Second order Doppler

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... 

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. 

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]... 

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...

Isomer shifts have been measured in a variety of nickel compounds. In most cases, however, the information concerning chemical bond properties was not very impressive. The reason is that the second-order Doppler (SOD) shift is, in many systems, of comparable magnimde as the real chemical isomer shift, which causes... 

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]. 

Fig. 10. Extrapolation of the half maximum center (o) and of the line position corrected for the light-shift, second order Doppler effect and 8D hyperfine structure ( ) versus the light power P in the case of the 2 >i/2 (F = 1) — 8D5/0 transition of deuterium...

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