Mossbauer-Lamb factor

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

In anisotropic crystals, the amplitudes of the atomic vibrations are essentially a function of the vibrational direction. As has been shown theoretically by Karyagin [72] and proved experimentally by Goldanskii et al. [48], this is accompanied by an anisotropic Lamb-Mossbauer factor/which in turn causes an asymmetry in quadra-pole split Mossbauer spectra, for example, in the case of 4 = 3/2, f = 1/2 nuclear transitions in polycrystalline absorbers. A detailed description of this phenomenon, called the Goldanskii-Karyagin effect, is given in [73]. The Lamb-Mossbauer factor is given by... 

Inserting (4.65) into (4.57), we find a modified relation for the relative intensity ratio I2H1 for crystals with anisotropic (but axially symmetric) Lamb-Mossbauer factor/... 

Here, L(v) is a lineshape function that integrates to unity, v is the frequency,/ is the Lamb-Mossbauer factor, and the desired side bands have an area fraction / that is proportional to which hence determines the relative peak heights in a NIS spectrum. More details are provided in Appendix 2 (Part III, 3 of CD-ROM). An equivalent and often more suggestive display of the NIS spectrum is the PVDOS approach, which describes the NIS signal in terms of the partial vibrational density of states ... 

The relatively low Lamb-Mossbauer factors encountered in Ni Mdssbauer spectroscopy (a diagram showing the Lamb-Mdssbauer factor as a function of Debye temperature is given in [3]) require the cooling of both source and absorber, preferentially to temperatures S80 K. The cryogenic systems have been described, e.g., in [2, 4]. 

Carbon-supported Ru-Sn catalyst Ru and Sn Mossbauer measurements were performed to investigate catalysts of ruthenium and tin supported on activated carbon (Ru-Sn/C). The samples were subjected to different reducing and oxidizing treatments. The presence of tin leads to a substantial increase of the Lamb-Mossbauer factor of the metallic Ru-particles showing that tin strengthens the attachment of the particles to the support. The close contact between the two metals appears to be decisive for the formation of catalytically active sites (Ru-Sn and Ru-SnOj,-)... 

NIS of synchrotron radiation yields details of the dynamics of Mossbauer nuclei, while conventional MS yields only limited information in this respect (comprised in the Lamb-Mossbauer factor /). NIS shows some similarity with Resonance Raman- and IR-spectroscopy. The major difference is that, instead of an electronic resonance (Raman and IR), a nuclear resonance is employed (NIS). NIS is site-selective, i.e., only those molecular vibrations that contribute to the overall... 

The primary parameters that can be extracted from conventional Mossbauer spectra are the Lamb-Mossbauer factor,/, as well as the various hyperfine parameters that provide information about the state of the electronic enviromnent of the Mossbauer... 

Lamb-Mossbauer Factor in Spin-Crossover Compounds (Example [Fe (tpa)(NCS)2])... 

Lamb-Mossbauer Factor in Single Crystals (Example ... 

From the individual contributions of the modes to the msd along the c-axis ( 6 pm ) and along the a-axis ( 8 pm ), the corresponding calculated molecular Lamb-Mossbauer factors for the c-cut crystal (/Lm,c = 0.90) and for the a-cut crystal = 0.87) were derived. Comparison with the experimental /-factor, i.e., / P = 0.20(1) and/ N> = 0.12(1) [45], indicates that by far the largest part of the iron msd must be due to intermolecular vibrations (acoustic modes) of the nitroprusside anions and its counter ions. This behavior is reflected in the NIS spectrum of GNP by the considerable onset of absorption probability density below 30 meV in Fig. 9.36a. 

In molecular crystals, the Lamb-Mossbauer factor can be regarded (in good approximation) as a product of a molecular part,/ °, and a lattice part,/ [101] ... 

Table 9.3 Factorization of the Lamb-Mossbauer factor/lm into a molecular part/JIi" and into a lattice part/j()Jj at different temperatures...

Lamb-Mossbauer factor of the absorber, the number of resonant isotope species... 

This technique, besides allowing determination of the Lamb-Mossbauer factor, provides direct access to the density of phonon states for the probe isotope in a solid. It thus provides information about lattice dynamics that is excluded by the limitations of Mossbauer spectroscopy. This technique could be valuable in investigations of adsorption with the adsorbing element as the probe and showing the modifications brought about by the adsorbate on the dynamic properties of the probe. 

Because Miattice is much larger than the nnclear mass M, the recoil energy is negligible and resonance absorption takes place. In terms of quantum mechanics, there is a probability / for a zero-phonon process, where no lattice vibrations are excited. Rudolf Mossbauer showed that the probability / for a zero-phonon process is given by the subsequently named Lamb-Mossbauer factor ... 

The quantity (x ) denotes the mean-square displacement of the Mossbauer nucleus in the direction of the incoming y-ray and k = lit jl. is the wave number of the y-ray. The decrease of the mean-square displacement (x ) with decreasing temperature leads to an increase of the Lamb-Mossbauer factor / and thus the Mossbauer effect. 

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