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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!

Note by Priyansh Sangule
1 year, 8 months ago

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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 · 1 year, 8 months ago

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@Kushal Patankar Indeed. This is what my teacher explained me. BTW what about the other cases? Do all of them have the same reasoning? Priyansh Sangule · 1 year, 8 months ago

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@Priyansh Sangule 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 · 1 year, 8 months ago

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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 · 1 year, 8 months ago

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@Agnishom Chattopadhyay Hey, thanks for your views bro! :D Priyansh Sangule · 1 year, 8 months ago

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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 · 1 year, 8 months ago

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The answer to the first could be that only 14 bravais lattice are seen practically yet. Naman Kapoor · 1 year, 8 months ago

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@Naman Kapoor Nope, they are mathematical structures - not empirically derived structures in chemistry. Agnishom Chattopadhyay · 1 year, 8 months ago

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@Agnishom Chattopadhyay You both are somewhat right. They do are mathematical structures but we exclude the ones which are practically not possible. Priyansh Sangule · 1 year, 8 months ago

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@Priyansh Sangule Which ones are they? Agnishom Chattopadhyay · 1 year, 8 months ago

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@Agnishom Chattopadhyay Like, Edge Centric and other cases... Priyansh Sangule · 1 year, 8 months ago

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@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 · 1 year, 8 months ago

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@Agnishom Chattopadhyay Yes. I think this is more appropriate. Thanks! Priyansh Sangule · 1 year, 8 months ago

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@Agnishom Chattopadhyay Your understanding is, well, alright as always! \(\ddot\smile\) Pranjal Jain · 1 year, 8 months ago

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