Impossibility of observing a continuous commensurate-incommensurate transition

It has been shown previously [J. Lajzerowicz and A. P. Levanyuk, Phys. Rev. B 49, 15 475 (1994)] that in a perfect three-dimensional crystal the commensurate-incommensurate transition is always discontinuous due to a fluctuation-induced long-range attraction between domain walls. At the same time it has been concluded long ago that in real crystals a strong repulsion between the domain walls exists due to impurity-induced roughening of the domain walls. This poses the question whether the commensurate-incommensurate transition becomes continuous in real crystals. We argue that the most important point to answer this question is that a very slow activation motion of domain walls over disorder-induced potential barriers is necessary for making the repulsion operative, so that over a reasonable time of experiment this interaction ''propagates'' to distances that are no more than one order of magnitude bigger than the domain-wall width. Meanwhile the attractive interactions between the domain walls ''propagate'' almost instantaneously, thus leading to the conclusion that experimentally observed commensurate-incommensurate transition will always be discontinuous. Our estimates show that for structural incommensurate systems the ''equilibrium'' distance between the domain walls near transition point has the same order of magnitude as the domain-wall width.