Incommensurate response

Fig. 21. Images of the magnetic scattering from YBajCujOj above and below at 34 and 24.5 meV in the two-dimensional reciprocal space of the CUO2 planes. At the lower energy (e,f) an incommensurate response, described by the model shown in (d), appears at the positions noted in the schematic map (a). The resonance that appears at the position (b,c) in the superconducting state is described in sect 4.2. From Mook et al.

Figure 5 Typical velocity relationship of kinetic friction for a sliding contact in which friction is from adsorbed layers confined between two incommensurate walls. The kinetic friction F is normalized by the static friction Fs. At extremely small velocities v, the confined layer is close to thermal equilibrium and, consequently, F is linear in v, as to be expected from linear response theory. In an intermediate velocity regime, the velocity dependence of F is logarithmic. Instabilities or pops of the atoms can be thermally activated. At large velocities, the surface moves too quickly for thermal effects to play a role. Time-temperature superposition could be applied. All data were scaled to one reference temperature. Reprinted with permission from Ref. 25. 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that... 

A more recent study has examined the SH response from a Au(lll) electrode during UPD of a variety of metals, Ag, Cu, Pb, T1 and Sb [155]. In situ x-ray diffraction techniques have examined Ag, Cu and Pb on this substrate [159-161]. Silver is shown to form an epitaxial overlayer with Ag atoms sitting in 3-fold hollow sites forming a (1X1) commensurate overlayer. The lattice mismatch between lead and the substrate is shown to prevent formation of a commensurate overlayer but forms a hexagonal close-packed overlayer contracted by 0.7% from bulk lead. Although T1 and Sb on Au (111) have not been examined by x-ray diffraction, a close packed structure would necessitate an incommensurate overlayer due to the lattice mismatch. 

A parallel combination of comb filters with incommensurate delays is also a useful structure, because the resulting frequency response contains peaks contributed by all of the individual comb filters. Moreover, the combined echo density is the sum of the individual densities. Thus, we can theoretically obtain arbitrary density of frequency peaks and time echoes by combining a sufficient number of comb filters in parallel. 

Impuritiesand the a P-quartz tranition. The a- 3-quartz transition was the basis for one of the earliest systematic investigations of the variation of transition temperatures in response to impurities. Pure a-quartz undergoes a first-order transition to a microtwinned incommensurate structure at 573°C, and this modulated phase transforms to P-quartz at 574.3°C with second-order behavior (Van Tendeloo et al. 1976, Bachheimer 1980, Dolino 1990). Tuttle (1949) and Keith and Tuttle (1952) investigated 250 quartz crystals and observed that Tc for natural samples varied over a 38°C range. In their examination of synthetic specimens, substitution of Ge for Si raised the critical temperature by as much as 40°C, whereas the coupled exchange of Ar +Li o Si depressed Tc by 120°C. They concluded from their analyses that the departure of the a-P-quartz inversion temperature from 573°C could be used to assess the chemical environ-ment and the growth conditions for natural quartz. 

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