Supplementary Publication.

Resolution and Boundary Recognition.

Both with optical transmission microscopy and the scanning technique adopted for the present work, the light passes through the thickness of the specimen slide and so can be influenced by the intersection boundaries of the component grains at both surfaces of the thin-section. The resulting uncertainty, often known as the “Holmes” effect, has been reviewed (as referenced in the body of our paper) with respect to the effects of “over-projection” on the results of modal analyses. It has been shown that these effects depend on the ratio of the “grain size” to the thickness of the thin-section.
Similarly a separation of the two intersection boundaries, as seen normal to the thickness of the thin-section, could lead to a ‘blurring’ of the overall grain section outlines with a consequent effect on the grain-section skeleton produces to represent the two-dimensional texture of the specimen. Some estimate of the effect can be obtained by examining the geometry for the sectioning of spherical particles, both in its most symmetrical formulation and its limiting cases.

Geometric parameters - General formulation

  
  sin θ1 = (d - ½ T)/R : sin θ2 = (d + ½ T)/R : cos θ1 = x1/R : cos θ2 = x2/R

The separation of the intersections of the particle with the two surfaces of the thin-section, s = x1 - x2, and for the particle to intersect with both surfaces:-     0 ≤ d  ≤ (R - ½ T)

Then s = R cos θ1 - R cos θ2 = R{cos θ1 - cos θ2} and
    since  cos θ = (1 - sin2 θ) ½ then, s = R{(1 - sin2 θ1) ½ - (1 - sin2 θ2) ½}

          s = R{(1 - [(d - ½ T)/R]2) ½ - (1 - [(d + ½ T)/R]2) ½} ----- (1)

From this general equation (1) it can be seen that the separation of the intersections of a spherical particle with the two surfaces of the thin-section is influenced by both the size of the particle (R) and the displacement (d) of the particle relative to thickness of the thin-section.

When d → ½ T, the expression (1) reduces to the simplest symmetrical case to evaluate for a spherical particle.

i.e.  s = R{(1 - (1 - [T/R]2) ½} = R{(1 - (1 - sin2 θ2) ½}

When d = 0 (i.e. the particle is centred at the mid-point of the thin-section thickness:-    s = R{(1 - [- ½ T/R]2) ½ - (1 - [+ ½ T/R]2) ½} = 0

i.e. The particle intersects the two opposite thin-section surfaces immediately one above the other and there is no separation between them in the sense of the light passing normally through the thin-section.
Note however that this does not address the situation of a significantly opaque particle, where the buried diameter could affect the passage of the light and make a contribution to the apparent thickness of the observed boundary. The larger the particle the smaller will be the contribution to the apparent boundary width from this source. The maximum affect will be the limiting case when 2R = T. The particle just fits within the thickness of the thin-section and the distance between buried diameter and the point intersection of the particle with the thin-section surface would be ½ T (i.e. 0.015 mm). Whether with such a small particle this would be observed or across the width of the whole buried diameter (0.03mm), might depend on the optical properties of the particle but it does not seem likely that these limiting situations would have much affect on the overall assessment of the thin-section.

When d → (R - ½ T), the limiting case of intersection with both boundaries, s is a maximum for the give particle size.

	

Whilst the larger the particle in general, the greater the separation in this limiting (or approaching limiting) case, the smaller the particle the more likely the limiting case is to occur. Bigger particle could be expected to provide a greater number of standard thin-sections with d less than (R - ½ T).
	The challenge is to evaluate, from the general equation (1), the influence of the geometric parameters between the limiting cases, in order assess the possible effects on grain boundary recognition uncertainty. One approach (a spread sheet computation solution) is to calculate boundary separations (s) at different particle sizes (R) for a series of displacement values (d) in relation to the thickness of the standard thin-section (0.03mm). These values, plotted against the radius R (Figure attached) demonstrate increasing tendency for boundary separation and uncertainty as R approached T. Even once the particle diameter starts to be accommodated within the thickness of the thin-section, the effect only falls off markedly as d approached closely to zero (¼ & ⅛ T).

Geometric parameters Approximate solution and Frequency of Occurrence.

From the ‘general formulation’
          s = R{(1 - [(d - ½ T)/R]2) ½ - (1 - [(d + ½ T)/R]2) ½} ----- (1)

for [(d - ½ T)/R]2 and [(d + ½ T)/R]2 ≤ 1, then expanding as Taylor’s series and collecting only the first terms of the polynomial:-

          s ≈ R{(1 - ½ [(d - ½ T)/R]2)  - (1 - ½ [(d + ½ T)/R]2) }
	  
	   ≈ R/2{[(d + ½ T)/R]2 -  [(d - ½ T)/R]2}
	  
	   ≈ R/2{[(d2 + dT + ¼T2 - d2 + dT - ¼T2)/R]2} = dT/R ----- (2)

i.e. Since T (the thickness of a thin-section) is a constant (0.03 mm) then s is proportional to the ratio d/R, as implied in the general formulation (1). However in order to assess even this approximate solution it is still necessary to evaluate this ratio in terms of particle - thin-section displacement for d = k (R - ½ T) where 0 ≤ k ≤ 1 (Figure attached). 
Once again, a consequence of the limiting value of d = (R - ½ T) is that the larger the particle, the greater the potential to give rise to a larger boundary separation. However, the smaller the particle the more likely it is to approach the limiting condition, whilst the larger particle will give rise to more particle sections that do not. Even without considering the full mathematics of random sectioning, it can be seen from the simple example of successive cuts each of T thickness, a particle of 0.09mm diameter could provide three sections, two of which would be of the limiting orientation, whilst a particle of 0.6mm diameter would provide twenty, with still only two at the limit. With larger particles it is only the sections near to or at the limiting condition that show significant boundary separations (Figure attached). Furthermore the larger particles will provide more sections of larger area so that they will be fewer in number in a given area and the limiting condition will be consequently again be less likely to be observed.
Although one can only examining the simple symmetrical case of spherical particle in this way, it does give support to the view that, as a consequence of the scanning beam passing through the thickness of the thin-section, there is a practical lower limit to the measurement of particle section areas. Increasing magnification beyond a limit does not give rise to greater resolution and discrimination.