One-dimensional density wave

On passing from the least ordered smectic A phase down to the more ordered crystal H and K phases the layer planes tend to sharpen up. In the smectic A and C phases the layers are therefore very diffuse, and can be thought of as one-dimensional density waves relative to the director. Thus, locally these two phases are very similar in structure to the nematic phase. 

The mesophases differ from each other regarding the positional order of the molecules (Fig. l). In the nematic phase there is no long range positional order at all just as in isotropic liquids. Nematics are normally uniaxial, however biaxial nematics were discovered very recently. In the smectic phases the centre of masses of the molecules are concentrated in layers forming a one-dimensional density wave. In the smectic A and C phases there is no long-range positional order within the layers. The smectic A phase is uniaxial, the director (n) is parallel with the layer normal, 1. In the C phase the director is tilted with respect to the layer normal. This phase is biaxial although the deviation from uniaxiality is usually small. There are further smectic phases in which the molecules form two-dimensional lattices within the layers (ordered smectic phases). The difference between ordered... 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

Orthogonal Hexagonal B Orthorhombic E Weakly coupled layers Short-range order One-dimensional density wave liquid layers ... 

The simplest smectic phase is the smectic A(S ) phase. This phase has traditionally been described as a system that is a solid in the direction along the director and a fluid normal to the director, or equivalently, as stacked two-dimensional fluids it is more properly described as a one-dimensional density wave in a three-dimensional fluid with the density wave along the nematic director. The phase is similar except the density wave vector makes a finite angle with the director. In both and phases there is complete translational symmetry perpendicular to the density wave vector. 

The smectic phase is envisaged as a one-dimensional density wave of the form... 

It should be noted that the term shock waves refers to a pressure wave of large amplitude that arises from sharp and vioient disturbances when the velocity of wave propagation exceeds the veiocity of sound propagation. Characteristicaiiy, an abrupt change of the medium properties (e.g., pressure, stress, density, particie velocity, temperature, etc.) takes piace in a limited space across the shock wave (Schetz and Fuhs, 1996 Shapiro, 1953 Anderson, 1982 Saad, 1992). In the case described in this chapter, the physicai phenomenon of shock wave is restricted to one-dimensional plane wave propagation, in which properties of air in the resonant tube of the wave generator... 

This error is large in the case of bubbles within a resonant field. In the light of this, Yosioka and Kawasima extended King s theory to allow for compressible spheres in 1955. They demonstrated that the time-averaged acoustic radiation force on a spherical particle of radius a, at position X within a one-dimensional standing wave of acoustic energy density e is... 

Figure 3.4 Particle-in-a-one-dimensional box. (a) The four lowest allowed energy levels (n = 1, 2, 3 and 4). (b) The corresponding wave functions i//n. (c) Probability densities ip 2.

Consider the thermal wave given in Fig. 4.4. If a differential control volume is taken within this one-dimensional wave and the variations as given in the figure are in the x direction, then the thermal and mass balances are as shown in Fig. 4.5. In Fig. 4.5, a is the mass of reactant per cubic centimeter, Cj is the rate of reaction, Q is the heat of reaction per unit mass, and p is the total density. Note that alp is the mass fraction of reactant a. Since the problem is a steady one, there is no accumulation of species or heat with respect to time, and the balance of the energy terms and the species terms must each be equal to zero. 

The creation of eddies in a combustion zone is dependent on the nature of the flow of the unburned gas, i. e., the Reynolds number. If the upstream flow is turbulent, the combustion zone tends to be turbulent. However, since the transport properties, such as viscosity, density, and heat conductivity, are changed by the increased temperature and the force acting on the combustion zone, a laminar upstream flow tends to generate eddies in the combustion zone and here again the flame becomes a turbulent one. Furthermore, in some cases, a turbulent flame accompanied by large-scale eddies that exceed the thickness of the combustion wave is formed. Though the local combustion zone seems to be laminar and one-dimensional in nature, the overall characteristics of the flame are not those of a laminar flame. 

The combustion wave of a premixed gas propagates with a certain velocity into the unburned region (with flow speed = 0). The velocity is sustained by virtue of thermodynamic and thermochemical characteristics of the premixed gas. Figure 3.1 illustrates a combustion wave that propagates into the unburned gas at velocity Mj, one-dimensionally under steady-state conditions. If one assumes that the observer of the combustion wave is moving at the same speed, Wj, then the combustion wave appears to be stationary and the unburned gas flows into the combustion wave at the velocity -Wj. The burned gas is expelled downstream at a velocity of-M2 with respect to the combustion wave. The thermodynamic characteristics of the combustion wave are described by the velocity (u), pressure (p), density (p), and temperature (T) of the unburned gas (denoted by the subscript 1) and of the burned gas (denoted by the subscript 2), as illustrated in Fig. 3.1. 

Rommer, S., Ostlund, S. Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group. Phys. Rev. B 1997, 55(4), 2164. 

One-dimensional electrical conductors, platinum complexes, 26 235-268 band theory, 26 237-241 charge density waves, 26 239-240 Kahn-Teller effect, 26 239-240 and superconductivity, 26 240-241 One-electron reactions, oxo-molybdenum centers, 40 56-57... 

Drozdova O, Yakushi K, Yamamoto K, Ota A, Yamochi H, Saito G, Tashiro H, Tanner DB (2004) Optical characterization of 2kp bond-charge-density wave in quasi-one-dimensional 3/4-filled (ED0-TTF)2X (X = PFg and AsFg). Phys Rev B70 075107-1/8... 

The velocity of advance of the front is super sonic in a detonation and subsonic in a deflagration. In view of the importance of a shock process in initiating detonation, it has seemed difficult to explain how the transition to it could occur from the smooth combustion wave in laminar burning. Actually the one-dimensional steady-state combustion or deflagration wave, while convenient for discussion, is not easily achieved in practice. The familiar model in which the flame-front advances at uniform subsonic velocity (v) into the unburnt mixture, has Po> Po> an[Pg.249]

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