We are finally ready to consider the general case where we take a multidimensional function that is approximately periodic (quasi-periodic). Consider a crystalline material whose unit cell and Bravais lattice structures are specified by M. If this material is unbounded in all directions, and we consider a function which is periodic (i.e., invariant under translation by a lattice vector), then it is the superposition of waves whose wavenumber vectors are necessarily precisely lattice vectors in the reciprocal lattice specified by .
However, a real crystalline material has finite extent, and has imperfections of perturbations. The ideally periodic function is constrained to satisfy certain boundary conditions. We illustrate the consequences of this by considering the situation where the crystal is comprised of only a finite number of translates of the unit cell. Let denote the finite region occupied by the crystal, and consider the window function defined as:
If is the ideal, truly periodic function (with periodic specified by M) and is the truncated function:
then has a continuous spectrum given by:
where is the Fourier transform of .
It can be shown that has a continuous spectrum that has infinite extent, but which dies out proportionately fast with . Moreover, becomes ``narrower'' and causes (1.38) to be a better ``approximation'' to the discrete spectrum (1.35) as the volume of grows to encompass all space.
The point is that quasi-periodic functions have quasi-discrete spectra, with the spectral energy concentrated at points in the reciprocal lattice.