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The analysis of Laue zones can provide detailed information regarding the samples. For instance, √2-TTB phase of Pb_xNb_1.17W_1.0O_5.93+x (x > 0.15) in Figure 2325a presented asymmetric electron diffraction patterns and exhibited systematically weak odd-order (First order here) Laue layers lying half-way between the positions of the strong even-order (zero and second order here) layers. The even-order layers correspond to the basic 3.8 Å c_TTB (c axis of TTB structure) repeat. Analyses of multiple samples indicate that these weak odd-order reflections were present in the same positions as the maxima in the even layers and at the midpoint of each edge of the basic TTB square.

Figure 2325a. Asymmetric diffraction pattern of the √2-TTB phase of Pb_xNb_1.17W_1.0O_5.93+x (x > 0.15). The inset enlargement of the first-order Laue layer shows that the black mesh defines the √2-TTB reciprocal lattice periodicity [2], and the dashed square illustrates the position of the basic TTB cell repeat with respect to the √2-TTB cell. Adapted from [1]

Figure 2325b shows the average lead-TTB structure (e.g. Pb_xNb_1.17W_1.0O_5.93+x) obtained by XRD technique and projected onto the ab plane. Note that only the main lead positions are shown.

Figure 2325b. Average lead-TTB structure obtained by XRD technique and projected onto the ab plane. [3]

The asymmetric diffraction pattern can be explained by the basis of structure factor. For instance, the statistical structure factor of the Pb_xNb_1.17W_1.0O_5.93+x materials can be expressed by the contributions from the cation and oxygen atom frameworks: F(hkl) = F_M(hkl) + F_O(hkl). Their supercell contains four equivalent (MO)₂M₂₀ units of the o-√2TTB structure at (0, 0, 0), (1/2, 1/2, 0), (0, 0, 1/2), and (1/2, 1/2, 1/2) [1], therefore, the overall structure factor from the framework cations can be assumed by,

          ------------- [2325a]

where,
          N_M -- The number of Nb and W atoms

On the other hand, the framework anions are located at the positions (0, 0, 0) (0, 1/2, 0), (1/2, 0, 1/2), and (1/2, 1/2, 1/2). Therefore, the structure factor from the framework anions can be assumed by,

          ------------- [2325b]

where,
          N_O -- The number of oxygen atoms

Therefore, oxygen scattering contributes to both odd and even Laue layers. When l is even, both h and k must be even for maxima to be observed and thus only the basic TTB reflections occur. For the cases that l is odd, to observe the maxima lying half-way along one side of the TTB square in the even layer, k must be even and h odd.

The difference between Equations 2325a and 2325b determines the difference of the reflection intensities in the odd- and even-order Laue layers because oxygen gives weaker scattering while Nb and W atoms give stronger.

[1] Sarah K. Haydon and David A. Jefferson, Quaternary Lead-Niobium-Tungsten Oxides Based on the Tetragonal Tungsten Bronze Structure, Journal of Solid State Chemistry 161, 135 - 151 (2001).
[2] S. K. Haydon, Ph.D. Thesis, University of Cambridge, (2000).
[3] S. T. Triantafyllou, P. C. Christidis, and Ch. B. Lioutas, J. Solid State Chem. 130, 176 - 183 (1997).