Examples of lognormal distributions. Left hand plot shows the probability density function for the number of spheres [see equation 5], and right hand plot shows the same multiplied by the cube of the sphere diameter. Curves have (right to left) (the delta-function spike), 1.05, 1.1, 1.2 and 1.4. These values correspond to σ = 0, 0.22, 0.31, 0.43 and 0.58, respectively.
Confidence intervals for predictions (estimates) of sphere size parameters from circular sections, compared to the known exact values. These data are for a computer generated random distribution of monodisperse spheres, and show that the estimates converge to the correct values as the number of sampled circles increases.
Relative errors in predictions of sphere size parameters from circular sections. The data are from computer generated random distributions of lognormal sphere sizes, with log widths (bottom to top in each plot) σ = 0, 0.22, 0.31, 0.43 and 0.58 (as for Fig. 1), which correspond to 1, 1.05, 1.1, 1.2 and 1.4, respectively. The estimates converge to the correct values as the number of sampled circles increases.
Relative errors in predictions of sphere size parameters from circular sections. The data are from the same procedure as Fig. 3, but circles with diameters smaller than a fraction α of have been omitted from the statistics. The plots show the cases (the same as Fig. 3), 0.1, 0.2 and 0.5.
Relative errors in predictions of cube size parameters for randomly positioned and randomly oriented cubes in space. The equivalent spheres and equivalent circles for polygonal cross sections are calculated, and the same procedure is used as in Fig. 3.
Left: photomicrograph (plane-polarized light) of sample SC459 from the Shiant Isles Main Sill, containing loose clusters and chains of olivine phenocrysts set in a matrix of interstitial plagioclase and augite. Note the rare grains of chromite (opaque) that were also part of the crystal load of the incoming magma. Width of image is 4.5 mm. Right: same image segmented by hand. We measured the area of each olivine crystal and determined the diameter of the circle with the same area.
Back-scatter electron image of experimental charge SI-5 (see text for details), showing plagioclase (plag), pyroxene (px) and oxide grains (ox) set in a silica-rich glass containing droplets of an immiscible Fe-rich liquid (now quenched to glass). Most of the droplets appear to have nucleated homogeneously, but several droplets attached to the plagioclase grain at the far right probably nucleated heterogeneously on the plagioclase surface. The scale bar is 20 µm long.
Statistics for equivalent diameters of a total of olivine grain cross-sections from sample SC459 from the Shiant Isles Main Sill.
The smallest and largest equivalent diameters in the sample are and .
Statistics on emulsion droplet size distribution in immiscible basaltic glasses from the experimental charges of Charlier and Grove (2012).
Final T (°C)
Equilibration time (hours)
3.94 ± 1.54
2.22 ± 0.87
1.77 ± 0.69
2.26 ± 0.30
1.59 ± 0.21
1.42 ± 0.19
1.36 ± 0.04
1.28 ± 0.04
1.06 ± 0.03
2.42 ± 0.10
1.80 ± 0.07
1.35 ± 0.05
1.85 ± 0.11
1.51 ± 0.09
1.23 ± 0.07
2.22 ± 0.09
1.75 ± 0.07
1.27 ± 0.05
The number of droplets measured is . The equilibration time gives the time for which the charge was held at the final temperature after having been cooled from a starting temperature of 1100°C at 1°C/hr. Charges SI-5, SI-8 and SI-13 have a bulk composition identical to that of a dyke cutting the Sept Iles intrusion and charges M-5, M-6 and M-9 have a bulk composition typical of an intermediate basalt from the Mull Tertiary volcano (for further details see Charlier and Grove (2012)).