where (Kronecker’s delta) is 1 if = and 0 otherwise. The Fourier coefficients are thus the overlap n m integrals of with each complex exponential component: V(x) The condition is required to ensure normalization. If , the function is identical to a cosine wave of V(x) wavelength If , the function contains no component with wavelength a/n. V(x) a/n. 1.4 The Reciprocal Lattice. In a similar way, a periodic function in three dimensions may be invariant under translation by a set of Bravais lattice vectors R: where represents some periodic property of the lattice (e.g., electron density) and R takes the form V . Here, { are integers and {} are primitive lattice vectors. Now we introduce a set of primitive {} that satisfy the following condition: reciprocal lattice vectors Explicit formulas for the primitive reciprocal lattice vectors are given by the expressions Note that the reciprocal vectors are defined with reference to a particular Bravais lattice, often called the or direct real lattice. When because the product of two vectors is normal to both. i j, ≠ It follows that for = since . i j A general reciprocal lattice vector is then K . Here, { are integers and {} are primitive vectors of the reciprocal lattice.
where (Kronecker’s delta) is 1 if n = m and 0 otherwise. The Fourier coefficients are thus the overlap integrals of V(x) with each complex exponential component: The condition is required to ensure normalization. If , the function V(x) is identical to a cosine wave of wavelength a/n. If , the function V(x) contains no component with wavelength a/n. 1.4 The Reciprocal Lattice. In a similar way, a periodic function in three dimensions may be invariant under translation by a set of Bravais lattice vectors R: where V represents some periodic property of the lattice (e.g., electron density) and R takes the form . Here, { are integers and {} are primitive lattice vectors. Now we introduce a set of primitive reciprocal lattice vectors {} that satisfy the following condition: Explicit formulas for the primitive reciprocal lattice vectors are given by the expressions Note that the reciprocal vectors are defined with reference to a particular Bravais lattice, often called the direct or real lattice. When i ≠ j, because the product of two vectors is normal to both. It follows that for i = j since . A general reciprocal lattice vector K is then . Here, { are integers and {} are primitive vectors of the reciprocal lattice.
Reciprocal%20Lattice.docx