Fig 1.
Types of convolution considered in this study.
The pseudo-Voigt profile (not shown) is a linear combination of Gaussian and Lorentzian profiles.
Fig 2.
Screenshots from PACE, PACE-GD and PACE-IC.
(a) shows the start screen, from which the others are accessed. (b) is the main screen from PACE, following fitting and deconvolution of a random profile. The top right hand panel shows the initial fit (blue line) to the data (points), then the lower panel shows the deconvoluted (solid red line) and convoluted (dashed black line) fits, incorporating a Gaussian beam with a 15 μm full width at half-maximum (c) shows PACE-GD following extraction of a beam size from a Gaussian convoluted profile. (d) shows PACE-IC following fitting and deconvolution of a profile with a stepped initial condition, and a 15 μm wide square/rectangular beam. The top panel shows the initial fit using the imported initial condition (red dashed line), then the deconvoluted profile, as in (b).
Fig 3.
Method for convoluting diffusion profiles, described in the text.
Fig 4.
The effect of EPMA accelerating voltage on a step function measured over a Fo85-Fo90 couple, simulated using the Monte Carlo method implemented in CASINO (20).
(a) profiles of Fe Kα (counts, normalised) as a function of accelerating voltage, showing an increase in width, but also a change in shape, as the kV increases. (b) the profiles from (a), deconvoluted using PACE-GD to give the FWHM as a function of kV. This was done both assuming a purely Gaussian interaction (circles) as well as a pseudo-Voigt interaction (squares). The FWHM of the pseudo-Voigt line is shown (V), as well as those of the Gaussian (V(G)) and Lorentzian (V(L)) components comprising the Voigt line. Lines (dashed, dotted and solid) represent increasing intervals of 1 kV.
Fig 5.
Deconvoluting Ti profiles in quartz determined by SEM-CL.
Images were acquired using a Zeiss Gemini 1530 field emission gun scanning electron microscope equipped with an ellipsoidal mirror and an ASK SEM-CL View VIS (250–900 nm) imaging spectrometer. The SEM was operated at 7 kV, 10 nA with a working distance of 14.2 mm. (a) CL image of an un-annealed high Ti quartz-low Ti quartz couple, with an extracted greyscale profile in (c). (b), same sample as in (a), after annealing at 1600°C, 20 kbar for 89.5 h, showing a wider transition zone, with the extracted profile in (d). (c) assuming that the gradient in (a) is purely convolution (i.e. (a) is a step function) and the beam-sample interaction is Gaussian, the FWHM is determined by fitting a curve with the form erf(x/(√2σ)) to the data, giving FWHM = 322 nm. (d) extracted profile from (b), deconvoluted using PACE and the FWHM from (c). The ‘Ti-rich’ section contains ~3000 wt. ppm Ti, versus <0.2 wt ppm for the ‘Ti-poor’ section.
Fig 6.
Data from Gualda et al [42], fitted and deconvoluted using PACE, assuming a 5 μm FWHM Gaussian beam.
The measured Dt (D*t) is just 0.06 log units greater than the true Dt (Dt), i.e. convolution has almost no effect on the measured profile.
Fig 7.
An experimental Cr diffusion profile in forsterite, measured using LA-ICP-MS with a 6x100 μm, deconvoluted.
The measured D (D*) is 0.11 log units higher than the deconvoluted D. Note the deviation between data and model in the near-interface region—this is due to the way that PACE treats boundaries (fixed composition equal to the rim composition).
Fig 8.
Deconvoluting a Pb in zircon profile from [48].
(a) shows an apparent Pb Lα profile between Pb-doped and Pb-free zircon, generated by CASINO [20], using an incident beam at 25 kV with a nominal 2 μm diameter. From the fit, σ = 693 nm, hence FWHM = 1633 nm. However, because [48] analysed a section 30° from normal to the interface, the relevant FWHM is 1633/~2 ≈ 816 nm. (b) shows the [48] EPMA profile, deconvoluted. Convolution has an effect of <0.01 log10D units in this case.