Linear Doppler effect

The first term represents the absorption frequency coq = Ek — Ei) of an atom at rest if the recoil of the absorbing atom is neglected. The second term describes the linear Doppler shift (first-order Doppler effect) caused by the motion of the atom at the time of absorption. The third term expresses the quadratic Doppler effect (second-order Doppler effect). Note that this term is independent of the direction of the velocity v. It is therefore not eliminated by the Doppler-free techniques described in Chaps. 2-5, which only overcome the linear Doppler effect. 

In most real cases, emitter and absorber transition energies Ey are different, because the emitter and absorber nuclei interact with their different chemical surroundings. In order to meet the criteria for resonance, the sample and source can be put in motion relative to one another. This leads to a (linear) Doppler Effect of a magnitude suitable for matching the emission and absorption lines. This can be described as an energy change AEs, gained when source and absorber move towards one another, and lost if they move apart, such that... 

Mossbauer spectra are usually recorded in transmission geometry, whereby the sample, representing the absorber, contains the stable Mossbauer isotope, i.e., it is not radioactive. A scheme of a typical spectrometer setup is depicted in Fig. 3.1. The radioactive Mossbauer source is attached to the electro-mechanical velocity transducer, or Mossbauer drive, which is moved in a controlled manner for the modulation of the emitted y-radiation by the Doppler effect. The Mossbauer drive is powered by the electronic drive control unit according to a reference voltage (Fr), provided by the digital function generator. Most Mossbauer spectrometers are operated in constant-acceleration mode, in which the drive velocity is linearly swept up and down, either in a saw-tooth or in a triangular mode. In either case. 

As mentioned in the introduction, meandering and resonant drift of spiral waves in the photosensitive BZ reaction can be achieved experimentally by a periodically changing light intensity [9, 90]. The Doppler effect imposes superspiral structures in the case of spiral wave dynamics other than rigid rotation. These superspiral structures have been observed in experiments see for example [11, 41, 55, 56, 89]. See [70] for a mathematical analysis based on linearized analysis and eigenfunctions. 

The basic reason for the persistent discrepancy is the obstinate refusal to admit that cosmological redshifts could arise from anything but the Doppler effect. As in the Stonier (1990) model many observed redshifts may well be of the Doppler type, but unrelated to distance. In other cases the redshift may be distance dependent, but caused by curvatme and not expansion. It would clearly be impossible to find a common proportionality constant for two such imrelated linear relationships. In one instance maybe Hq = 100 compared to Hq = 50 in the other. Who knows ... 

During the start-up of Superphenix, an experiment related to the Doppler effect has been performed, on the CMP core, decreasing slowly the temperature from 400 to 180°C while maintaining isothermal conditions in the reactor. The increase in reactivity was compensated by control rod insertion. The contributions of the expansion reactivity coefficient (linear with respect to temperature) and of the Doppler effect (logarithmic with respect to temperature) have been separated. The model took into account the effective temperature, using the Debye temperature. The comparison of experiment and calculation, using the reference scheme is given in Table 7. 

The velocities of very cold atoms are very small, i.e., the linear and the quadratic Doppler effects both become small and the recoil term becomes significant. It turns out that for cold Ca atoms at T = 10 pK, the recoil effect leads to large asymmetries in the Lamb dips of absorption spectra taken with short pulses [1119]. These are not found in experiments performed at room temperature, where the broad Doppler background masks these asymmetries, and they are based on the fundamental asymmetry between absorption and stimulated emission with short pulses. 

Note Equations (3.38) and (3.39) describe the linear Doppler shift. For higher accuracies, the quadratic Doppler effect must also be considered (Sect. 14.1). 

The relative accuracy of the transition frequencies, which can be achieved within a sympathetically cooled ensemble, is in principle better than with buffer gas cooled ions or with fast ion beams. Molecular dynamics simulations of the motion of the trapped HD+ ions lead to an estimate of a Doppler broadening of only 10 MHz under ideal conditions, whereas a thermal 10K ion ensemble leads to a Doppler width (FWHM) of 280 MHz. In reality the reported line broadening was 40 MHz indicating an effective temperature of 0.2 K. Note that this temperature describes only the axial motion of the HD+ ions, the radial motion does not lead to a first-order Doppler effect. The explanation given by the authors is based on a non-linear coupling between the axial and radial motion of the ions via Coulomb interaction. In a more detailed analysis, Koelemeij et a/. ° distinguish between translational (secular) temperature of 50 mK resulting... 

Whereas the resolution in linear Raman spectroscopy is limited in principle by the slit width of the spectrometer, a considerable improvement in the instrumental resolution was attained through the development of the techniques of nonlinear or coherent Raman spectroscopy, where the interaction of two laser beams with the third-order susceptibility of the sample creates the spectrum. In this case, the resolution is determined by the convoluted linewidth of the two lasers, the Doppler effect, and pressure broadening of the spectral lines. 

Experimental Validation. The following types of measurement have been used to evaluate the accuracy of Doppler effect calculations (a) the South-west Experimental Fast Oxide Reactor (SEFOR) was built and operated specifically to measure Doppler effects (or fast-acting fuel reactivity feedback effects with expansion effects minimised) (b) the dependence of reactivity on temperature in operating power reactors, such as PHENDC and SUPER-PHENDC (fi-om the non-linearity of the temperature coefficient, for example) (c) the ZEBRA 5 Doppler Loop experiments, in which a test zone was heated. Experiments were performed with and without sodium present (d) the temperature dependence of the reactivity worths of small samples oscillated at the centre of critical assemblies (e) the differences in reaction rates in samples irradiated at different temperatures and (f) temperature dependent thick sample transmission and self-indication measurements, which are usually analj ed together with the differential nuclear data to provide average resonance parameter data. The uncertainties in extrapolating fi-om these comparisons to the conditions in an operating power reactor must also be taken into account. 

Doppler effect measurements have been made in PHENDC [4.94] and SUPER-PHENDC [4.95]. The measurements were of the isothermal temperature changes at very low power. The Doppler effect is identified with the non-linear component of the temperature coefficient assuming a dependence of the form b/T for the Doppler effect. The uncertainties in the analysis of the measurements arise from uncertainties in control rod reactivity worth and rod worth profile and from the statistical analysis of the data. It is concluded that the maximum uncertainty is 15% and the probable error 10%. A more recent analysis is presented in [4.96]. 

A plot of V vs —1/ 2 should be linear, with equal slope and intercept. The most reliable and accurate data available fail this test statistically [24], The discrepancy is not large but significant and reminiscent of the spectroscopic red shift measured in galactic light. The common origin of these discrepancies cannot be a Doppler effect and most likely is due to space-time curvature. 

The first term wq = (Eg -Ej )/h represents the eigen frequency of the atom in rest if recoil is neglected. The second term is the linear (first-order) Doppler effect, describing the well-known Doppler shift Aw =J< in the absorption frequency of a moving atom. The third term represents the second-order Doppler effect. Note that this term is independent of the direction of V and cannot be eliminated by the methods, discussed in Chap.10, which only overcome the first-order Doppler effect. The last term in (13.14) describes the photon recoil effect, where has been approximated by wq. 

The advent of lasers in spectroscopy has made possible highly precise measurements of spectroscopic as well as of fundamental interest, Particular emphasis has been put onto the elimination of the Doppler effect, which was one of the main obstacles in classical spectroscopy. This can be achieved using well collimated atomic beams or non-linear field/atom interactions, which, combined with quantum interference methods, are capable of yielding a resolution beyond the natural linewidth. In historical perspective, these methods were developed because of the problems associated with the Doppler effect, the possibilities offered by the high intensity and narrow spectral band width of lasers and, most important, an ever persistent wish to obtain very high optical resolution. 

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-... 

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