Proving Bravais Lattice Volume?

In summary, to prove the volume of a Bravais lattice, you start by understanding the properties of parallelograms and prisms, and then relate them to the triple scalar product of the three basis vectors.
  • #1
FONE
4
0
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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  • #2
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.
 
  • #3
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.
 

Related to Proving Bravais Lattice Volume?

1. What is a Bravais lattice volume?

A Bravais lattice volume is the volume of the unit cell of a Bravais lattice, which is a mathematical concept used to describe the repeating pattern of atoms in a crystal structure. It is a fundamental parameter in crystallography and is often used to calculate the density of a crystal.

2. How is Bravais lattice volume calculated?

Bravais lattice volume is calculated by multiplying the lengths of the three sides of the unit cell. The unit cell is the smallest repeating unit of a crystal lattice and can be visualized as a cube with three different edge lengths. The volume is calculated using the formula V = a * b * c, where a, b, and c are the lengths of the three edges.

3. Why is it important to prove Bravais lattice volume?

Proving Bravais lattice volume is important because it helps validate the accuracy of experimental data and theoretical calculations. It also provides a better understanding of the crystal structure and can aid in predicting the physical and chemical properties of crystals.

4. What methods are used to determine Bravais lattice volume?

There are several methods used to determine Bravais lattice volume, including X-ray crystallography, neutron diffraction, and electron diffraction. These techniques involve analyzing the scattering patterns of X-rays, neutrons, or electrons as they pass through a crystal, which provides information about the lattice structure and allows for the calculation of volume.

5. Can Bravais lattice volume be changed?

No, Bravais lattice volume cannot be changed. It is a fundamental property of a crystal lattice and is determined by the arrangement of atoms in the lattice. However, the volume can be affected by changes in temperature, pressure, or chemical composition, which can cause the lattice to expand or contract.

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