Doppler transformation

The factor of 2y comes from transformation to the rest frame, Y, and from the Doppler transformation, (1+8)y ... 

For all of the eases eonsidered earlier, a C(t) funetion is subjeeted to Fourier transformation to obtain a speetral lineshape funetion I(co), whieh then provides the essential ingredient for eomputing the net rate of photon absorption. In this Fourier transform proeess, the variable co is assumed to be the frequeney of the eleetromagnetie field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to I(co) is ... 

This result, when substituted into the expressions for C(t), yields expressions identieal to those given for the three eases treated above with one modifieation. The translational motion average need no longer be eonsidered in eaeh C(t) instead, the earlier expressions for C(t) must eaeh be multiplied by a faetor exp(- co2t2kT/(2me2)) that embodies the translationally averaged Doppler shift. The speetral line shape funetion I(co) ean then be obtained for eaeh C(t) by simply Fourier transforming ... 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

To analyze the data, first perform the vx- and ty-corrections and the time-to-speed transformation to make the velocity volume element the same for all data points, and then normalize each Doppler-selected TOF spectrum according to the averaged ID Doppler profile from several independent scans, I vz) = )T)(, vyS(vy vx 0,vz)dvy. Due to the large exothermicity of this reaction, the problematic density-to-flux transformation is not negligible (despite the large probe laser size used to minimize its effects) and needs to be accounted for (the wy-correction , see Sec. 3.3). By combining all the resulting TOF spectra, the product 3D velocity flux contour... 

It was pointed out by Lorentz[36] that the same transformation equations (24) were obtained for (22) by Voigt (Gottinger Nachrichte, 1887, p.41) in an analysis of the Doppler... 

The above equation is very similar to the cross-ambiguity function, but here the time-delay is introduced only in the transmitted signal. This form of the transform is more convenient for digital implementation and will be referred to below as a range-Doppler correlation function. 

For this library the corresponding transformation in range-Doppler space is given by the rotation... 

It is of interest to compare the half-widths at half-intensity of the spectral functions of the three systems shown in Fig. 3.2. These amount to roughly 140, 80 and 50 cm-1 for He-Ar, Ne-Ar and Ar-Kr, respectively, which are enormous widths if compared to the widths of common Doppler profiles, etc. The observed widths reflect the short lifetimes of collisional complexes. From the theory of Fourier transforms we know that the product of lifetime, At, and bandwidth, A/, is of the order of unity, Eq. 1.5. The duration of the fly-by interaction is given roughly by the range of the induced dipole function, Eq. 4.30 (1/a = 0.73 a.u. for He-Ar), divided by the mean relative speed, Eq. 2.12. We obtain readily ... 

Some very important surface properties of solids can be properly characterized only by certain wet chemical techniques, some of which are currently under rapid improvement. Studies of adsorption from solution allow determination of the surface density of adsorbing sites, and the characterization of the surface forces involved (the energy of dispersion forces, the strength of acidic or basic sites and the surface density of coul-ombic charge). Adsorption studies can now be extended with some newer spectroscopic tools (Fourier-transform infra-red spectroscopy, laser Raman spectroscopy, and solid NMR spectroscopy), as well as convenient modern versions of older techniques (Doppler electrophoresis, flow microcalorimetry, and automated ellipsometry). 

Portal vein thrombosis (see chapter 39.3.3) with gradual or incomplete obstruction merely produces anastomoses and develops asymptomatically. After complete obstruction, ultrasonography will showparaportal, angiomatous anastomoses, predominantly in the porta hepatis (= cavernous transformation). (95) Splenomegaly is evidenced with a normal-sized liver. The portal vein is not detectable. Colour Doppler sonography has become of paramount importance in portal vein system diagnostics. 

Weltin, G., Taylor, K.J.W., Carter, A.R., Taylor, C.R. Duplex Doppler identification of cavernous transformation of the portal vein. Amer. X Roentgenol. 1985 144 999-1001... 

Here, (...) = J2pn ri. .. ri) is the appropriate combinedquantal and thermodynamic average (over the classical probabilities pn) related with the condensed matter system. M and n(p) are the mass and momentum distribution of the scattering nucleus, respectively, and ior = q2 /2 M is the recoil energy. For convenience, h = 1. Eq. (2) is of central importance in most NCS experiments, since it relates the SCS directly to the momentum distribution. Furthermore, n(p) is related to the nuclear wave function by Fourier transform and therefore, to the spatial localization of the nucleus. It takes into account the fact that, if the scattering nucleus has a momentum distribution in its ground state, the 5-function centered at uor will be Doppler broadened. 

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