In a "perfect" crystal, the crystal is built repeating  the unit cell by transaltion along the 3 directions of space.

For modulated structures, same case + addition of a periodic modulation (atomic positions, occupations, thermal factors...) along at least one direction of space.

When the peridocity of this modulation is not commensurate with lattice parameters, it becomes incommensutate modulation

Thank you so much Professor Marchet for your straightforward and clear explanation!

Thank you so much Dr. Andreas Schönleber for your helpful answer and for the suggested articles!

can you please tell how to characterize incommensurate phase. I have read that X-ray diffraction pattern of an incommensurate would contain satellite peaks in addition to peaks of the main phase. How to index these satellite peaks and how to arrive at the space group of the modulated phase ?

Hi Manoj Kumar Jana,

Indexing of satellite peak in incommensurate phase can be done with the help of complementary XRD and indexing different zone axis SAED in TEM.

Different zone axis will deconvolute the peaks either it is coming from (3+d) superspcae where d-1,2,3.

For detail you can go through following article.

DOI- 10.1039/C6QI00219F