How Do You Prove the Volume of a Bravais Lattice?

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SUMMARY

The volume of a Bravais lattice is calculated using the formula V = a1 · (a2 × a3), where a1, a2, and a3 are the basis vectors. To prove this, one must first establish that the area of a parallelogram is determined by the product of the lengths of adjacent sides multiplied by the sine of the included angle. Subsequently, the volume of any prism can be derived as the area of the base multiplied by the height, which connects to the concept of the triple scalar product.

PREREQUISITES
  • Understanding of vector operations, specifically dot and cross products
  • Familiarity with the geometric interpretation of volume and area
  • Knowledge of Bravais lattices and their properties
  • Basic principles of linear algebra
NEXT STEPS
  • Study the geometric interpretation of the cross product in three dimensions
  • Learn about the properties of the triple scalar product
  • Explore the relationship between parallelograms and prisms in vector spaces
  • Investigate different types of Bravais lattices and their applications in crystallography
USEFUL FOR

Students and researchers in physics, materials science, and crystallography who are looking to deepen their understanding of lattice structures and their mathematical foundations.

FONE
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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!
 
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FONE said:
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.
 
That was very helpful, thanks!



OlderDan said:
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.
 

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