Carlo Palmisano

Vectorial ray-based diffraction integral

Laser interferometry, as applied in cutting-edge length and displacement metrology, requires detailed analysis of systematic effects due to diffraction, which may affect the measurement uncertainty. When the measurements aim at subnanometer accuracy levels, it is possible that the description of interferometer operation by paraxial and scalar approximations is not sufficient. Therefore, in this paper, we place emphasis on models based on nonparaxial vector beams. We address this challenge by proposing a method that uses the Huygens integral to propagate the electromagnetic fields and ray tracing to achieve numerical computability. Toy models are used to test the methods accuracy. Finally, we recalculate the diffraction correction for an interferometer, which was recently investigated by paraxial methods.

Convective forces in high precision mass measurements

This paper deals with a numerical simulation of the natural convection air flow and its influence on apparent mass in high precision mass measurements. Results are in reasonable agreement with the experimental values obtained on a 1 kg silicon sphere and a 1 kg mass standard. There is clear evidence that the three-dimensional finite element model studied, allowing inclusions of the boundary and initial conditions determined experimentally, captures the most important features of convection and estimates the relevant corrections.

Interval estimations in metrology

This paper investigates interval estimation for a measurand that is known to be positive. Both the Neyman and Bayesian procedures are considered and the difference between the two, not always perceived, is discussed in detail. A solution is proposed to a paradox originated by the frequentist assessment of the long-run success rate of Bayesian intervals.

Influence of surface stress in the determination of the (2 2 0) lattice spacing of silicon

We have examined the influence of surface stress in the measurement of the (2 2 0) lattice plane spacing of a Si crystal by combined x-ray and optical interferometry. By means of a finite element simulation we have obtained first a quantitative description of various non-linear displacement fields of the diffracting planes; the simulation indicates that surface stress does not cause a significant difference between bulk and surface lattice spacing. Subsequently, to quantify the effect of the above fields on the period of the interference fringes, we have solved numerically the perturbed Takagi–Taupin equations; these results indicate that the fringe period is determined by the atom spacing on the crystal entrance surface, whereas, intuitively, we might have expected this period to be a kind of average of the (2 2 0) lattice plane spacing. To corroborate our numerical calculations, we have found analytical expressions which allow us to isolate the contribution of the non-linear superficial terms on the measurement of the plane spacing.

X-ray and γ-ray propagation in bent crystals with flat and cylindrical surfaces

In this paper, X-ray and gamma-ray propagation in crystals having a constant strain gradient and flat or cylindrical surfaces is investigated. When a displacement field is present, the Takagi-Taupin equations are solved either by the Riemann-Green method or by a numerical method. The results are applied to study the operation of a double-crystal Laue-Laue diffractometer having a flat collimating crystal followed by a bent analyzer crystal. In particular, the effect of the analyzer strain on the location of the diffraction peaks in the dispersive and non-dispersive set-up is examined, thus confirming the previously reported peak location as being set only by the diffracting-plane spacing on the analyzer entrance surface.

Two-wave approximation in surface effects in asymmetric Laue crystals

This paper is a study of surface effects, e.g. roughness or asymmetrical cut, in the Laue diffraction of X-rays by crystals, based on the Takagi-Taupin equations. By means of Riemann-Green integrals, first a formal solution has been obtained when the entrance and the exit surfaces are arbitrary. Then a coordinate transformation mapping a propagation domain with arbitrary boundaries into a rectangular domain with straight boundaries is given. Potential measurement errors in \gamma-ray wavelength and silicon lattice-parameter measurements by double-crystal diffractometry and X-ray interferometry, respectively, are outlined and anticipated by studying, in the two-wave approximation, the reflection peak shift and extra phase originating from an asymmetrically cut crystal. A relationship between analyser displacement, interferometer-signal phase and relative uncertainty in lattice-parameter measurement is also given.

Effects of analyser deformation in scanning x-ray interferometry

Measurement of the (220) Bragg-plane spacing of Si using combined x-ray and optical interferometry with an uncertainty lower than 10−8 requires the study of influence factors not considered in the past. In this paper the Bragg-plane spacing definition itself is re-examined and the influence of a constant strain gradient is studied by means of Takagis equations. Their analytical and numerical solutions indicate that, contrary to our previous assumption, the period of the travelling fringes is affected by the Si atom spacing at the analyser entrance surface.

Lattice strain effects in the measurement of the Si lattice parameter by Laue‐case double‐crystal diffractometry

The measurement of the (220) Bragg-plane spacing of Si by Laue-case double-crystal diffractometry with an uncertainty lower than 10 -8 requires a detailed theoretical framework that includes the study of lattice strain. In the present paper, the propagation of X-rays through a deformed crystal is re-examined and the influence of a constant strain gradient on the centre of the reflection domain is studied by means of Takagis equations. Their analytical and numerical solutions indicate that the measured lattice spacing refers to the crystal entrance surface.

A Fourier optics approach to the dynamical theory of X-ray diffraction--continuously deformed crystals.

X-ray diffraction in continuously deformed crystals is considered by application of Fourier optics and from the viewpoint of the analogy between X-ray dynamics and the motion of two-level systems in quantum mechanics. Different forms of Takagis equations are traced back to a common framework and it is shown that they are different ways to represent the same propagation equation. A novel way to solve Takagis equations in the presence of a constant strain gradient is presented and approximation methods derived from quantum mechanics are considered. Crystal deformation in X-ray interferometry and two-crystal spectrometry are discussed and it is demonstrated that Si lattice-parameter measurements depend on the diffracting plane spacing on the crystal surface.

The ellipsoidal nested sampling and the expression of the model uncertainty in measurements

The measurand value, the conclusions, and the decisions inferred from measurements may depend on the models used to explain and to analyze the results. In this paper, the problems of identifying the most appropriate model and of assessing the model contribution to the uncertainty are formulated and solved in terms of Bayesian model selection and model averaging. The computational cost of this approach increases with the dimensionality of the problem. Therefore, a numerical strategy to integrate over the nuisance parameters and to compute and to sample the measurand post-data distribution is also outlined.