Biorthogonality vector construction

Let us now describe this graphical construction in more mathematical terms. Although the reference axes R ) are generally nonorthogonal, one can always construct the associated set of conjugate vectors R ) that are biorthogonal to the R ), namely,... 

Fig. 9.3. Construction of the inverse lattice in 2-D. In order to satisfy the biorthogonality relatirais Eq. (9.15), the vector b has to be orthogonal to 02. while f>2 must be perpendicular to i. The lengths of the vectors b and b2 also follow from the biorthogonalify relations b a = b2 02 = 2n.

Let us now construct the so called biorthogonal basis bi,b2,b3 with respect to the basis vectors a, a2, of the primitive lattice, i.e. the vectors that satisfy the... 

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