Fourier termination/truncation

FIGURE 9.12. Series-termination errors, (a) A normal atomic scattering factor curve and (b) the atomic peak obtained by Fourier transformation, (c) A truncated atomic scattering factor curve, such as that used for data that are measured to a lower sin 6/ value than advisable. The missing portion of the scattering curve is indicated, (d) The atomic peak obtained by Fourier transformation. Note the ripples caused by loss of the missing portion of the atomic scattering curve. 

Series-termination errors Errors that result from a limitation in the number of terms in a Fourier series. Ideally an infinite number of data is required to calculate a Fourier series. In practice, the number of data depends on the resolution (reciprocal radius or sin0/A) to which the data have been measured. Because of truncation of the Fourier series at the highest value of sin 0/X of the data, peaks in the resulting Fourier syntheses are surrounded by a series of ripples. These are especially noticeable around a heavy atom because its scattering factor is still appreciable at the highest values of sin 9/X measured. Difference maps (q-v.) can be used to obviate most of the effects of series-termination errors. 

Apodization is a mathematical procedure used to overcome the fact that a recorded inter-ferogram is truncated (i.e., does not extend to an infinite distance) and to ensure that the interferogram to be Fourier-transformed terminates smoothly without a step. An explanation of apodization is given in Section 4.4.1. In Figure 5.12 , a trapezoidal apodization function is shown overlaid with a measured interferogram to be weighted by this apodization function. 

---