry.
At liquid to solid phase transition, continuous symmetry is broken,resulting a crystalline solid with discrete symmetry.
Thus, when we talk about symmetries of CRYSTALLINE SOLIDS, wetalk about DISCRETE SYMMETRIES.
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Why to care about symmetry?
Quartz crystal
Symmetry of a crystal is manifested in it’s external shape!
But we are not interested about shape (unless it is diamond), butabout properties of a crystal.
In general a crystal is not isotropic and it’s properties are determinedby the symmetries of the crystal.
Symmetry helps to describe the structure and properties of a crystalin a systematic way.
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Take home message
We discussed about five symmetry elements – translation, reflection,glide, rotation, inversion.
Gas and liquid have continuous translation and rotation symmetries.
Crystals have discrete symmetries.
Golden rule in crystallography: a symmetry operation act on all of thespace, including the other symmetry elements.
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