How do I find the reciprocal primitive vector for a lattice?

• delrepublica
In summary, the problem at hand is to find the primitive vectors of the reciprocal lattice, b1, b2, and b3, given the primitive vectors of the hexagonal lattice, a1, a2, and a3. The equation for b1 can be written as b1 = [(2*pi) (a2 X a3)] / [a1 * (a2 X a3)]. The individual components of the vectors can be found by using dot and cross products, which the person is unsure of how to do. They are asking for help on how to find b1 as a first step.
delrepublica

Homework Statement

Here's a problem I'm having:

The primitive vectors of the hexagonal lattice are:

a1 = ck

a2 = (a/2)i + ([a√3]/2)j

a3 = (-a/2)i + ([a√3]/2)j

Find the primitive vectors of the reciprocal lattice, i.e. b1, b2, and b3.

Homework Equations

I do know that the equation for b1 would be

b1 = [(2*pi) (a2 X a3)] / [a1 * (a2 X a3) ]

The Attempt at a Solution

I really have no idea how to do dot-products and cross-products in this case. Could someone please help me go through just the first step, which is to find b1, and then I could take it from there?

Thank you very much in advance!

anybody??

why don't you know how to do dot and cross products? What is the problem? Can you show us?

there is no need to post twice "please help" and "anybody"... calm down, and show attempt to solution.

I mean how would YOU do to find b1?

1. How do I determine the primitive vectors for a lattice?

The primitive vectors for a lattice can be determined by taking any two non-collinear vectors that span the lattice and then finding their reciprocal vectors.

2. What is a reciprocal vector?

A reciprocal vector is a vector that, when dotted with a primitive vector, results in an integer.

3. How do I find the reciprocal primitive vector for a lattice?

To find the reciprocal primitive vector for a lattice, you can use the formula a* = (b x c) / (a * (b x c)), where a, b, and c are the primitive vectors and a* is the reciprocal primitive vector.

4. Can a lattice have more than one set of primitive vectors?

Yes, a lattice can have multiple sets of primitive vectors. However, only one set will be considered the primitive vectors for that specific lattice.

5. How do I check if my chosen primitive vectors are correct?

To check if your chosen primitive vectors are correct, you can use the dot product to see if they are perpendicular to each other and the cross product to see if they span the lattice. Additionally, you can use the formula a* x a = 2𝜋, where a* is the reciprocal vector of a, to confirm that the vectors are reciprocal.

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