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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: bsomnath@iitk.ac.inI 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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