Ossbauer Line Research Articles

Interpretation of low-temperature Mössbauer spectra in the presence of Kondo deviations. I. General considerations

In this paper, we first derive the general expression for the M\"ossbauer line shape of a powder source when cascade effects come into play. Then we consider the case of a paramagnetic impurity---in a metal---which exhibits an incipient Kondo effect, and we examine the problems which arise when the relaxation of this impurity by the conduction electrons is studied by M\"ossbauer spectroscopy, i.e., in the presence of hyperfine coupling. We extend conventional relaxation theory up to third order in ${\mathcal{H}}_{1}=\ensuremath{-}2{J}_{\mathrm{sf}}\ensuremath{\alpha}\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$ and investigate under what conditions the third-order damping terms can be expressed as products of first- and second-order transition amplitudes. We find that this occurs when either "extreme narrowing" or "secular" approximations are valid; in these two cases, the Kondo correction to the relaxation matrix reduces to the standard computation of the second-order transition amplitude. In the following paper, these results are applied to the study of relaxation effects of the hyperfine populations at low temperatures, in the M\"ossbauer cascade of an Au $^{170}\mathrm{Yb}$ source.

Diffusion Broadening of the Mössbauer Line in Wüstite

The temperature dependence of the diffusion broadening for ${\mathrm{Fe}}^{57}$ in w\"ustite, $\mathrm{Fe}{\mathrm{O}}_{x}$, from 800 to 1050 \ifmmode^\circ\else\textdegree\fi{}C is reported. Measurements were done at various compositions $x$ by adjusting the ratio of partial pressures of carbon dioxide to carbon monoxide in which the samples were annealed. Our data indicate that the M\"ossbauer line broadening is only about half of that predicted by the Singwi-Sj\"olander theory. Our results support the recent tracer-diffusion measurements of Chen, Peterson, and Robinson, which contradict several earlier studies on the diffusion of iron in w\"ustite. The temperature dependence of the recoil-free fraction is reported in the region where diffusion broadening is observed.

Mössbauer Line Shift in Iron

The M\ossbauer line shift of iron in Fe${\mathrm{F}}_{3}$ is reexamined from the point of view of its temperature dependence. The time-averaged correction to the isomer shift arising from the virtual excitation of an electron from $3{d}^{5}$ to $4s$ with the simultaneous emission or absorption of one phonon is calculated and compared with recent experiments.

Mössbauer Study of Brownian Motion in Liquids: Colloidal Cobaltous Hydroxy Stannate in Glycerol, Ethanol-Glycerol, and Aqueous-Glycerol Solutions

We have measured the resonant absorption of the 23.9-keV $\ensuremath{\gamma}$ rays of ${\mathrm{Sn}}^{199m}$ in a colloidal suspension of CoSn${(\mathrm{OH})}_{6}$ in glycerol in the temperature range 110-323 \ifmmode^\circ\else\textdegree\fi{}K. We measured the colloidal-particle radii with a scanning electron microscope, and found them to be 650\ifmmode\pm\else\textpm\fi{}150 \AA{}. We have also studied the fixed-temperature M\ossbauer line broadening of our glycerol-based colloid, as a function of ethanol and water dilution. We find that the diffusive broadening of the M\ossbauer line is proportional to the absolute temperature divided by the viscosity. The constant of proportionality is the same within the experimental errors, whether the sample viscosity is changed by varying the sample temperature, or by diluting the sol with ethanol or water. These results are in good agreement with the Singwi-Sj\olander theory and the Einstein-Stokes relation. For this system we observed a rapid falloff in the recoil-free fraction in the temperature region where diffusive broadening effects were observed, similar to other earlier observations of this effect. Attempts to prepare large-particle sols of FeSn${(\mathrm{OH})}_{6}$ in glycerol were made to try to see a breakdown in the Singwi-Sj\olander theory. Some evidence for a failing in the theory was found for particle sizes of 1500 \AA{}, but these results were not conclusive, as the sols were always unstable for these large colloidal particles.

Mössbauer Study of Diffusion in Liquids: DispersedFe2+in Glycerol and Aqueous-Glycerol Solutions

M\"ossbauer parameters, diffusion constants, and coefficients of viscosity for ${\mathrm{Fe}}^{2+}$ ions, dissolved from Fe${\mathrm{Cl}}_{2}$ \ifmmode\cdot\else\textperiodcentered\fi{} 4${\mathrm{H}}_{2}$O, in nominally dry glycerol, glycerol-5-wt% water, and glycerol-10-wt% water have been measured. The broadening of the M\"ossbauer line is observed to have a complicated temperature dependence, which we explain by assuming two mechanisms of diffusion. At low temperatures we postulate that diffusion is via a jumping process described by an Arrhenius relation, while at high temperatures, diffusion is via a gaslike motion obeying the Einstein-Stokes and Litovitz relations. A quantitative fit to the observed line broadening is excellent. A very rapid falloff in the quadrupole splitting and the logarithm of the recoil-free fraction is observed in the temperature region where diffusive broadening is measurable; this falloff is proportional to the observed broadening, with two distinct proportionality constants for the low- and high-temperature regions. These observations are explained by a phenomenological theory due to Jensen and by postulating that the average electric field gradient at the iron nucleus reduces in proportion with the iron diffusivity.

Stochastic Theory of Line Shape: Generalization of the Kubo-Anderson Model

A general solution is presented for the line shape of radiation emitted by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms ${V}_{1}, {V}_{2}, \ensuremath{\cdots}, {V}_{n}$. The solution is valid even if these forms do not commute with one another ($[{V}_{i}, {V}_{j}]\ensuremath{\ne}0$), so that the Hamiltonian need not commute with itself at different times: $[\mathcal{H}(t), \mathcal{H}({t}^{\ensuremath{'}})]\ensuremath{\ne}0$. This is a generalization of the Kubo-Anderson model in which it is assumed that the Hamiltonian does commute at different times. The treatment given here thus extends this adiabatic or random-frequency-modulation theory to include the nonadiabatic effects of transitions induced by the fluctuating Hamiltonian. The solution involves the inversion of a matrix and is similar to Sack's solution of the Kubo-Anderson model. The matrix found in the present case is labeled by quantum-mechanical as well as stochastic indices, and it reduces to the form found by Sack when the possible forms of the Hamiltonian commute with one another. Numerical evaluation of line shapes can be accomplished easily with a computer, and in the simplest cases analytical expressions can be found. The applicability of the theory to NMR and M\"ossbauer line shapes and to perturbed angular correlations is discussed. A specific example of the NMR line shape of a spin-\textonehalf{} nucleus in a fixed magnetic field and a fluctuating field perpendicular to it is considered in detail as an illustration of the utility of the derived expressions. The solution uses the Liouville-operator notation and this is discussed in an Appendix.

Evidence of Two Forms of Cobaltous Oxide

The existence of two forms of cobaltous oxide, CoO(I) and CoO(II), has been established by M\"ossbauer, x-ray, and chemical techniques. CoO(I) has the NaCl structure as determined by a well-resolved powder x-ray pattern. Its density is measured to be 6.4 g/${\mathrm{cm}}^{3}$ from both x-ray and direct macroscopic experiments. ${\mathrm{Fe}}^{57}$ M\"ossbauer lines in CoO(I), doped with ${\mathrm{Co}}^{57}$, show only ${\mathrm{Fe}}^{2+}$ when determined by either the isomer shift or the hyperfine splitting below the N\'eel temperature. We find a N\'eel temperature of 288\ifmmode^\circ\else\textdegree\fi{}K, which is 3\ifmmode^\circ\else\textdegree\fi{}K less than the value reported by earlier workers. CoO(I) is completely inert in air at room temperature, and its stoichiometry does not change over time intervals as long as two months. CoO(II) could be prepared at low temperatures only. Form II also had the NaCl structure, but the x-ray line of the powder pattern was very broad. The density of CoO(II) is 3.2 (+1.0, -0.0) g/${\mathrm{cm}}^{3}$, about half of the value found for CoO(I). CoO(II) shows only ${\mathrm{Fe}}^{3+}$ lines in the M\"ossbauer pattern, with the appropriate isomer shift and hyperfine field. CoO(II) is very reactive with oxygen, at room temperature, although the x-ray pattern is not appreciably altered after oxygen absorption. CoO(II) has a N\'eel temperature of 270\ifmmode\pm\else\textpm\fi{}10\ifmmode^\circ\else\textdegree\fi{}K, slightly less than that of CoO(I), and an effective M\"ossbauer characteristic temperature of 320\ifmmode^\circ\else\textdegree\fi{}K, substantially less than the 510\ifmmode^\circ\else\textdegree\fi{}K value found for CoO(I). The results of studies of thermally activited transformation of II\ensuremath{\rightarrow}I and pressure and quenching experiments are given. CoO(I, II), a mixture of forms I and II, is obtained by the usual chemical recipes. This mixed material was also studied. At low temperature, CoO(I, II) behaves as a simple mixture of I and II, but at temperatures above 200\ifmmode^\circ\else\textdegree\fi{}K there is an interplay between the two forms which is responsible for the observed increase in the relative fraction of ${\mathrm{Fe}}^{3+}$ to ${\mathrm{Fe}}^{2+}$ observed in the M\"ossbauer patterns. This interplay is also connected with the shift in the apparent N\'eel temperature for ${\mathrm{Fe}}^{3+}$ in pure form II compared with the value obtained for CoO(I, II). A detailed model is proposed which explains all of the properties observed thus far. The data preclude the possibility of Auger after-effects in cobaltous oxide, as suggested by Wertheim, and they also show that stoichiometry fluctuations are not the principal cause of the observed ${\mathrm{Fe}}^{3+}$ M\"ossbauer line, as was recently claimed by Triftsh\"auer and Craig.

Mössbauer Spectra in a Fluctuating Environment II. Randomly Varying Electric Field Gradients

We derive an expression for the M\"ossbauer line shape in the presence of an electric field gradient which jumps at random between the $x$, $y$, and $z$ axes. This Hamiltonian represents an idealized model for the effects on a nucleus of Jahn-Teller distortions, jump diffusion of vacancies, or electronic relaxation. A simplified calculation based on a model in which the field gradient jumps between positive and negative values along the $z$ axis is also given. In certain limiting circumstances the two calculations give similar results: The M\"ossbauer line shape consists of a single unsplit line for fast jumping, and of a quadrupole doublet for slow transitions. The results of the calculation agree with experiments of Pipkorn and Leider and of Chappert, Frankel, and Blum, as interpreted by Ham.

Mössbauer Spectra in a Fluctuating Environment

The M\"ossbauer line shape in the presence of time-dependent electric field gradients and magnetic fields is considered. Two specific soluble stochastic models are treated: (1) a static electric field gradient with a randomly fluctuating magnetic field which takes on values $+h$ and $\ensuremath{-}h$, each directed along the axis of the field gradient, and (2) as in (1), but with the fluctuating magnetic field perpendicular to the axis of the field gradient. Example (2) is more complex than (1), since the fluctuating field is in this case capable of inducing transitions between the nuclear levels, while in (1) this is not possible. Specific calculations for the two cases illustrate the differences between them.

Simple Binary Collision Model for Van Hove'sGs(r, t)

It is noted that the Van Hove self-correlation function ${G}_{s}(r, t)$ for a dilute fluid is determined by a linearized Boltzmann equation identical to that occuring in the theory of neutron diffusion. A simple model for the collision integral, in which a molecule emerges from a collision with a Maxwellian distribution, allows some interesting analytic results to be obtained. The double Fourier transform ${S}_{s}(\ensuremath{\kappa}, \ensuremath{\omega})$, of ${G}_{s}(r, t)$ is expressible in terms of the probability integral for complex arguments. Since ${S}_{s}(\ensuremath{\kappa}, \ensuremath{\omega})$ is directly proportional to the M\"ossbauer line shape, the transition of the line shape from Doppler broadening to simple diffusion broadening is explicitly exhibited as a function of momentum transfer. The spatial moments of ${G}_{s}(r, t)$ are calculated as a function of time. The model used gives the same mean-square displacement as does the Langevin equation. The resulting ${G}_{s}(r, t)$ is not, however, Gaussian as is shown by the mean fourth power of the displacement which is 25% greater at intermediate times than would be predicted by the Langevin equation. The non-Gaussian effects lead to an appreciable narrowing of ${S}_{s}(\ensuremath{\kappa}, \ensuremath{\omega})$ for intermediate values of $\ensuremath{\kappa}$.