Interactive Tutorial about Diffraction Phase problem I: Shift in real space 

Phase problem
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In this exercise we look again at the Fourier transform of a single atom.
The calculations are performed twice, once for the
blue atom at the origin of real space and second for the red atom shifted to 0.5, 0, 0. Note that this is
not a periodic crystal. The first image shows the two different
positions.
The Fourier transforms were calculated along a line through reciprocal space from 5.5,0,0 to +5.5,0,0. The left image shows the real(blue) and imaginary (red) part for the atom at 0,0,0. The corresponding intensities reduced by a factor of 10 are shown in green color. The middle image shows the corresponding values for the atom shifted in real space. The right image shows the phase of the wave scattered by the atom at 0,0,0 (blue) and the shifted atom (red).
In the above left image the real part of the amplitude is positive, while the imaginary part is zero, as expected from a centrosymmetric structure. In the middle image, the Fourier transform of the atom shifted in real space, the real and the imaginary part oscillate. The resulting intensity, however, is identical in both cases! The information about the phase is lost. This marks a good opportunity to discuss the crystallographic phase problem. The phase of the wave scattered by the atom at 0,0,0 (blue) is zero at all points in reciprocal space, as expected. With the atom at 0.5,0,0 (red), however, the phase oscillates between 180 and 180 degrees with a wave length of two reciprocal lattice units. Both structures are centrosymmetric, the only difference is the shift of origin by 0.5,0,0 ! A quick look at the properties of Fourier transforms reveals the cause. A real space shift results in multiplication with a phase factor in reciprocal space:
Possible extensions of this exercise:

© Th. Proffen and R.B. Neder, 2003 