Doppler first-order

The measurements of 2s — Is transitions in magnetically trapped hydrogen have achieved a relative accuracy of one part in 1012 [21] by means of two-photon spectroscopy which eliminates the first-order Doppler broadening. It is hoped that this technique will allow the measurement of the Is — 2s transition with the accuracy limited only by the shape of the transition line dictated by quantum electrodynamics, i.e. to a few parts in 1015. Further, if the center of the Is — 2s line could be determined with the accuracy of a few parts in 103 of its width, the relative accuracy for this transition would increase to a few parts in 1018. 

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)... 

The first-order Doppler frequency shift, Av, of a wave emitted by a source moving at a velocity, v, is given by the following expression... 

To eliminate the residual first order Doppler shift due to the failure of the direction of propagation of the rf field to be precisely perpendicular to the fast beam, measurements were taken with the rf drive on both the right and left sides of the beam. To eliminate the frequency shift due to phase errors in the rf drive system, measurements were made with the entire rf system, including the spectroscopy region, rotated 180° about an axis passing through the midpoint between the two... 

First order Doppler broadening can be eliminated by using a standing wave geometry (i.e. oppositely running waves) to excite the two-photon transition. The fractional second order Doppler shift, v1 c2, is less than 2 x 10-16 at a temperature of 1 mK. 

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 1953 Dicke [18] proposed the use of an inert buffer gas such as helium to reduce the first order Doppler width of a radiating system by partially averaging the velocities when the times between collisions are short compared to the radiative lifetime. 

Once one has succeeded in trapping but a single ion, then the ion must be placed in the lower vibration levels (n = 1,2,3...) in the trap. First, micro-motion reduction is required. The amplitude of this micro-motion can be controlled thanks the so-called RF photon correlation . This procedure consists of modulating the output signal from the photomultiplier at the frequency of the RF drive potential. The amplitude of the micro-motion is then revealed by the amplitude of the fluorescence which depends on the first-order Doppler shift. It is, therefore, relatively easy to use precisely either laser detuning or to modify the potential voltages so as to reach the maximum reduction of the amplitude that corresponds to the lowest KE (Figure 11.10). 

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 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... 

Once the property perturbations are known and the unperturbed field equations solved for N and Q, this first-order expression is easily evaluated as a straightforward piece of matrix algebra the operators (once determined) are composed of scalar elements only. It is, admittedly, no trivial task to estimate the change in cross sections and collision probabilities associated with, for example, a problem in Doppler broadening. It is not the place of this article to enter into details of the collision... 

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 -. ... 

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-... 

Detailed neutronics calculations were performed in spherical geometry using the transport code DTF-IV in the P1-S4 approximation with 27 broad energy groups. A space-independent perturbation cross section was also computed at the center of the system for comparison with measured results. The first-order transport perturbation theory program GAPER was used in evaluating central material worths as well as Doppler and sodium-void reactivity coefficients. 

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