Whats up with the Bravais Lattices?

Hello everyone.

We're doing 'The Solid State' in our school where I first went through the concept of Unit Cells Lattices. Can someone help me through with -

1) Why are there only 14 such lattices?

2) How can one give a rigorous proof about the non existence of unit cells of 'End-centered cubic lattice' and 'face centered tetragonal lattice'.

Thank you!

#Chemistry #PSQuest #BravaisLattice

Easy Math Editor

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Comments

2) Even if you put particles on edge centers, it will again form same type of but smaller simple cubic unit cells. And same is the case of face centered tetragonal lattice.

Kushal Patankar - 5 years, 11 months ago

Indeed. This is what my teacher explained me. BTW what about the other cases? Do all of them have the same reasoning?

Priyansh Sangule - 5 years, 11 months ago

May be they exist but , their properties may be similar to one of those 14. And this can be reason for having only 14 bravais lattice.

Kushal Patankar - 5 years, 11 months ago

1 is a bit difficult to show.

For 2, the main idea is that if you did place lattice points at those places, you'd get smaller unit cells as repeating patterns.

I came to know about this from the coursera course on "Symmetry: Beauty, Form and Function"

To begin with, I suggest you look up the Wallpaper Symmetry Groups first.

(by the way, I do not have the full answer to your question)

Agnishom Chattopadhyay - 5 years, 11 months ago

Hey, thanks for your views bro! :D

Priyansh Sangule - 5 years, 11 months ago

The answer to the first could be that only 14 bravais lattice are seen practically yet.

Naman Kapoor - 5 years, 11 months ago

Nope, they are mathematical structures - not empirically derived structures in chemistry.

Agnishom Chattopadhyay - 5 years, 11 months ago

You both are somewhat right. They do are mathematical structures but we exclude the ones which are practically not possible.

Priyansh Sangule - 5 years, 11 months ago

@Priyansh Sangule – Which ones are they?

Agnishom Chattopadhyay - 5 years, 11 months ago

@Agnishom Chattopadhyay – Like, Edge Centric and other cases...

Priyansh Sangule - 5 years, 11 months ago

@Priyansh Sangule – As far as my understanding goes, there are no edge centric bravais lattices because inserting lattice points at the midpoint of every edge it instead transform into another kind of Bravais Lattice

Agnishom Chattopadhyay - 5 years, 11 months ago

@Agnishom Chattopadhyay – Your understanding is, well, alright as always! $\ddot\smile$

Pranjal Jain - 5 years, 11 months ago

@Agnishom Chattopadhyay – Yes. I think this is more appropriate. Thanks!

Priyansh Sangule - 5 years, 11 months ago

See, both of the things you have asked are based on observations so there is actually no answer. The reason agnishom gave is what I don't know.

Vishal Ch - 5 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$