One-dimensional modulation

D. de Fontaine. A theoretical and analogue study of diffraction from one-dimensional modulated structures. In J.B. Cohen and J.E. Hillard, editors, Local Atomic Arrangements Studied by X-Ray Diffraction, pages 51-88. The Metallurgical Society of AIME, Warrendale, PA, 1966. 

A characteristic feature of these partially oxidized bis(oxalato)platinate salts of divalent cations is the coexistence of two modulations of the lattice over a wide temperature range (a) a one-dimensional modulation, as detected by the appearance of diffuse lines on X-ray films, perpendicular to the [Pt(C204)2] anion stacking direction and surrounding the even Bragg reflection layer lines of non-zero order (b) a three-dimensional modulation which gives rise to a complicated pattern of fine satellite spots in the neighbourhood of every reciprocal layer line. 

In photonic materials, the band gap is determined by geometric arrangement of a dielectric material. The underlying principle of how photonic materials work is best explained using Maxwell s equations (Joannopoulos et al., 1995). Once again, the central importance of Maxwell s equations is confronted when optical properties of materials are discussed. In photonic materials, a periodic stmcture is produced in one, two, or three dimensions. The periodic property is a dielectric constant. A trivial macroscopic onedimensional example would be a collection of individual microscope shdes separated by layers of Saran Wrap . This would produce a one-dimensional modulation in the... 

In addition to one-dimensional modulations considered above, both two-and three-dimensional modulations are possible. Furthermore, atomic parameters affected by modulations may be one or several of the following positional (as shown in Figure 1.52 and Figure 1.53), occupancy, thermal displacement, and orientation of magnetic moments. The latter, i.e. commensurately or incommensurately modulated orientations of magnetic moments are quite common in various magnetically ordered structures (e.g. pure lanthanide metals such as Er and Ho), and both the value of the modulation vector and the amplitude of the modulation function often vary with temperature. 

Conventional lattices may be perturbed by functions with different periodicity, e.g. by sinusoidal or saw-tooth-like modulations, see section 1.21 in Chapter 1. In the simplest case (one-dimensional modulation), Eq. 5.1 becomes d / = h + kb + k + wq assuming that the perturbation function is periodic and has the modulation vector q. In a case of three-dimensional modulation, a total of six indices (k, k, I, m, n, and p) are required to identify every point observed in reciprocal space = ha + kb + /c +... 

All superspace groups for one dimensionally modulated structures (756 superspace... 

The (1 -xlBiFeOj-xLaFeOj system shows a one-dimensional modulated structure over the composition range x=0.19-0.3 at 1075°C. The basic unit cell of this... 

For simplicity, we consider a one-dimensional modulation." In lowest approximation, the Gibbs energy of the solution is... 

Figure 3. One-dimensional /-modulated selective INEPT H- C NMR correlations that...

Figure Cl.4.5. Population modulation as the atom moves through the standing wave in the Tin-periD-lin one dimensional optical molasses. The population lags the light shift such that kinetic is converted to potential energy then dissipated into the empty modes of the radiation field by spontaneous emission (after 1171).

Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

The smectic X phase in the pure compound I shows additionally to the smectic C layering the two-dimensional (modulated) structure in which the smectic C layers are periodically shifted with a respect to one another by half a layer spacing. 

Additional information may by obtained from the modulation depth of the signal. For the simple case of one-dimensional, angle-resolved scattering into a single, uncoupled continuum (see Section IVC), the modulation depth is given by (see Eq. (4))... 

Example 1. One-dimensional simulation of apcm module in a water tank with MathCad... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

The TDE solute module is formulated with one equation describing pollutant mass balance of the species in a representative soil volume dV = dxdydz. The solute module is frequently known as the dispersive, convective differential mass transport equation, in porous media, because of the wide employment of this equation, that may also contain an adsorptive, a decay and a source or sink term. The one dimensional formulation of the module is ... 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Figure 2. Datacollapses forrelaxation of one-dimensional sinusoidal modulations, 0.28Tp.

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... 

Figure 4.15 CDW formation in a one-dimensional metal. Sinusoidally modulated electron density is shown above. The dots along the x-axis represents the ions. (Following Di Salvo, 1977.)...

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