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The methods for determining incommensurate modulated structures (IMSs) in high-resolution transmission electron microscopy have been studied and improved to a level more perfect than before. This is demonstrated by means of the IMS determination for Bi2.31Sr1.69CuO6+δ as an example. First, as previously, the projected potential map (PPM) of the IMS with resolution 0.2 nm was obtained after image deconvolution from a [100] image. Secondly, the resolution of the PPM was enhanced to 0.1 nm through phase extension combined with the electron-diffraction data so that the substitutional and displacive modulation functions could be determined. Thirdly, a (2+1)-dimensional structure model that corresponds to the [100] projected IMS was built for calculating the related partial structure factors that were utilized to correct the experimental electron-diffraction intensities for both main and satellite reflections. After three cycles of diffraction-intensity correction and phase extension, all unoverlapped atoms projected along the [100] direction in Bi2.31Sr1.69CuO6+δ were resolved, and the modulations of substitution and displacement could be observed clearly. The substitution of Bi for Sr atoms at the Sr(O) columns was seen in the final PPM and verified by high-dimensional image simulation.

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