Dimensionless wavenumber

Fig. 8. LDF valence band structure of [9,2] chiral nanotube. The Fermi level lies at midgap at -3.3 eV. Dimensionless wavenumber coordinate k ranges from 0 to t.

The following calculations are based on a linear stability analysis taking into account only the first order of. Furthermore, we use two approximations. One is the long wavelength approximation the wavelength of perturbation of the solid-liquid interface is much larger than the mean thickness of the liquid film, then we can define a small dimensionless wavenumber The other is the quasistationary approximation we... 

Electrically driven convection in nematic liquid crystals [6,7,16] represents an alternative system with particular features listed in the Introduction. At onset, EC represents typically a regular array of convection rolls associated with a spatially periodic modulation of the director and the space charge distribution. Depending on the experimental conditions, the nature of the roll patterns changes, which is particularly reflected in the wide range of possible wavelengths A found. In many cases A scales with the thickness d of the nematic layer, and therefore, it is convenient to introduce a dimensionless wavenumber as q = that will be used throughout the paper. Most of the patterns can be understood in terms of the Carr-Helfrich (CH) mechanism [17, 18] to be discussed below, from which the standard model (SM) has been derived... 

Both Qj and Ax are in dimensionless units and therefore the force constants kj are equal to the excited state vibrational energies in wavenumbers. The latter are obtained from the spectrum (Fig. 10 and Table 1). The At denote the excited state distortions and are the parameters to be determined. The values of both E00 (22613 cm-1) and F(19 cm-1) are obtained from the spectrum in Fig. 10. The initial wavepacket 0 is calculated by using the literature values for the ground state vibrational energies along the 3 modes. 

It is typical in the study of cosmological perturbations to use not P k) and Ct themselves, but rather the quantities A2 k) = k3P(k)/(2ir2) and 8T2 = ( + 1 )CtJ(27r). These dimensionless quantities give the contribution to the total variance in density or temperature from a given 3D or spherical wavenumber, or even more heuristically, the mean-square fluctuation at wavelength A 27x/k or angular scale 0 180°/ . In addition, the Sachs-Wolfe effect has ( + l)Ct/(2ir) = const at low . 

The force constants in dimensionless normal coordinates are usually defined in wavenumber units by the... 

Redlich-Peterson heterogeneity factor (dimensionless) radiation wavenumber (nm )... 

Figure 2. Threshold voltage Uth/Uo and the critical wavenumber qc versus the dimensionless dielectric anisotropy eajev calculated from Eqs. (7) and (8). a b Planar alignment with a a > 0, c d homeotropic ahgnment with a a < 0. Dashed lines correspond to the Freedericksz transition, solid lines to the direct EC transition.

On the contrary, for M 0, we observe the existence of a cut-off wavenumber which is function of the three dimensionless parameters W, M and Re and in a such case, each of these parameters can play the role of a bifurcation parameter (in contrast with (8.27), where the bifurcation parameter is only W ). 

Now we have closed homogeneous system (49, 51) for u. /i, c, jT, (f. By assuming the determinant of this system to be zero, the dispersion equation could be obtained. This dispersion equation determines four eigenvalues ujk depending on the wavenumber a and on the free dimensionless parameters Bi, G, T, Di, 6, k., cq, Ma, e. 

Emission from a body occurs from thermally excited atoms and molecules within the body. The basic principle of thermal emission is described by Kirchhoff s law which states that the ratio between the energy of radiation emitted by a body in a thermal equilibrium and its absorptance is a function of only the temperature of the body and the wavenumber (or wavelength) of the radiation it does not depend on the material constituting the body. The absorptance mentioned above may be defined as follows. When a body is irradiated, the radiation is partly reflected, partly absorbed, and the remainder passes through the body, if scattering by the body is ignored. If the proportions of the reflection, absorption and transmission are expressed, respectively, by reflectance (r), absorptance (a), and transmittance (t), the following relation holds r -i- a -t- t = 1. It is clear that each of the three quantities is a dimensionless constant with a value between 0 and 1. (As each of them is a function of wavenumber v, they are expressed as r(v), a(v), and t(v) when necessary.) Usually, transmittance is denoted by T, but it is not used in this chapter to avoid confusion with temperature T. 

On the right-hand side of the constitutive equation, Eq. (1.3), a diffusion term has been added, as proposed by Sureshkumar and Beris [81], so that in turbulent simulations the high wavenumber contributions of the conformation tensor do not diverge during the numerical integration of this equation in time. This parallels the introduction of a numerical diffusion term in any scalar advection equation (e.g., a concentration equation with negligible molecular diffusion) that is solved along with the flow equations under turbulent conditions [82]. In Eq. (1.3), Dq is the dimensionless numerical diffusivity [54-56]. The issue of the numerical diffusivity is further discussed in Sections 1.3.2 and 1.4.3. 

Experimental and calculated spectral parameters for propyne in the gas phase (wavenumbers in cm, absolute differential Raman scattering cross sections (da/dn)j in I0 36 the depolarization ratios pj are dimensionless)... 

Fig. 2. Dimensionless curves of critical permeability P versus wavenumber k. Corrugations will appear in the region marked UNSTABLE , where the permeability is sufficiently large. Equivalently, for a given permeability there is a critical flow speed, or shear rate, at which corrugations will form. Ablating sea ice is more unstable than advancing sea ice.

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