Reciprocal Lattice

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{imathbf{K}cdotmathbf{R}}=1 for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. For an infinite three dimensional lattice, defined by its primitive vectors (mathbf{a_{1}}, mathbf{a_{2}}, mathbf{a_{3}}) , its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae mathbf{b_{1}}=2 pi frac{mathbf{a_{2}} times mathbf{a_{3}}}{mathbf{a_{1}} cdot (mathbf{a_{2}} times mathbf{a_{3}})} mathbf{b_{2}}=2 pi frac{mathbf{a_{3}} times mathbf{a_{1}}}{mathbf{a_{2}} cdot (mathbf{a_{3}} times mathbf{a_{1}})} mathbf{b_{3}}=2 pi frac{mathbf{a_{1}} times mathbf{a_{2}}}{mathbf{a_{3}} cdot (mathbf{a_{1}} times mathbf{a_{2}})}. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion: left[mathbf{b_{1}}mathbf{b_{2}}mathbf{b_{3}}right]^T = 2pileft[mathbf{a_{1}}mathbf{a_{2}}mathbf{a_{3}}right]^{-1}.