Head-tail ordering

Judging from purely energetic arguments the dipole moment of CO is expected to induce a head-tail ordered stmcture at very low temperatures in thermal equilibrium. A reasonable candidate stmcture for such a hypothetical fully ordered stmcture within the a-phase would be antiferroelectric with space group P2i3(7 ) (see, e.g.. Refs. 117 and 159). Based on a mean-field treatment of the dipole-dipole interactions the transition temperature to the fully ordered solid was estimated [230] to be smaller than 5 K. However, the nonvanishing residual entropy of 4.6 J/K mol reported in Ref. 76 has... 

Evidence in favor of dynamic disorder in a-CO was also obtained from nuclear quadrupole resonance studies [199, 369] and from dielectric measurements [251], which also indicate short-range antiferroelectric order [251]. The rate of molecular reorientation leading to head-tail flips was found to become vanishingly small at low temperatures A head-tail reorientation time of about 5 X lO hours was estimated [369] at a temperature of 10 K. Thus, the extremely slow kinetics at the low temperatures which are energetically necessary to allow for head-tail ordering seems to prevent head-tail ordering in the bulk CO phase. 

Figure 54. Possible head-tail ordered herringbone structures for a complete commensurate CO monolayer on the basal plane of graphite only the projection on the surface plane is shown and the directions of the molecular dipole moments are indicated by arrows, (a) Ferrielectric stmcture with the net dipole perpendicular to the herringbone symmetry axis, (b) Ferrielectiic structure with the net dipole parallel to the herringbone symmetry axis, (c) An-tiferroelectric structure with no net dipole. The principal axes of the ellipses correspond to the 95% electronic charge density contour given in Table I. Note that each type of head-tail ordering can be combined with any of the six herringbone ground states from Fig. 6.

This situation can be compared [153-155] to N2O on graphite, a molecule which has slightly larger quadrupole and dipole moments and a solid phase which is structurally very similar to the bulk a-CO solid. There is a residual entropy of about In 2 at low temperatures, and thus the bulk N2O solid does not undergo a head-tail ordering transition [153, 155]. Also no heat capacity anomaly was detected for N2O on graphite [153, 155] down to 3-4 K, which could be rationalized [153] in terms of the striking difference in the dynamical behavior of the N2O and a-CO solids [251]. 

It is found [138] that the increase of the corrugation due to the inclusion of axially symmetric (experimentally determined bulk) quadrupole moments located at the carbon sites [361] which model the aspheiical charge distribution in the graphite substrate [see (3.9) and (3.10) in Section III.D.l] stabilizes the commensurate herringbone structure. This structure is head-tail-ordered as in Ref. 17 (see Fig. 53a or Fig. 54Z>, where the molecular axes have a systematic out-of-plane tilt) the unit cell is deformed because of the displacement of the molecular centers on the two sublattices. The Brillouin-zone-center frequency gap in the phonon spectrum is estimated [138] to amount to about 10 K in the ground state,... 

The order of the head-tail transition was determined based on systematic finite-size scaling of various data [214, 215]. A head-tail order parameter is defined in closest analogy to the herringbone order parameter (3.11)-(3.12) as... 

Figure 59. Fourth-order cumulants(3.22)forthe head-tail order parameter (5.8) obtained from Monte Carlo simulations of complete monolayer ( /3 x -J3)R30° CO on graphite as a function of temperature (5 = 0.13 A). The dashed line marks the trivial value in the ordered phase, the dotted line marks the universal value U for the two-dimensional Ising universality class [47, 53], and = 11.9 K is determined from the intersection point. Symbols for the linear dimension of the L x L system L = 18 (asterisks), 24 (diamonds), 30 (triangles), 42 (squares), and 60 (circles). (Adapted from Fig. 2 of Ref. 215.)...

Similarly, the isothermal susceptibility xr associated to the head-tail order parameter can be obtained ftom its fluctuation relation and scaled according to the scaling relation [11]... 

Figure 60. Scaling plot (5.9) of the head-tail order parameter susceptibility obtained from Monte Carlo simulations of complete monolayer (-J3 x yfi)R30° CO on graphite (6 = 0.13 A) with two-dimensional Ising exponents y = 7/4 and y = 1, and = 11.9 K from the cumulant intersection in Fig. 59. Only the scaling regime 1 - T/T L " 0 is shown, and the data above (asterisks) and below (circles) the transition are superimposed for all system sizes I = 18. .. 60 solid and dotted lines are the amplitude fits (5.11) of the data above and below the transition, respectively. (Adapted from Fig. 4c of Ref. 215.)...

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