The recoil-free fraction

The Mossbauer recoil-free fraction, the anisotropy in the recoil-free fraction and the thermal shift, when studied as a function of temperature, may all yield information on lattice dynamics which is as useful as that obtained from heat capacity or X-ray scattering measurements. One obvious advantage of the Mossbauer technique is the fact that it measures the vibrational modes of the Mossbauer probe, thereby yielding information on local modes which is unobtainable by other techniques. 

The subject of the recoil-free fraction was initially discussed in Mossbauer s original paper (Mossbauer, 1958). Since then, many books and review articles (e.g. Nussbaum, 1966) have been devoted to describing experimental measurements and theoretical calculations of the recoil-free fraction f Hence this discussion will be limited to some of the major aspects of the information obtainable from /-factor measurements. The measurement itself is relatively easy, since for thin absorbers the Mossbauer spectral area is proportional to /, and thus the relative change in / with temperature is readily obtainable. With thick absorbers one can still obtain accurate values, although the analysis requires transmission integrals (Shenoy, 1973). 

The fraction of gamma rays emitted or absorbed without recoil by a nucleus bound in a crystal is given to a good approximation by the expression  

Where k is the wavevector of the gamma ray and x is the mean-square displacement of the Mossbauer nucleus in the direction of the gamma ray. The exponent 2W s the Debye-Waller factor, which is also relevant to X-ray and neutron scattering line intensities. Thus a measurement of the recoil-free fraction at a particular temperature /(T) yields the value of x at this temperature, which can be compared with theoretical calculations of 

The distribution function G(f2) is generally derived using a specific model. In the Einstein oscillator model there is only one frequency Qg and thus G(Q)=5K(i -i E) where is the Kronekker delta function. Hence, defining an Einstein temperature 0, given by this gives 

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The example is typical for many applications of Mossbauer spectroscopy in catalysis a catalyst undergoes a certain treatment, then its Mossbauer spectrum is measured in situ at room temperature. Flowever, if the catalyst contains highly dispersed particles, the measurement of spectra at cryogenic temperatures becomes advantageous as the recoil-free fraction of surface atoms increases substantially at temperatures below 300 K. Secondly, spectra of small particles that behave superparamagne-... 

Independent of specific theoretical models for the phonon spectrum of a solid matrix, the recoil-free fraction can be given in terms of the y-energy Ej and the mean local displacement of the nucleus from its equilibrium position ([2] in Chap. 1) [5] ... 

The recoil-free fraction / is an important factor for determining the intensity of a Mossbauer spectrum. In summary, we notice from inspecting (2.14)-(2.17) and Fig. 2.5a that... 

For a more detailed account of the recoil-free fraction and lattice dynamics, the reader is referred to relevant textbooks ([12-15] in Chap. 1). 

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. 

However, in contrast, the resonance effect increased by cooling both the source and the absorber. Mdssbauer not only observed this striking experimental effect that was not consistent with the prediction, but also presented an explanation that is based on zero-phonon processes associated with emission and absorption of y-rays in solids. Such events occur with a certain probability/, the recoil-free fraction of the nuclear transition (Sect. 2.4). Thus, the factor/is a measure of the recoilless nuclear absorption of y-radiation - the Mdssbauer effect. 

So far we have considered only the recoil-free fraction of photons emitted by the source. The other fraction (1 —/s), emitted with energy loss due to recoil, cannot be resonantly absorbed and contributes only as a nonresonant background to the transmitted radiation, which is attenuated by mass absorption in the absorber... 

The recoil-free fraction /a of transition metal complexes or proteins in frozen solution can be as small as 0.1-0.3, when measured just below the melting point, but the /-factor increases strongly when the temperature is lowered to fiquid nitrogen temperatures (77 K), and at fiquid helium temperatures (4.2 K) it may reach values of 0.7-0.9 [35]. This makes a substantial difference to the acquisition time of the spectra because of the square dependency on the signal (3.1). 

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]... 

Kaltseis et al. [234] have investigated the recoil-free fraction of the 46.5 keV transition of W in anhydrous lithium tungstate. Their results can be expressed by an effective Debye temperature of 172 9 K which is in good agreement with a value of 205 40 K derived from X-ray diffraction measurements of Li2W04 powder. 

As is well known, the recoil-free fraction of very small crystals differs markedly from that of bulk material. Roth and Horl [236] observed a decrease of the/-factor from 0.61 to 0.57 in going from 1 p,m crystals to microcrystals with a diameter of about 60 A. Two effects will contribute to this decrease (1) the low frequency cutoff, because the longest wavelength must not exceed the dimensions of the crystal, and (2) high frequency cut-off caused by the weaker bonds between surface atoms. 

Wender and Hershkowitz [237] used the sensitivity of the recoil-free fraction in tungsten Mossbauer spectroscopy to deduce the effect of irradiation of tungsten compounds by Coulomb excitation of the resonance levels (2 states of I82,i84,i8 y with 6 MeV a-particles. While no effect of irradiation on the/-factors could be observed for tungsten metal in agreement with [233], a decrease of/was measured for WC, W2B, W2B5, and WO3 after irradiation. 

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... 

The intensity of the Mossbauer effect is determined by the recoil-free fraction, or /factor, which can be considered as a kind of efficiency. It is determined by the lattice vibrations of the solid to which the nucleus belongs, the mass of the nucleus and the photon energy, Ea and is given by ... 

Figure 5.2 The recoil-free fraction,/, of iron as a function of temperature for different values of the Debye temperature, 6 y Bulk iron compounds have Debye temperatures on the order of 450-500 K surface phases, however, have significantly lower Debye temperatures, implying that measurements may have to be carried out at lower temperatures...

The intensity of a Mossbauer spectrum depends not only on the recoil-free fractions of the source and the absorber and on the number of absorbing nuclei, but also on the linewidth of the absorption lines and on whether or not saturation effects occur. The following approximate expression is valid for relatively thin absorbers [17] ... 

Thus making samples not too thick helps in getting sharper spectra and facilitates the quantitative interpretation. Finally, particularly in the Mossbauer spectra of small catalyst particles, one should be aware of the temperature dependence of the absorption area through the recoil-free fraction. If the spectrum contains contributions from surface and bulk phases, the intensity of the former will be greatly underestimated if the spectrum is measured at room temperature. The only way to obtain reliable concentrations of surface and bulk phases is to determine their spectral contributions as a function of temperature and make an extrapolation to zero Kelvin [13]. 

Surface phases have low Debye temperatures. As a result, the recoil-free fraction may be low at room temperature (see Fig. 5.2). Thus, measuring at cryogenic temperatures will increase the Mossbauer intensity of such samples considerably. But there can also be other circumstances which call for low temperature experiments. 

The Mossbauer effect involves the resonance fluorescence of nuclear gamma radiation and can be observed during recoilless emission and absorption of radiation in solids. It can be exploited as a spectroscopic method by observing chemically dependent hyperfine interactions. The recent determination of the nuclear radius term in the isomer shift equation for shows that the isomer shift becomes more positive with increasing s electron density at the nucleus. Detailed studies of the temperature dependence of the recoil-free fraction in and labeled Sn/ show that the characteristic Mossbauer temperatures Om, are different for the two atoms. These results are typical of the kind of chemical information which can be obtained from Mossbauer spectra. 

Let us now discuss some recent work by Sano and myself on completely characterizing barium stannate, a material first proposed by Plotnikova, Mitrofanov, and Shpinel (21), as a source for tin Mossbauer spectroscopy. It is easily prepared, is a stoichiometric compound and has all the properties one desires in a Mossbauer matrix. The recoil-free fraction at room temperature is about 0.55 with about a 10% error. The line width extrapolated to zero absorber thickness is about 6% larger than natural—i.e., the line width observed is ca. 0.318 mm./sec. at zero ab-... 

Dr. Stockier What, if any, is the effect of the difference in the recoil-free fraction of the two compounds of tin ... 

The recoil-free fraction /, while strictly speaking not the result of a chemical interaction, can indirectly provide useful chemical, as well as structural, information. As shown earlier, / is related to , the mean square vibrational amplitude of the resonant atom in the direction of the y ray. The temperature dependence of is often approximated using the... 

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