First-order 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 order to dissipate the recoil energy Mossbauer was the first to use atoms in solid crystal lattices as emitters and also to cool both emitter and absorber. In this way it could be shown that the 7-ray emission from radioactive cobalt metal was absorbed by metallic iron. However, it was also found that if the iron sample were in any other chemical state, the different chemical surroundings of the iron nucleus produce a sufficient effect on the nuclear energy levels for absorption no longer to occur. To enable a search for the precisely required absorption frequency, a scan based on the Doppler effect was developed. It was noted that a velocity of 102 ms-1 produced an enormous Doppler shift and using the same equation (7) it follows that a readily attainable displacement of the source at a velocity of 1 cms-1 produces a shift of 108 Hz. This shift corresponds to about 100 line-widths and provides a reasonable scan width. 

Resonance gamma spectrometry or Mossbauer spectrometry can be used to study the hyperfine interactions between a nucleus and its chemical neighborhood [142], In order to examine these interactions with the help of a Mossbauer spectrometer, the first-order Doppler effect shift of the wave emitted by a moving source is applied. The arrangement used for a Mossbauer spectrometer consists of a radioactive source containing a Mossbauer isotope in an excited state (see Figure 4.54)... 

These three fluctuating effects, with their respective orders of magnitude, account for the observed standard deviation 6xi0-9 ( 2.5 kHz), which therefore represents the effective uncertainty on our measurement. Residual first order Doppler effect and relativistic corrections are negligible at this scale. 

In Sect. 2.4 we saw that the first-order Doppler effect can be exactly canceled for a two-photon transition if the two photons hcoi = fuo2 have opposite wave vectors, i.e., k = —k2. A combination of Doppler-free two-photon absorption and the Ramsey method therefore avoids the phase dependence (p Vx) on the transverse velocity component. In the first interaction zone the molecular dipoles are excited with the transition amplitude a and precess with their eigenfrequency (jl> 2 = ( 2 — E )/h. If the two photons come from oppositely traveling waves with frequency u), the detuning... 

The first teirni in Eq. (7) contains the factor (see (5)), i.e. the inverse of the difference factor in ( t), and is therefore independent of whether the minimal-coupling or the multipolar Hamiltonian is used. The second teimi represents a velocity effect on emission with q ui. Thi effect is rather curious since such emission can be perpendicular to V, i.e. its frequency can be Doppler-free. This term can also be obtained directly from the minimal-coupling Hamiltonian, if (ri) is expanded to first order in r -. ... 

A parametric study carried out in order to establish the design orientations of burner cores taught us first that a considerable reduction of the fuel inventory (or dilution ) is always necessary to be able to operate a large core with a high plutonium content and therefore with an attractive plutonium burning performance. This dilution results in a decrease in in-pile fuel residence time as well as in a reduction (favourable) of the sodium void reactivity, whereas a decrease of the uranium content of the fuel brings about a reduction of the Doppler effect, a decrease of the conversion ratio which causes a daily reactivity loss ttiat makes it difficult to achieve long irradiation cycles, as well as a reduction of tiie delayed neutron fraction. 

In Sect. 7.4 we saw that the first-order Doppler effect can be exactly canceled for a two-photon transition if the two photons ha>] = ha>2 have opposite wave vectors, i.e., ki = — 2. A combination of Doppler-free two-photon absorp-... 

Also, the residual Doppler effect has not been considered in the theoretical calculations for the nonlinear dispersion. Although two-photon Doppler-free configuration [19] has been used in all the experiments described in this Chapter, it can only cancel the first-order Doppler effect. To consider the residual Doppler effect due to the ground-state frequency difference, integration over the veloeity distribution for the hot atoms should be carried out [19], which was not done in our simple treatments [16,17]. 

The inherent resolution of collinear-beam spectroscopy is still limited by the residual Doppler broadening. In beams with a broad velocity distribution the labeling of one velocity class by optical pumping, probed in a second Doppler-tuning zone, was exploited already before narrow Doppler widths were achieved. The complete elimination of the first-order Doppler effect in resonant two-photon absorption on Ne I has been discussed in Section 3.3, in connection with a precision measurement of the relativistic Doppler effect. A similar experiment was performed on In I, where the 29p Rydberg state was excited from 5p Pi/2 via 6s Si/2 and detected by field ionization. The linewidth caused by the laser jitter can be reduced to the transit-time limit of a few hundred kilohertz. 

As is well known, the first-order Doppler effect can be eliminated by irradiating a well-collimated atomic beam perpendicular to a laser beam. Since tunable, narrow-band cw dye laser radiation of sufficiently short wavelength is not yet available, in most cases the excitation of atomic Rydberg states by a one-photon transition from the ground state is not feasible. The population of Rydberg states, however, might proceed from excited, metastable or even short-lived states which are continuously pumped by an additional laser beam. The latter approach was chosen by... 

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... 

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