Incommensurability

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

When a mismatch is inevitable, as in the combination Gej-Sii j. — Si, it is found that up to a value of jc = 0.4, there is a small mismatch which leads to a strained silicide lattice (known as commensurate epitaxy) and at higher values of jc there are misfit dislocations (incommensurate epitaxy) at the interface (see p. 35). From tlrese and other results, it can be concluded that up to about 10% difference in the lattice parameters can be accommodated by commensurately strained thin films. 

Considering the changes of E upon fi for different values of the parameters, Villain has concluded that imperfections in the surface structure may have a very strong influence on the behavior of incommensurate phases and on the C-IC transition. The usual lowering of symmetry during the C-IC transition does not occur here and the nature of the C-IC transition may be quite different from that on the surface free of defects. 

Of course, the above discussion apphes only to systems exhibiting domain wall structure, i.e., to weakly inhomogeneous phases formed on surfaces with low corrugation of the gas-solid potential and characterized by the presence of more then one type of equivalent sublattices. When this is not the case, i.e., when the dense incommensurate phase can be considered to be... 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

Landau then identifies turbulence as the limit state consisting of an infinite number of incommensurate frequencies. 

Figure 15. Arrangement of the Mn - O layers and separating sheets according to Giovanoli [3]. The layer structure can be (a) completely ordered or (d) completely disordered (turbostratic disorder). The cases (h) and (c) represent situation between the two extremes, (b) Disorder of the interlayer atoms or molecules but an ordered stacking of the Mn - O layers with constant layer distance, (c) Disorder of the interlayer atoms and an incommensurate shift of the complete Mn - O sheet within the layer plane, resulting in an incommensurate superstructure along the r -direction (perpendicular to the layer) and in a diffuse distribution of the electron density in this layer, resulting in a lower contribution of this layer to the 0 0 / reflections. (Adapted from Ref. [47]).

The adsorption of alkali metals on single crystal surfaces can result in the formation of ordered structures (commensurate or incommensurate super-... 

What are the objections to these ideas The equivalence between students ideas and scientific conceptions had often been doubted. The main argument of many authors has been that both sides are incommensurable. Students conceptions with limited empirical foundation, often ad hoc formulated or post hoc reconstmcted by researchers, have a completely different ontological status to empirically based ideas that are carefully formulated and sharpened by debate among scientific peers. Gault (1991) sketches some cmcial differences as follows ... 

These structures were firstly observed for terminally polar mesogens [11, 12]. However, recent experiments give clear evidence of the presence of smectic A layering [37, 38], re-entrant nematic behaviour [39], two-dimensional lattices [40, 41] and smectic layering with incommensurate periodicities [42] for non-polar sterically asymmetric LCs. 

Fig. 19—Shear stress and chain angle as a function of sliding distance, from simulations of alkanethiolates on Au(111) at temperature 1 K (a) results from commensurate sliding show a stick-slip motion with a period of 2.5 A, (b) in incommensurate case both shear stress and chain angle exhibit random fluctuations with a much smaller average friction [45],...

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