Attenuation length

Abstract: We examine substrate/overlayer experiments and the equations commonly used to quantify overlayer thicknesses. Comparisons with accurate Monte-Carlo simulations show that using attenuation lengths (rather than inelastic mean free paths) eliminates most of the error due to elastic scattering without increasing the complexity of the quantification. We give attenutation lengths for 27 elements, calculated by the criterion that systematic errors in such quantifications should be minimized. These are therefore the best attenuation length values to use in layerwise quantification. We show that, provided these attenuation length values are used, the error in estimation of the thickness of an overlayer due to elastic scattering can be limited to +(5% +1 A) for an emission angle ≤58° from the surface normal, and +(10%a + 1 A) for an emission angle ≤63° from the surface normal. This accuracy is acceptable for most analytical work. Other methods (such as analytical transport theory) are much more complicated, and achieve a high precision that is often unnecessary in view of other uncertainties typically present in these experiments (such as errors due to surface morphology and diffraction effects). The results presented here, using the full theory, show that the analyst's simple straight-line approximation is in fact of adequate accuracy, provided that the correct values of attenuation length are used. Simple semi-empirical equations are presented, which allow the analyst to estimate the attenuation length for electrons of kinetic energy between 50 and 2000 eV, to a standard uncertainty of 6%.

Abstract: The energetics and length scales associated with the interaction between point defects (vacancies and self-interstitial atoms) and grain boundaries in bcc Fe was explored. Molecular statics simulations were used to generate a grain boundary structure database that contained $\ensuremath{\approx}$170 grain boundaries with varying tilt and twist character. Then, vacancy and self-interstitial atom formation energies were calculated at all potential grain boundary sites within 15 \AA{} of the boundary. The present results provide detailed information about the interaction energies of vacancies and self-interstitial atoms with symmetric tilt grain boundaries in iron and the length scales involved with absorption of these point defects by grain boundaries. Both low- and high-angle grain boundaries were effective sinks for point defects, with a few low-$\ensuremath{\Sigma}$ grain boundaries (e.g., the $\ensuremath{\Sigma}3$${112}$ twin boundary) that have properties different from the rest. The formation energies depend on both the local atomic structure and the distance from the boundary center. Additionally, the effect of grain boundary energy, disorientation angle, and $\ensuremath{\Sigma}$ designation on the boundary sink strength was explored; the strongest correlation occurred between the grain boundary energy and the mean point defect formation energies. Based on point defect binding energies, interstitials have $\ensuremath{\approx}$80$%$ more grain boundary sites per area and $\ensuremath{\approx}$300$%$ greater site strength than vacancies. Last, the absorption length scale of point defects by grain boundaries is over a full lattice unit larger for interstitials than for vacancies (mean of 6--7 \AA{} versus 10--11 \AA{} for vacancies and interstitials, respectively).

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Abstract: SUMMARY We perform a theoretical study of the effect of fine layering on the compressional-wave velocities and attenuation coefficients in fluid-saturated rocks. This effect in a permeable rock differs from that in a purely elastic solid because of the local flow of the pore fluid across the interfaces, which is caused by the passing wave. For analytical calculations. Biot theory is applied to non-homogeneous (randomly and periodically layered) porous media leading to Biot's equations with variable coefficients. By analysing these equations with the help of a statistical perturbation technique we obtain the velocity and normalized attenuation 1/Q of the fast compressional wave as a function of frequency f. Both attenuation and velocity dispersion are found to obtain their maximum values near some frequency f0, at which the Biot slow-wave attenuation length equals the mean inhomogeneity size (mean layer thickness or characteristic length). In the low-frequency limit, 1/Q is proportional to f1/2 for random and to f for periodic layering. At frequencies higher than f0, attenuation decreases with increasing frequency as f−1/2, regardless of the particular type of layering. The results for periodic layering are in a good agreement with recently published exact results. The results for the more realistic case of random layering with exponential correlation reveal more gradual changes of velocity and attenuation versus frequency than those for a periodically layered medium.