International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 4.4, p. 455   | 1 | 2 |

Section 4.4.3.2. Modulated smectic-A and smectic-C phases

P. S. Pershana*

aDivision of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA
Correspondence e-mail: pershan@deas.harvard.edu

4.4.3.2. Modulated smectic-A and smectic-C phases

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Previously, we mentioned that, although the reciprocal-lattice spacing [|{\bf q}|] for many smectic-A phases corresponds to [2\pi /L], where L is the molecular length, there are a number of others for which [|{\bf q}|] is between [\pi /L] and [2\pi /L] (Leadbetter, Frost, Gaughan, Gray & Mosley, 1979[link]; Leadbetter et al., 1977[link]). This suggests the possibility of different types of smectic-A phases in which the bare molecular length is not the sole determining factor of the period d. In 1979, workers at Bordeaux optically observed some sort of phase transition between two phases that both appeared to be of the smectic-A type (Sigaud et al., 1979[link]). Subsequent X-ray studies indicated that in the nematic phase these materials simultaneously displayed critical fluctuations with two separate periods (Levelut et al., 1981[link]; Hardouin et al., 1980[link], 1983[link]; Ratna et al., 1985[link], 1986[link]; Chan, Pershan et al., 1985[link], 1986[link]; Safinya, Varady et al., 1986[link]; Fontes et al., 1986[link]) and confirmed phase transitions between phases that have been designated smectic-[\hbox{A}_{1}] with period [d \approx L], smectic-[\hbox{A}_{2}] with period [d \approx 2L] and smectic-[\hbox{A}_{d}] with period [L\lt d\lt 2L]. Stimulated by the experimental results, Prost and co-workers generalized the De Gennes mean-field theory by writing [\rho ({\bf r}) = \langle \rho \rangle + \hbox{Re} \{\Psi_{1} \exp (i{\bf q}_{1} \cdot {\bf r}) + \Psi_{2} \exp (i{\bf q}_{2} \cdot {\bf r})\},] where 1 and 2 refer to two different density waves (Prost, 1979[link]; Prost & Barois, 1983[link]; Barois et al., 1985[link]). In the special case that [{\bf q}_{1} \approx 2{\bf q}_{2}] the free energy represented by equation (4.4.2.3)[link] must be generalized to include terms like [(\Psi_{2}^{*})^{2} \Psi_{1} \exp [i({\bf q}_{1} - 2{\bf q}_{2}) \cdot {\bf r}] + \hbox{c.c.}] that couple the two order parameters. Suitable choices for the relative values of the phenomenological parameters of the free energy then result in minima that correspond to any one of these three smectic-A phases. Much more interesting, however, was the observation that even if [|{\bf q}_{1}|\lt 2|{\bf q}_{2}|] the two order parameters could still be coupled together if [{\bf q}_{1}] and [{\bf q}_{2}] were not collinear, as illustrated in Fig. 4.4.3.1(a)[link], such that [2{\bf q}_{1} \cdot {\bf q}_{2} = |{\bf q}_{1}|^{2}]. Prost et al. predicted the existence of phases that are modulated in the direction perpendicular to the average layer normal with a period [4\pi /[|{\bf q}_{2}|\sin (\varphi)] = 2\pi /|{\bf q}_{m}|]. Such a modulated phase has been observed and is designated as the smectic-A (Hardouin et al., 1981[link]). Similar considerations apply to the smectic-C phases and the modulated phase is designated smectic-[\tilde{\rm C}]; (Hardouin et al., 1982[link]; Huang et al., 1984[link]; Safinya, Varady et al., 1986[link]).

[Figure 4.4.3.1]

Figure 4.4.3.1 | top | pdf |

(a) Schematic illustration of the necessary condition for coupling between order parameters when [|{\bf q}_{2}|\;\lt\; 2|{\bf q}_{1}|]; [|{\bf q}| = (|{\bf q}_{2}|^{2} - |{\bf q}_{1}|^{2})^{1/2} = |{\bf q}_{1}| \sin (\alpha)]. (b) Positions of the principal peaks for the indicated smectic-A phases.

References

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