Debye-Waller factors for incommensurate structures

Abstract

The Debye-Waller factors for displacive-type incommensurate (IC) structures are calculated taking into account the correlation of the phases of the modulation. It is shown that and that IC satellites should not be observed in the immediate vicinities of transitions.

The role of the incommensurate phase in the opalescence of quartz

“…However, as we have proposed that the SAS is due to non-equilibrium rotation patches, the argument of the weakness of the ferroelastic effect is now not very relevant. Finally we mention our strong disagreement with two other claims of Aslanyan et al on a larger range of the 1q inc phase [56] and on the components of the inc modulation [61].…”

“…Then with a large wavevector q = b * /3 + κ, the quadratic expansion gives a variation of the refractive index larger by three orders of magnitude, which can easily produce the quartz opalescence. To explain why satellite peaks around b * /3 have never been observed, they used their recent theory of the Debye-Waller factor of inc phases [56]. In the inc phase of quartz, a complete extinction of the first-and second-order satellites is predicted; only the third-order satellites, enhanced by a coupling with strain, would be observed, again with a small wavevector 3κ.…”

On the origin of the inhomogeneities and birefringence in the incommensurate phase of quartz

“…The Debye-Waller factors for the IC structures have been calculated by the present authors [19], and it has been shown that they should be very small in the immediate vicinities of displacive-type IC transitions, which can lead to the IC satellites not being observable in diffraction experiments. In our paper [19], we also discussed earlier calculations of the Debye-Waller factors for the IC structures carried out by Overhauser [20] and by Axe [21]. The situation in which the IC third-harmonic modulation amplitude is larger than the amplitude of the fundamental modulation b/3 + κ has been also discussed by the present authors [22].…”

Numerical analysis of atomic motion in the incommensurate phase of quartz (SiO 2 ) in the vicinity of phase transition

“…The existence of the six satellites with almost equal intensity around G=(022) implies that u z is included in the modulation pattern, and its magnitude is comparable to that of u x and u y . To solve these and other problems, a new model was proposed by Aslanyan et al, which claims that the appearance of u z is attributed to the higher order interactions between η and the TA 1 branch (lowest frequency phonon) such as η{[(∂η/∂x) 2 −(∂η/∂y) 2 ]u yz +2(∂η/∂x)(∂η/∂y)u xz } [6]. It is also claimed that the actual temperature interval of the IC phase is wider than [T c ,T i ], but in the interval [T i ,T x ] the diffraction satellites are not observed because of large phase fluctuation [7].…”