Mossbauer second-order Doppler shift

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. 

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... 

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). 

The second-order Doppler shift, (5sod> which is often called temperature shift, is a peak shift related to the relativistic Doppler effect originating from the thermal motion of the nuclei. If the Mossbauer atom has a speed u, and moves in a direction making angle a with the direction of the y ray it emits, then the v frequency of the emitted y ray will differ from the Vo frequency it had if the atom had been at rest. The v frequency is related to Vo in the following way ... 

In the Mossbauer spectrum, the second-order Doppler shift is measured as a peak shift in velocity units ... 

These three types of hyperfine interactions with the relevant Mossbauer parameters are most important in solid state research, in addition, one often extracts further helpful information from the temperature and pressure dependence of the Mossbauer parameters, the shape and width of the resonance lines (relaxation phenomena), and the second-order Doppler shift (lattice dynamics). 

In the above analysis it is assumed that the atoms vibrate with a single frequency Q, as in the Einstein theory, whereas in a real crystal there is a distribution of vibrational frequencies with an appropriate cut-off, as considered in the Debye theory. In any case, it is clear that a measurement of the /-factor by Mossbauer spectroscopy can provide knowledge concerning phonon properties, such as their frequency distribution and density of states. Similar information can also be obtained from an analysis of the second-order Doppler shift. Unfortunately, the restriction imposed by the relative timescales, typically as discussed earlier, normally... 

The possible contributions of the Mossbauer technique to the study of phase transitions has been outlined previously (Shenoy, 1973). Almost all phase transitions cause changes in the lattice dynamics of the crystal and these changes can be studied through measurements of the recoil-free fraction, its anisotropy and the second-order Doppler shift. The phase transition itself is in many cases also observed through changes in the hyperfme interaction parameters. 

Fig. 1.6 Shift and splitting of nuclear levels of nucleus by the hyperfine interactions and expected Mossbauer spectra, a Center shift of the observed absorption line. In this case the observed shift is the sum of isomer shift and second order Doppler shift, b Electric quadrupole splitting AEq and c magnetic h5 perfine splitting that is a nuclear Zeeman splitting. is the magnitude of the hyperfine magnetic field at nucleus...

The Mossbauer fraction /of each kind of iron species is mainly governed by the lattice dynamics in the crystal. Therefore, / is dependent on the coordination and can differ slightly from mineral to mineral. Moreover, the Mossbauer fraction is particularly very sensitive to the valence state of iron and is for Fe " considerably lower than for Fe ". Because of the relationship with lattice vibrations, / is also strongly temperature dependent. This means that at RT a large difference in /values is observed and only to a lesser extent at 80 K. The /values for some iron-containing minerals, determined from the temperature dependence of the isomer shift (second order Doppler shift), are fisted in Table 3.1. 

The observed lineshift is not entirely caused by the chemical isomer shift. There is another generally smaller contribution termed the second-order Doppler shift. The emitting or absorbing nucleus is not stationary - it is vibrating at its position in the lattice. The period of its vibration is much shorter than the Mbssbauer lifetime, so the average displacement and velocity are effectively zero, but the mean-squared values of the velocity, (v ), are finite, v) is zero, so only the second-order term in (v ) can influence the Mossbauer resonance. This gives rise to a shift in the Mossbauer line of... 

Besides the IS there exists a shift of the Mossbauer lines due to the second-order Doppler effect (i3, 14). This shift is given by... 

Mossbauer Spectroscopy. Mdssbauer spectroscopy has been used as a powerful technique to probe the electronic structure of the five- and six-coordinate ferrous nitrosyl porphyrins. The isomer shifts of both types of complexes (5 0.35 mm s ) are similar and show temperature dependences that are consistent with a second-order Doppler effect, but are slightly smaller than those of most other iron(II) porphyrin complexes (5 0.45 mm Unlike the isomer... 

The thermal shift of Mossbauer lines arises from the second-order Doppler effect (Pound Rebka, 1959). If is the mean-square velocity of the Mossbauer nucleus in the crystal, then the relative shift in energy of the Mossbauer line will be given by AE/Eq= — /2c. In the Debye model approximation this leads to AE... 

The existence of a relativistic temperature-dependent contribution to the chemical isomer shift was pointed out independently by Pound and Rebka [6] and by Josephson [7]. The emitting or absorbing atom is vibrating on its lattice site in the crystal. The frequency of oscillation about the mean position is of the order of 10 per second, so that the average displacement during the Mossbauer event is zero. However, there is a term in the Doppler shift which depends on v, so that the mean value is non-zero. 

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