Herringbone ordering

Linear N2 molecules adsorbed on graphite show a transition from a high-temperature phase with orientational disorder to a low-temperature phase with herringbone ordering of the orientational degrees of freedom (see Sec. lie and Fig. 11). 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

FIG. 13 Herringbone order parameter and total energy for N2 (X model with Steele s corrugation). Quantum simulation, full line classical simulation, dotted line quasiharmonic theory, dashed line Feynman-Hibbs simulation, triangles. The lines are linear connections of the data. (Reprinted with permission from Ref. 95, Fig. 4. 1993, American Physical Society.)... 

Figure 4.14. Phase diagram, coverage vs. temperature, of N2 physisorbed on graphite. Symbols used fluid without any positional or orientational order (F), reentrant fluid (RF), commensurate orientationally disordered solid (CD), commensurate herringbone ordered solid (HB), uniaxial incommensurate orientation-ally ordered (UlO) and disordered (UID) solid, triangular incommensurate orientationally ordered (lO) and disordered (ID) solid, second-layer liquid (2L), second-layer vapour (2V), second-layer fluid (2F), bilayer orientationally ordered (2SO) and disordered (2SD) solid. Solid lines are based on experimental results whereas the dashed lines are speculative. Adapted from Marx Wiechert, 1996.

Figure 15.2 Thin film phase unit cell, (a) The side view illustrates the layered structure of the thin film phase, (b) The top view emphasises the herringbone ordering motive, which is a common feature of all pentacene polymorphs.

The so-called herringbone ordering occurring at low coverages and temperatures is the most extensively studied orientational transition of N2 on graphite including experiment, theory, and simulation. 

The aim of the first exploratory Monte Carlo investigation [274] was to introduce a more simplified generic model for further studies of herringbone ordering (see Section II.B and Section in.D.l). The systems with 6x6 and 12 x 12 molecules were clearly too small and the statistical averaging insufficient to establish the order of the transition. 

The first Monte Carlo simulation [244] (see also the broad discussion in Section 5.3 of Ref. 246) focusing on the order of the herringbone transition was based on the strictly two-dimensional anisotropic-planar-rotor model (2.5), which is similar to the planar model used in Ref. 274 except that the quadrupole interactions are treated only approximately (see Section II.B for forther details). The instrument of diagnostics was the three-component herringbone order parameter [244, 246] defined as... 

Figure 32. Herringbone order parameter for the anisotropic-planar-rotor model (2.5) as a function of the reduced temperature T = TIK. Circles Monte Carlo results [244]. Dotted line mean-field approximation [62, 141]. Solid line triangular cluster-variational method [62]. Arrow first-order transition temperature obtained from a real-space renormalization group treatment of a planar quadrupolar six-state model [345]. (Adapted from Fig. 2 of Ref. 345.)...

Figure 38. Striped domain wall in a model of uniaxially compressed N2 monolayers on graphite (a) observed in Monte Carlo simulations of 52 X 12 molecules at 10 K and a coverage of 1.026 monolayers. Dots denote the centers of the honeycomb hexagons of the graphite basal plane, and crosses mark the mean positions of the molecular centers of mass. The inset of (a) shows the herringbone order in the commensurate region at the left and right boundaries of (a). The center-of-mass distribution in the region of the domain wall of (a) sampled from the Monte Carlo trajectory is magnified in (b). (Adapted from Fig. 1 of Ref. 283.)...

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... 

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