Particle diameter, signal-to-noise ratio and beam requirements for extended Rayleigh resolution measurements in the scanning electron microscope

The extended Rayleigh resolution measure was introduced to give a generalized resolution measure that can be readily applied to imaging and resolving particles that have finite size. Here, we make a detailed analysis of the influence of the particle size on this resolution measure. We apply this to scanning electron microscopy, under simple assumption of a Gaussian electron beam intensity distribution and a directly proportional emitted signal yield without detailed consideration of scattering internal to the sample, other than being proportional to the sample thickness. From this, we produce beam-width normalized characteristics relating the particle diameter and resolution measure, while also taking consideration of the reduced signal yield that occurs from smaller particles. From our analysis of these characteristics, which we fit to experimental image data, we see that particle diameters <0.7 times the beam 1/e full width, d, give agreement better than 10% with the true extended Rayleigh resolution. Furthermore, we consider the signal current that must be collected to reliably distinguish between the mid-gap and peak intensity regions in the particle images. This leads to a practical guide that the signal-to-noise ratio (SNR) occurring between large area, continuous regions made of the same materials as the particle and background should typically be 10–30 times greater than the SNR that is desired to be achieved between the peak and mid-gap regions of just resolved adjacent identical particles having diameters in the size range 0.4–0.7d.

Generalized resolution matrix for neutron spin-echo three-axis spectrometers

This article describes the energy resolution of spin-echo three-axis spectrometers (SE-TASs) by a compact matrix formalism. SE-TASs allow one to measure the line widths of elementary excitations in crystals, such as phonons and magnons, with an energy resolution in the µeV range. The resolution matrices derived here generalize prior work: (i) the formalism works for all crystal structures; (ii) spectrometer detuning effects are included; these arise typically from inaccurate knowledge of the excitation energy and group velocity; (iii) components of the gradient vector of the dispersion surface dω/dq perpendicular to the scattering plane are properly treated; (iv) the curvature of the dispersion surface is easily calculated in reciprocal units; (v) the formalism permits analysis of spin-echo signals resulting from multiple excitation modes within the three-axis spectrometer resolution ellipsoid.