How Do You Prove the Volume of a Bravais Lattice?

[FONE]

Proving Bravais Lattice Volume?!?

Hi guys,

So with a Bravais lattice, you have 3 basis vectors: a1, a2, and a3.

I know that you would get the volume of the lattice as a scalar product of the three: V = a1 dot [a2 x a3].

How would you start going about PROVING this? A little direction to start would be helpful.

Thanks!

 

[OlderDan]

Hi guys,

So with a Bravais lattice, you have 3 basis vectors: a1, a2, and a3.

I know that you would get the volume of the lattice as a scalar product of the three: V = a1 dot [a2 x a3].

How would you start going about PROVING this? A little direction to start would be helpful.

Thanks!

You start by recognizing or proving that the area of a parallelogram is the product of the lengths of adjacent sides times the sine of the included angle. Then you recognize that the volume of any prism (right prism or not) is the area of the base times the height. Then you relate that to the triple scalar product.

 

[FONE]

That was very helpful, thanks!

You start by recognizing or proving that the area of a parallelogram is the product of the lengths of adjacent sides times the sine of the included angle. Then you recognize that the volume of any prism (right prism or not) is the area of the base times the height. Then you relate that to the triple scalar product.