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

International Tables for Crystallography (2006). Vol. B, ch. 4.4, pp. 457-458   | 1 | 2 |

Section 4.4.4.1. Hexatic phases in two dimensions

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

4.4.4.1. Hexatic phases in two dimensions

| top | pdf |

The hexatic phase of matter was first proposed independently by Halperin & Nelson (Halperin & Nelson 1978[link]; Nelson & Halperin 1979[link]) and Young (Young, 1979[link]) on the basis of theoretical studies of the melting process in two dimensions. Following work by Kosterlitz & Thouless (1973[link]), they observed that since the interaction energy between pairs of dislocations in two dimensions decreases logarithmically with their separation, the enthalpy and the entropy terms in the free energy have the same functional dependence on the density of dislocations. It follows that the free-energy difference between the crystalline and hexatic phase has the form [\Delta F = \Delta H - T\Delta {\bi S} \approx T_{c}{\bi S} (\rho) - T {\bi S} (\rho) = {\bi S} (\rho) (T_{c} - T)], where [{\bi S} (\rho) \approx \rho \log (\rho)] is the entropy as a function of the density of dislocations [\rho] and [T_{c}] is defined such that [T_{c}{\bi S} (\rho)] is the enthalpy. Since the prefactor of the enthalpy term is independent of temperature while that of the entropy term is linear, there will be a critical temperature, [T_{c}], at which the sign of the free energy changes from positive to negative. For temperatures greater than [T_{c}], the entropy term will dominate and the system will be unstable against the spontaneous generation of dislocations. When this happens, the two-dimensional crystal, with positional QLRO, but true long-range order in the orientation of neighbouring atoms, can melt into a new phase in which the positional order is short range, but for which there is QLRO in the orientation of the six neighbours surrounding any atom. The reciprocal-space structures for the two-dimensional crystal and hexatic phases are illustrated in Figs. 4.4.4.3(b)[link] and (c)[link], respectively. That of the two-dimensional solid consists of a hexagonal lattice of sharp rods (i.e. algebraic line shapes in the plane of the crystal). For a finite size sample, the reciprocal-space structure of the two-dimensional hexatic phase is a hexagonal lattice of diffuse rods and there are theoretical predictions for the temperature dependence of the in-plane line shapes (Aeppli & Bruinsma, 1984[link]). If the sample were of infinite size, the QLRO of the orientation would spread the six spots continuously around a circular ring, and the pattern would be indistinguishable from that of a well correlated liquid, i.e. Fig. 4.4.4.3(a)[link]. The extent of the patterns along the rod corresponds to the molecular form factor. Figs. 4.4.4.3(a)[link], (b)[link] and (c)[link] are drawn on the assumption that the molecules are normal to the two-dimensional plane of the phase. If the molecules are tilted, the molecular form factor for long thin rod-like molecules will shift the intensity maxima as indicated in Figs. 4.4.4.3(d)[link] and (e)[link]. The phase in which the molecules are normal to the two-dimensional plane is the two-dimensional hexatic-B phase. If the molecules tilt towards the position of their nearest neighbours (in real space), or in the direction that is between the lowest-order peaks in reciprocal space, the phase is the two-dimensional smectic-I, Fig. 4.4.4.3(d)[link]. The other tilted phase, for which the tilt direction is between the nearest neighbours in real space or in the direction of the lowest-order peaks in reciprocal space, is the smectic-F, Fig. 4.4.4.3(e)[link].

[Figure 4.4.4.3]

Figure 4.4.4.3 | top | pdf |

Scattering intensities in reciprocal space from two-dimensional: (a) liquid; (b) crystal; (c) normal hexatic; and tilted hexatics in which the tilt is (d) towards the nearest neighbours as for the smectic-I or (e) between the nearest neighbours as for the smectic-F. The thin rods of scattering in (b) indicate the singular cusp for peaks with algebraic line shapes in the HK plane.

Although theory (Halperin & Nelson, 1978[link]; Nelson & Halperin, 1979[link]; Young, 1979[link]) predicts that the two-dimensional crystal can melt into a hexatic phase, it does not say that it must happen, and the crystal can melt directly into a two-dimensional liquid phase. Obviously, the hexatic phases will also melt into a two-dimensional liquid phase. Fig. 4.4.4.3(a)[link] illustrates the reciprocal-space structure for the two-dimensional liquid in which the molecules are normal to the two-dimensional surface. Since the longitudinal (i.e. radial) width of the hexatic spot could be similar to the width that might be expected in a well correlated fluid, the direct X-ray proof of the transition from the hexatic-B to the normal liquid requires a hexatic sample in which the domains are sufficiently large that the sample is not a two-dimensional powder. On the other hand, the elastic constants must be sufficiently large that the QLRO does not smear the six spots into a circle. The radial line shape of the powder pattern of the hexatic-B phase can also be subtly different from that of the liquid and this is another possible way that X-ray scattering can detect melting of the hexatic-B phase (Aeppli & Bruinsma, 1984[link]).

Changes that occur on the melting of the tilted hexatics, i.e. smectic-F and smectic-I, are usually easier to detect and this will be discussed in more detail below. On the other hand, there is a fundamental theoretical problem concerning the way of understanding the melting of the tilted hexatics. These phases actually have the same symmetry as the two-dimensional tilted fluid phase, i.e. the smectic-C. In two dimensions they all have QLRO in the tilt orientation, and since the simplest phenomenological argument says that there is a linear coupling between the tilt order and the near-neighbour positional order (Nelson & Halperin, 1980[link]; Bruinsma & Nelson, 1981[link]), it follows that the QLRO of the smectic-C tilt should induce QLRO in the near-neighbour positional order. Thus, by the usual arguments, if there is to be a phase transition between the smectic-C and one of the tilted hexatic phases, the transition must be a first-order transition (Landau & Lifshitz, 1958[link]). This is analogous to the three-dimensional liquid-to-vapour transition which is first order up to a critical point, and beyond the critical point there is no real phase transition.

References

Aeppli, G. & Bruinsma, R. (1984). Hexatic order and liquid crystal density fluctuations. Phys. Rev. Lett. 53, 2133–2136.
Bruinsma, R. & Nelson, D. R. (1981). Bond orientational order in smectic liquid crystals. Phys. Rev. B, 23, 402–410.
Halperin, B. I. & Nelson, D. R. (1978). Theory of two-dimensional melting. Phys. Rev. Lett. 41, 121–124, 519(E).
Kosterlitz, J. M. & Thouless, D. G. (1973). Ordering, metastability and phase transitions in two-dimensions. J. Phys. C, 6, 1181–1203.
Landau, L. D. & Lifshitz, E. M. (1958). Statistical physics. London: Pergamon Press.
Nelson, D. R. & Halperin, B. I. (1979). Dislocation-mediated melting in two dimensions. Phys. Rev. B, 19, 2457–2484.
Nelson, D. R. & Halperin, B. I. (1980). Solid and fluid phases in smectic layers with tilted molecules. Phys. Rev. B, 21, 5312–5329.
Young, A. P. (1979). Melting and the vector Coulomb gas in two dimensions. Phys. Rev. B, 19, 1855–1866.








































to end of page
to top of page