Phase problem
Shift in real space
Centrosymmetric vs. Acentric

Interactive examples
Single atom
Pair of atoms
One atom in unit cell
Two atoms in unit cell

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• The intensity is independent of the atom position. Since this is the only directly accessible value in a diffraction experiment, crystal structure determination is still an exciting field.
• The real and imaginary part of the diffraction pattern do change. With the atom at 0.0, the imaginary part is zero, while the real part is a smoothly varying function of h. As you start to shift the atom away from the origin, the real and imaginary part begin to oscillate. This is not to be confused with the oscillation that you observe when you add more atoms to the line. The real part stays symmetric, i.e. Re(h) = Re(-h), while the imaginary part is antisymmetric, i.e. Im(h) = -Im(-h). The phase angle varies in a saw-tooth like fashion. For an atom shifted to x, it is periodic with a periodicity of 1/|x|. Just like the imaginary part, it is antisymmetric.
• Switching from a positive shift to a negative shift switches the sign of the imaginary part and the phase angle.
• If you change the atom type while keeping the position, the intensity changes according to the square of the atomic form factor. The phase angle, however, remains unaffected.
• Changing from X-ray to neutron diffraction changes the intensity, the real and imaginary part. The X-ray intensities are equal to the square of the atomic form factor, while the neutron intensities are independent of h. Correspondingly, the real and imaginary part of the neutron diffraction pattern do not show the systematic decrease with increasing h. The phase angle is identical for X-ray and neutron diffraction.
• The phase angle increases linearly from zero degrees at h=0 to 180 degrees at h=1/(2x) for a positive x coordinate. It then jumps to -180 degrees and increases again to 180 degrees at h=3/(2x). If you shift an object in real space, its Fourier transform suffers a phase shift of e**(2pi i hx). Look at the argument 2pi i hx. This argument increases linearly with h. Taken modulo 2pi, it becomes a a periodic function of h.

Example 2:

• If you have two identical atoms at +x and -x, the imaginary part is zero. Since the real part oscillates between positive and negative values, the phase angle jumps between values of 0 and 180 degrees. As you start to shift the whole molecule along the x-axis, you get an additional phase angle due to this shift. Although the molecule conserves its internal center of symmetry, the overall shift creates an antisymmetric phase angle.
• Since a linear phase shift is added to all points in reciprocal space, the relative phase between two points does not change! Compare this linear shift to the shift you get in exercise 1 when you move the single atom by the same amount as you shifted the pair in exercise 2.
• Two important consequences arise: i) You can move your crystal a bit to the side during a diffraction experiment, the relative phases do not change. ii) You can artificially move the atoms within the unit cell by assigning a linear phase change to all reflections. This is referred to a "origin fixing" during crystal structure determination.
• If you maintain the positions at +x and -x, but change one of the atoms to a different type, the phase angle becomes a smoothly varying antisymmetric function.
• The periodicity of the phase angle for two identical atoms at +x and -x is 2/x.