# Reciprocal lattice basis

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## Summary:

I am trying to understand how to determine the reciprocal lattice basis from the lattice basis vectors

## Main Question or Discussion Point

may someone explain to me or show me the steps of how equations 7a, 7b, 7c were determined from equations 5 and 6  Related Linear and Abstract Algebra News on Phys.org
Infrared
Gold Member
You can check directly that those formulas for the ##b_i## satisfy the equations that you want. For example, ##b_1\cdot a_1=2\pi\frac{a_1\cdot (a_2\times a_3)}{a_1\cdot (a_2\times a_3)}=2\pi##.

Since the vectors ##a_i## are linearly independent, the equations ##b_i\cdot a_j=2\pi\delta_{ij}## can be uniquely solved for the ##b_i##, and hence the above are the unique solutions.

If you wanted to figure out these formulas from scratch, you could argue like this: since ##b_1## is orthogonal to both ##a_2## and ##a_3##, you know that ##b_1=c_1 (a_2\times a_3)## for some constant ##c_1##. Then the equation ##a_1\cdot b_1=2\pi## lets you solve for ##c_1##, etc.

hello,

thank you. yes, but my question is how are they solved for bi i.e. what are the steps to find b1 for example

thank you

Infrared
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