Doppler shift, second order

The experimentally observed isomer shift, (5exp, includes a relativistic contribution, which is called second-order Doppler shift, sod and which adds to the genuine isomer shift d. 

Experimental isomer shifts, exp, should be corrected for the contribution of soD in order to avoid misinterpretations. The value of (5sod drops with temperature and becomes vanishingly small at liquid helium temperature, because is proportional to the mean kinetic energy of the Mossbauer atom. In practice, sod may already be negligible at liquid nitrogen temperature it rarely exceeds —0.02 mm s at 77 K. At room temperature, 3soo rnay be as large as —0.1 mm s or more (Fig. 4.2). 

The temperature dependence of sod is related to that of the recoil-free fraction /(T) = Qxp[— x )Ey / Hc) ], where (x ) is the mean square displacement (2.14). Both quantities, (x ) and can be derived from the Debye model for the energy distribution of phonons in a solid (see Sect. 2.4). The second-order Doppler shift is thereby given as [20] 

In proteins, the value of 0 is usually in the low temperature range 100-300 K inorganic compounds may show values of 150-500 K, whereas metals have 0m as high as 1,000 K or more. 

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Appendix E An Introduction to Second-Order Doppler Shift. 547... 

The recoil-free fraction depends on the oxidation state, the spin state, and the elastic bonds of the Mossbauer atom. Therefore, a temperature-dependent transition of the valence state, a spin transition, or a phase change of a particular compound or material may be easily detected as a change in the slope, a kink, or a step in the temperature dependence of In f T). However, in fits of experimental Mossbauer intensities, the values of 0 and Meff are often strongly covariant, as one may expect from a comparison of the traces shown in Fig. 2.5b. In this situation, valuable constraints can be obtained from corresponding fits of the temperature dependence of the second-order-Doppler shift of the Mossbauer spectra, which can be described by using a similar approach. The formalism is given in Sect. 4.2.3 on the temperature dependence of the isomer shift. 

Cahbration spectra must be measured at defined temperamres (ambient temperature for a-iron) because of the influence of second-order Doppler shift (see Sect. 4.2.1) for the standard absorber. After folding, the experimental spectrum should be simulated with Lorentzian lines to obtain the exact line positions in units of channel numbers which for calibration can be related to the hteramre values of the hyperfine splitting. As shown in Fig. 3.4, the velocity increment per channel, Ostep, is then obtained from the equation Ustep = D,(mm s )/D,(channel numbers). Different... 

Fig. 4.2 Temperature dependence of the isomer shift due to the second-order Doppler shift, sod- The curves are calculated for different Mossbauer temperatures 0m by using the Debye model whereby the isomer shift was set to (5 = 0.4 mm s and the effective mass to Meff =100 Da, except for the dashed curve with Meff = 57 Da...

In order to elucidate the physical origin of second-order Doppler shift, sod, we consider the Mossbauer nucleus Fe with mass M executing simple harmonic motion [1] (see Sect. 2.3). The equation of motion under isotropic and harmonic approximations can be written as... 

From a chemical point of view, the second-order Doppler shift is very interesting with respect to its simple relation connecting (5sod, the recoil-free fraction/, and the... 

Note that we ignore higher order contributions such as v2/2c2 in (5-3) this has consequences, as we will see later when we discuss the second order Doppler shift. 

The isomer shift contains a contribution from the thermal motion of the individual atoms in the absorber, the second-order Doppler shift, which makes the isomer shift temperature-dependent ... 

The second term in (5-4) is the second-order Doppler shift. This is the higher-order term of the Taylor expansion that we ignored in (5-3). Like , it can be calculated in the Debye model. Figure 5.6 shows plots of the second-order Doppler shift for the case of iron and for different values of the Debye temperature. Soft lattice vibrations are expected to decrease the isomer shift, although the effect becomes only significant at temperatures well above 80 K. 

So by measuring the second-order Doppler shift of the Mossbauer nuclei in a material it is possible to determine their average velocity and thus their average vibrational kinetic energy, /2, where the mass of the Mossbauer nucleus. The... 

Polyakov 1997). Because the second-order Doppler shift is not the only factor controlling Mossbauer absorption frequencies, it is generally necessary to process data taken at a variety of temperatures, and to make a number of assumptions about the invariance of other factors with temperature and the form and properties of the vibrational density of states of the Mossbauer atom. Principles involved in analyzing temperature dependencies in Mossbauer spectra are extensively discussed in the primary literature (Hazony 1966 Housley and Hess 1966 Housley and Hess 1967) and reviews (e.g., Heberle 1971). 

Note that we ignore higher-order contributions such as d2/2c2 in Eq. (5-3) this has consequences, as we will see later when we discuss the second-order Doppler shift. In order to detect shifts and splitting in the nuclear levels due to hyperfine interactions in iron, one needs an energy range of at most 5 10 8 eV around E0, which is achieved with Doppler velocities in the range of —10 to +10 mm s 1. 

For an accurate data analysis, a detailed understanding of systematic effects is necessary. Although they are significantly reduced with the improved spectroscopy techniques described above, they still broaden the absorption line profile and shift the center frequency. In particular, the second order Doppler shift and the ac-Stark shift introduce a displacement of the line center. To correct for the second order Doppler shift, a theoretical line shape model has been developed which takes into account the geometry of the apparatus as well as parameters concerning the hydrogen atom flow. The model is described in more detail in Ref. [13]. 

It appears likely that the statistical uncertainty will eventually be reduced to around 100 kHz, so we consider sources of systematic error which may be expected to enter at this level. The uncertainty in the second-order Doppler shift (450kHz/eV) will be reduced to 100 kHz by a 5% measurement of the beam energy. The AC Stark shift of the 2S-3S transition will be around 70 kHz for the present laser intensity, and can be extrapolated to zero intensity by varying the UV power. Finally, as mentioned above, the systematic uncertainties will be quite different from those in the microwave and quench anisotropy measurements. 

With an appropriate choice of B the compensation is effective for all atomic velocities. As the velocity of the thermal beam of hydrogen atoms at 300 K is not well known, the magnetic field will be used to measure precisely the velocity distribution, and so the second order Doppler shift. 

The method of symmetric points was used to determine the center of the interference curve. Extensive calculations showed that the line profile should be symmetric about the center frequency. The line center was then corrected for the second order Doppler shift, The Bloch-Siegert and rf Stark shifts, coupling between the rf plates, the residual F=1 hyperfine component, and distortion due to off axis electric fields. A small residual asymmetry in the average quench curve was attributed to a residual variation of the rf electric field across the line and corrected for on the assumption this was the correct explanation. Table 1 shows the measured interval and the corrections for one of the 8 data sets used to determine the final result. 

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. 

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