Reciprocal Lattice - Arbitrary Collection of Atoms

Arbitrary Collection of Atoms

One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the Fraunhofer (long-distance or lens back-focal-plane) limit as a Huygens-style sum of amplitudes from all points of scattering (in this case from each individual atom). This sum is denoted by the complex amplitude F in the equation below, because it is also the Fourier transform (as a function of spatial frequency or reciprocal distance) of an effective scattering potential in direct space:

Here g = q/(2π) is the scattering vector q in crystallogapher units, N is the number of atoms, f_j is the atomic scattering factor for atom j and scattering vector g, while r_j is the vector position of atom j. Note that the Fourier phase depends on one's choice of coordinate origin.

For the special case of an infinite periodic crystal, the scattered amplitude F = M F_hkl from M unit cells (as in the cases above) turns out to be non-zero only for integer values of (hkl), where

when there are j=1,m atoms inside the unit cell whose fractional lattice indices are respectively {u_j,v_j,w_j}. To consider effects due to finite crystal size, of course, a shape convolution for each point or the equation above for a finite lattice must be used instead.

Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I, which relates to the amplitude lattice F via the usual relation I = F*F where F* is the complex conjugate of F. Since Fourier transformation is reversible, of course, this act of conversion to intensity tosses out "all except 2nd moment" (i.e. the phase) information. For the case of an arbitrary collection of atoms, the intensity reciprocal lattice is therefore:

Here r_jk is the vector separation between atom j and atom k. One can also use this to predict the effect of nano-crystallite shape, and subtle changes in beam orientation, on detected diffraction peaks even if in some directions the cluster is only one atom thick. On the down side, scattering calculations using the reciprocal lattice basically consider an incident plane wave. Thus after a first look at reciprocal lattice (kinematic scattering) effects, beam broadening and multiple scattering (i.e. dynamical) effects may be important to consider as well.