American Mineralogist
 © 2008 American Mineralogist
Abstract
The threedimensional shapes of plagioclase crystals in an experimentally cooled basaltic liquid have been reconstructed, with the aim of (1) better understanding crystal growth processes and the diversity of crystal shapes produced during cooling, and (2) to assess the validity of crystalsize distributions (CSDs) derived from 2D sections. The experimental charge was cooled from above the liquidus (~1175 °C) at a rate of 0.2 °C/h. It contained ~40% crystals at the quenching temperature of ~1120 °C. To quantify the crystals in 3D, photographs of a series of 2Dpolished sections were taken under an optical microscope using reflected light. Interpolation and 3D reconstruction of 261 individual crystals was performed using the gOcad geomodeling software, and their short (S), intermediate (I), and long (L) dimensions were measured. Plagioclase crystals are generally tabular, with a nearly constant I/L ratio. On the other hand, S/I and S/L shape factors are more variable, although both are found to be correlated with length of the S axis. These observations are believed to result either from crystal agglomeration and attachment, preferentially along (010) faces, or from varying thermodynamic or kinetic conditions during cooling. Growth rates along the S, I, and L axes have been calculated from the size of the largest crystals and vary from 1.5 × 10^{−10} to 5.1 × 10^{−10} and 7.2 × 10^{−10} m/s, respectively. The CSDs for the maximal length and short axes of 3D crystals are presented and compared with CSDs obtained from 2D sections. Published corrections for cutting effects are found to be generally very satisfactory.
Introduction
The use of crystalsize distribution (CSD) as a tool providing insight into the crystallization conditions of magmatic and metamorphic rocks has been advocated and promoted by several authors, most notably Marsh (1988) and Cashman and Marsh (1988). CSDs have the potential to provide quantitative constraints on the cooling history of a given rock because they are a function of nucleation and growth rates, these latter parameters both being temperature dependent. The relationship between CSD and thermal history has been used to estimate nucleation and growth rates from natural samples for which the cooling history is independently known (Cashman and Marsh 1988) and to constrain cooling history, assuming values for nucleation and growth rates (Garrido et al. 2001; Lentz and McSween 2000). Crystalsize distribution can be also used to assess the importance of other magmatic processes such as crystal settling or textural coarsening (Higgins 2002; Marsh 1998).
Crystalsize distribution is usually expressed as n_{v}(x), defined as the number of crystals per unit volume and per size interval Δx (i.e., for sizes in the range x to x + Δx). Crystalsize distributions are with few exceptions (e.g., Bindeman 2003; Castro et al. 2003; Mock and Jerram 2005) measured on 2D sections. The retrieval of the three dimensional CSD from rock sections generally relies on the following assumptions: (1) that the section is strictly two dimensional, which is the case for images obtained using the scanning electron microscope (SEM) or an optical microscope using reflected light, but not when using transmitted light (for which a 30 μm thick layer is observed when studying a standard petrographic thin section), and (2) that the crystal shape (aspect ratios) is constant and known.
If these conditions are fulfilled, then it is in principle possible to correct the 2DCSD for the cutting effect. In detail, two effects have to be taken into account. First, because the section plane cuts crystals at a random orientation and position, the measured 2D dimensions of a given crystal do not correspond to the true dimensions of the crystal (this is the socalled cutting effect). Second, the probability of cutting small crystals is smaller than that of cutting large crystals (this is the socalled probability of intersection effect). The stereological problem is fully solved mathematically for the distribution of solids of simple shapes, but for most crystal shapes, correction procedures proposed in the literature [from Wager (1961) to more recent methods from Peterson (1996), as summarized by Higgins (2000)] only give approximate solutions and all of the proposed methods fail to accurately reproduce the 3DCSDs for small size classes (Castro et al. 2003). Higgins (2000) has proposed a mathematical solution based on the Saltikov method (Saltikov 1967) and has developed a computer program (CSDCorrections) for the estimation of the size distribution of the largest dimension of 3D crystals. This computer program supersedes to some extent simplified correction methods based on intersection probability alone (e.g., Higgins 1994). On the other hand, the correction proposed by Higgins (1994) remains of potential interest as it may be used to compare the CSDs of the smallest 3D axis with size distributions derived from 2D measurements (as recently applied by Pupier et al. 2008). Although it is true that the distinction between largest and smallest dimensions is redundant for crystal populations of constant aspect ratio, comparison of various correction procedures becomes important for the case of a crystal population containing grains of variable aspect ratio, as is found to be the case here.
For each size class, the number of crystals per unit volume n_{v}(x) is obtained by dividing the number of crystals per unit surface area n_{a}(x) by a function of the length and aspect ratio. By numerical simulation of the probability of intersection effect, Higgins (1994) determined that
(1)
where S, I, and L are the short, intermediate and long axes of the parallelepipeds. Therefore, rigorous justification for the application of Equation 1 requires that the 3D aspect ratio is known and constant.
In this work, we present a reconstruction of the 3D shape of plagioclase in a synthetic basalt from successive serial sections with the help of the 3Dgeomodeller gOcad (Mallet 2002), as described below. This technique has already been used by Mock and Jerram (2005) to reconstruct the geometry of crystals in a natural rhyolitic sample, to quantify the 3D shape of phenocrysts, and to assess the validity of 3DCSD retrieval from 2D sections. In the study of Mock and Jerram (2005), the dimensions of the 3D crystals were defined making no a priori assumptions about crystal morphology. To some extent, this choice was made necessary by the fact that crystals in this rhyolitic sample have postcumulus, spacefilling, overgrowths, resulting in irregular shapes at crystal edges. The three major axes of each crystal were therefore calculated from the best fitting ellipse through each crystal polyhedron. On the other hand, crystal length was measured as the longest distance inside each crystal and these values were used to compute CSDs. Mock and Jerram (2005) have shown that crystal habit varies greatly in the sample they studied but that nevertheless, CSDs can be confidently retrieved from random 2D sections as long as the number of measured crystals exceeds 200. In the present work, we have chosen to study an experimentally cooled sample containing plagioclase crystals dispersed in a glassy matrix (Pupier et al. 2008). In this case, the glass content is high (60%) and postcumulus overgrowths are not present, so that a relatively simple 3D polyhedron may be drawn for each crystal. The 3D shapes are typical of known plagioclase morphology, and we can therefore directly measure the length of each crystal edge. We will discuss the measurements of crystal shape in terms of plagioclase crystallography, agglomeration processes, crystal growth rates, and discuss the influence of shape variability on CSD measurements. The 2D length and width measurements on polished sections are in an online data set.^{1}
3D reconstruction technique
A synthetic ferrobasaltic glass was prepared, of composition: SiO_{2} = 48.8, Al_{2}O_{3} = 14.9, FeO = 13.1, MgO = 6.5, CaO = 10.9, Na_{2}O = 2.7, and K_{2}O = 0.3 wt%. This composition is that of a basaltic dike (dike C) spatially associated with the Skaergaard intrusion, Greenland, and proposed as a possible parental magma (Brooks and Nielsen 1978; Toplis and Carroll 1995). The glass was ground to a powder, and then attached to a platinum wire loop, using polyvinyl alcohol as a binder. The platinum wire is 0.2 mm in diameter, and the loop ~3 mm across. Once molten, the experimental charge is more or less a sphere (~14 mm^{3} in volume), held on the wire loop by surface tension. This charge was introduced into a 1 atmosphere vertical dropquench furnace under controlled oxygen fugacity (Δ_{FMQ} = 0). It was initially held for 9 h at a superliquidus temperature of 1186 °C to achieve complete melting and to ensure redox equilibrium of the sample with the CO/CO_{2} gas mixture. The temperature of the furnace was then lowered at a rate of 0.2 °C/h until a final temperature of 1119 °C. Comparison with the experimental results of Toplis and Carroll (1995) shows that plagioclase is the expected liquidus phase (at a temperature of ~1175 °C) and that at ~1120 °C, 60% liquid should remain, plagioclase being the dominant mineral (25%) with less abundant olivine (8.5%) and clinopyroxene (6.5%). This proportion of liquid is high enough to ensure a good approach to equilibrium, while at the same time ensuring several plagioclase crystals sufficient for determination of the CSD.
After quenching, the sphere was mounted in epoxy in a 2.54 cm diameter aluminum holder and abraded to create a flat surface. Four holes with a diameter of 0.1 mm were drilled perpendicular to the surface around the sample as a reference frame. From this initial reference state, the sample was then sequentially polished. The spacing between successive sections was measured with an electronic micrometer and is typically 20 μm. A total depth of ~250 μm was abraded, such that ~9% of the bulk sphere could be reconstructed in 3D. The difference in thickness of the mount before and after each serial polish was measured not only on the sample itself but also at three other points on the mount, to assess and control tilting of the surface. Once polished, photographs were taken of each section under a binocular microscope in reflected light. The successive images were superposed and their position referenced in the gOcad software (Mallet 2002) (Fig. 1⇓).
The threedimensional shapes of crystals were constructed by hand one by one for 261 crystals. Each crystal was constrained to fit the position and 2Dshape of visible crystals over successive images and to respect an idealized crystal morphology. Although plagioclase feldspar can present skeletal, dendritic, or spherulitic morphology when formed under conditions of high supercooling, plagioclase morphology under conditions of low to moderate undercooling is most frequently acicular to tabular (Lofgren 1974; Smith and Brown 1988). Plagioclase morphology has been drawn (Fig. 2⇓) for the relevant plagioclase composition (labradorite) with WinXMorph software (2004–2005 Werner Kaminsky, University of Washington). α and γ angles are close to 90°, but the β angle is ca. 116°. The distance between the faces has been taken as the mean distance observed in the 3D reconstruction. Crystal shape is that of an 8faced polyhedron. The large faces are the (010) faces, perpendicular to the crystal axis b. Although it is true that (101) faces are predicted to be expressed (Fig. 2⇓), they are not predicted to be well developed, and could not be identified in our 2D images or 3D reconstructions. To simplify the reconstruction of crystal shape and associated dimensions, we have therefore chosen to draw crystals as parallelepipeds (see Fig. 2⇓). In this case, the short (S) axis of the crystal in 3D is defined as the distance measured perpendicular to the (010) faces, the intermediate (I) and long (L) axes being defined as the lengths of the edges of the (010) faces. Although the assumption that grains are parallelepipeds provides a simple and appropriate way to measure and compare the S, I, and L dimensions in our crystal population, it may be borne in mind that when defined in this way, the distance L does not correspond to the longest possible line that may be drawn through a given crystal. The maximum dimension has therefore also been considered, although the differences between L as defined here and the longest distance through a 3D crystal are not significant.
The accuracy of the reconstruction is limited both by tilting of the surface and spacing between successive sections. The tilting of the surface between successive sections has been found to be, at most, 0.2°. This value of tilt induces a maximum shift on the vertical positioning of 10 μm at the sample scale (i.e., over a horizontal distance of 3 mm), but of only 0.35 μm at the crystal scale (i.e., over a horizontal distance of 100 μm). This effect has therefore been ignored. The spacing between successive polished surfaces is on average 23 μm, whereas mean plagioclase length is found to be 150 μm. The uncertainty related to this spacing is therefore relatively large. The length of the axis measured perpendicular to the surface may therefore be underestimated by up to 46 μm, on average. In contrast, the error measured on the surface itself is only related to the size of the pixels used, and is ca. 1 μm. Note also that the spacing between (010) faces, i.e., the short axis of the crystals, was usually easily and precisely determined, because the probability that plagioclase crystals are cut perpendicular to the (010) surface is high.
Results and discussion
Results of the 3D reconstruction
The 3D reconstruction results in the definition of 261 crystals over a volume of ca. 0.8 mm^{3} (Fig. 3a⇓). The reconstruction is incomplete in the vicinity of olivine crystals because it was very difficult to assess the continuity of the crystals between sections, due to the interpenetration of the two mineral species. Moreover, the crystals that were touching the bottom or top sections have usually been discarded, with the exception of a few very large crystals that were considered as a measure of maximum size. Consequently, the total number of crystals per unit volume may be underestimated. The short axis could be recognized and measured on 257 of the 261 crystals. The crystals for which the measurement of the short axis is missing are those that are both small and nearly parallel to the plane of polishing. The intermediate and long axes have only been unambiguously recognized and measured on ca. 150 crystals each. The short, long, and intermediate axes have been measured simultaneously on 137 crystals. Of these 137 crystals, 42 have been considered as bestdefined (Fig. 3b⇓), meaning that the continuity of the crystals across different sections is unambiguous, that three families of crystal faces have been identified and that opposite faces are parallel. The dimensions are reported in Table 1⇓ for these 42 crystals. An estimate of the uncertainty for each dimension of each crystal is provided, taking into account both sources of uncertainty mentioned above. It is of note that, in general, the accuracy of the aspect ratio defined from these data (short/intermediate, short/long, and intermediate/long) depends on the orientation of the crystal, such that crystals whose long axis is perpendicular to the polished surface are better defined than those whose short axis is perpendicular to that surface. Note also that the relative error for the dimensions of small crystals is large, which limits the validity of the 3D reconstruction to the largest crystals.
The crystal size distribution of 3D grains has been determined for both the short axis, and maximum length (based on measurements of 257 crystals for the short axis and 143 crystals for maximum length). These two independent measures of “crystal size” have been considered for several reasons. For example, although maximum length is a commonly considered measure of “size,” the short axis is much better defined from our data and has a clear morphological meaning [the distance between the (010) faces]. Second, for crystal populations that may show significant shape variability, it is not necessarily possible to estimate the CSD of the short axis from the CSD of the maximum length and in this respect it is of interest to consider the CSDs of the three dimensions independently. Third, consideration of these two measures of crystal size allows assessment of correction procedures proposed by Higgins (2000), for the longest dimension, and the simplified correction of Higgins (1994), for the short axis. We are particularly interested in testing the latter as it was used to correct 2D data presented by Pupier et al. (2008). The studied volume has been estimated using gOcad, by integration of the observed surfaces contoured manually on each section. Note that the central volume occupied by the large olivine crystals has not been considered when calculating this volume. Because the long axis and maximal length could not be satisfactorily measured for all crystals, a correction factor was applied to the observed volume, equal to the ratio of the number of measurements to the total number of crystals, i.e., 143/257. Moreover, because the number of crystals is relatively small, the CSD is sensitive to the bin size used for the definition of the size classes. The slope and intercept have therefore been calculated for a range of bin size varying from dx = 0.006 to 0.01 mm for the short axis and 0.02 to 0.04 mm for the long axis and maximal length. When the data are represented in a loglinear plot and fitted with a linear function, the variation in values of slope and intercept due to variable binsize is found to be lower than the 2σ error on any given linear regression and has been neglected. On the other hand, the CSDs have been estimated from the 2D measurements of crystal widths and lengths made on two representative sections containing a total of 556 crystals. Width and length are measured as the minor and major axes of the best fitting ellipse by Scion Image. For the stereological corrections, we used constant mean shape factors of S/I = 0.31 and S/L = 0.21, values derived from the mean of the 3D measurements. The CSD estimated for short axes from 3D measurements has been compared to that calculated from 2D measurements of crystal width using the correction method based on Higgins (1994) and presented in Pupier et al. (2008). In this case, the 2DCSDs n_{a} were converted into the 3DCSDs n_{v} by applying Equation 1 using the shape factors stated above. The CSDs estimated for maximal length from 3D measurements have been compared to those calculated from 2D measurements of either crystal width or length using the CSDCorrections program of Higgins (2000). The errors indicated on the CSD parameters represent 2σ uncertainties derived from linear regression of the data, including individual errors on each point in the CSDCorrections program.
Plagioclase crystal shape
Despite the fact that cooling rate was constant during crystallization, the 3D shape of the crystals is highly variable (Figs. 3⇑ and 4⇓, Table 1⇑). This observation has already been made on microlites whose 3D shape was determined from thin sections observed in transmitted light (Castro et al. 2003), or in rhyolitic Kfeldspar phenocrysts (Mock and Jerram 2005). Two crystals are clearly acicular (I/L small, S/I long, i.e., close to 1) as visible on the 3D reconstruction (Fig. 3⇑) and on the Zingg diagram (Zingg 1935) (Fig. 4⇓). The shape of those crystals can unfortunately not be quantified in detail because they touch the lowermost section. A few crystals are either equant or elongated. All others are tabular (I/L long, i.e., close to 1, S/I small). There is no correlation between the short (S) and long (L), nor between the short (S) and intermediate (I) axes (Figs. 5a and 5b⇓). On the other hand, the long (L) and intermediate (I) axes are well correlated, with I/L ratio equal to 0.72 (±0.36) (Fig. 5c⇓). Interestingly, there is a positive correlation between the shape factor (S/I) and the short axis (S), (Fig. 6⇓), but no correlation between the shape factor S/I and the other axes (I and L) nor between I/L and any of the individual 3D dimensions.
Two interpretations are possible to explain shape variability. The first is that the shape is related to the temperature at which the crystal appeared (in other words, the time since crystal appearance). This is suggested by the positive correlation between the shape factor (S/I) and the short axis (S), because the largest crystals may be expected to be the oldest (Fig. 6⇑). The presence of a few large acicular crystals and many tabular crystals, may point to distinct episodes of crystal growth of different morphology. This variability in crystal shape could, for example, be the result of an evolution of plagioclase toward more albitic composition through time, or be the result of a change in diffusivities in the melt during cooling that may affect growth rates of individual crystal faces (Hammer 2006). Literature studies demonstrate that an acicular crystal habit of plagioclase is favored by higher degrees of undercooling, relative to conditions under which tabular crystals form (e.g., Lofgren 1974). In this respect, it is highly relevant to note that at the slow cooling rate of this experiment (0.2 °C/h), plagioclase appearance is significantly delayed relative to the equilibrium liquidus (by ~30 °C), as shown and discussed in detail by Pupier et al. (2008). Indeed, this level of undercooling corresponds to the limit between tabular and acicular shapes for plagioclase according to Lofgren (1974). In this case, the acicular crystals in our sample are believed to represent the very first crystals appearing immediately following the onset of delayed nucleation. Furthermore, it has been shown that after limited further cooling (<10° after the first crystals appear), nucleation and growth in experiments cooled at 0.2°/h are sufficient to establish thermodynamic equilibrium, and further crystallization would therefore take place at much lower levels of undercooling (Pupier et al. 2008). Most of the observed crystals in the sample, quenched at 1119 °C, i.e., 20 °C after the onset of crystallization, therefore grew under nearequilibrium conditions, consistent with the dominant tabular morphology. Such episodic crystal nucleation and growth upon cooling leading to distinct morphological populations has already been reported for clinopyroxene by Hammer (2006).
However, the absence of a correlation between the shape factor (S/I) and the other axes (I and L) contradicts the hypothesis of a morphological change related uniquely to changing thermodynamic or kinetic conditions through time. An alternative hypothesis is that the variation of crystal morphology is controlled to some extent by crystal attachment, a process called synneusis (Vogt 1921). It has been shown for plagioclase (Vance 1969) and olivine (Schwindinger and Anderson 1989), that crystal coalescence involves specific crystal faces, so as to minimize structural mismatch and interfacial energy. During synneusis, plagioclase crystals are attached by their large (010) faces (Vance 1969), thus coalescence may be expected to increase S, affecting the ratios S/I and S/L, but not the ratio I/L. This hypothesis can therefore satisfactorily explain the observation that there is a correlation between S/I and S, but no correlation between I/L and S. Direct evidence for crystal coalescence is difficult to obtain from 2D images when taken under reflected light or by SEM, because crystal boundaries are not visible. Some crystals that were clearly separated on some sections appear as one crystal in perfect continuity a few sections further on (Figs. 7a and 7b⇓). Others appear as crystals attached by their large faces in imperfect continuity, for example at their edges, as visible in the 3D reconstruction (Fig. 7c⇓) or in 2D images (Fig. 7d⇓). However, such cases where crystal coalescence can be unambiguously demonstrated are rare. The demonstration of synneusis in natural examples usually requires the recognition of misfits in crystal zoning (Vance 1969), a criteria that was not possible to apply here in the absence of such zoning. The importance of synneusis in natural and experimental systems is not clear, some authors favoring a significant role (Vance 1969; Pupier et al. 2008), others favoring alternative explanations such as epitaxial growth to explain the available observations (Dowty 1980).
Crystal growth rates
Growth rates in experimental charges can be calculated by dividing the size of the largest crystal by the time spent below the liquidus and by assuming that the largest crystal has nucleated at the liquidus (e.g., Kirkpatrick 1977). Performing that calculation from 2D measurements is difficult, because it is not possible to be sure of the crystallographic orientation of the largest apparent crystal. Crystalgrowth rates have been calculated for the S, I, and L axes from the 3D reconstruction (Table 2⇓). Considering the time spent below the liquidus temperature (1175 °C) as the total growth duration for the largest crystals, we find maximum growth rates of 6.8 × 10^{−11}, 2.4 × 10^{−10}, and 3.4 × 10^{−10} m/s along the S, I, and L axes respectively (considering only the best defined crystals). However, as noted above, it has been demonstrated that for the experiment in question, nucleation did not begin until a temperature of ~1150 °C (Pupier et al. 2008). The growth rates may therefore have been ca. twice as high as the values stated above (i.e., 1.5 × 10^{−10}, 5.1 × 10^{−10}, and 7.2 × 10^{−10} m/s). The mean I/L atio is 0.72 and, accordingly, the growth rate is 0.72× lower along the I axis. The mean S/I ratio is 0.26, and we find a concordant ratio of the growth rates along the S and I axes of 0.28. The growth rate along the I and L axes can probably be considered as the growth rate by atomic attachment along crystal faces. By contrast, we must keep in mind that because of agglomeration processes, the growth rate along the S axis may represent the sum of the growth rate related to atomic attachment along crystal faces and the increase in crystal size due to crystal agglomeration. These growth rates are higher than those measured in the Makaopuhi lava lake in Hawaii: 1.7 × 10^{−12} to 11 × 10^{−12} m/s according to Kirkpatrick (1977) and 5.4 × 10^{−12} to 10 × 10^{−12} m/s according to Cashman and Marsh (1988). It may be noted, however, that cooling rate in the lava lake is 10 to 20 times lower than in the experiment presented here. The growth rates determined in the present work are, however, in the range of other experimental determination of plagioclase growth rates, which can be as high as 10^{−5} m/s [e.g., Kirkpatrick et al. (1979) and review by Smith and Brown (1988)] depending on undercooling (for isothermal experiments) and cooling rate (for cooling experiments).
Crystalsize distribution
When plotted in semilogarithmic plots (Fig. 8⇓), the CSDs obtained from 3D measurements are bellshaped, with a maximum observed for a size of 20 μm for short axes, and ca. 100 μm for maximal lengths. The CSDs are linear for larger grain sizes, and the corresponding slope is −77 ± 12/mm with a Y intercept of 11.4 ± 1.0 for the short axis and −10.4 ± 2.6/mm with a Y intercept of 9.12 ± 0.77 for the maximal length.
The CSD obtained for maximal length in 3D can be compared with the CSDs obtained from corrected 2D measurements of crystals using the CSDCorrections program of Higgins (2000) (Fig. 8a⇑). Note that the CSD for the long axis would be very similar to that of the maximal length due to the tabular shape of plagioclase crystals. The CSD derived from corrected 2D measurement of crystal length do not show a downturn at small size, in contrast to the CSD obtained from the 3D reconstruction. This difference may indicate that the correction for the cutting effect is too great, leading to an overestimate of the number of very small crystals, a suggestion previously made by Castro et al. (2003). However, in light of the fact that the spacing between our serial sections is on the order of 20 μm, we cannot be sure that the downturn in the 3D CSD is not an artifact, due to incomplete sampling of the smallest crystals. Moreover, the number of small crystals is underestimated in the 3D reconstruction, because their long dimensions could not generally be measured. We find that the slope (–11.7 ± 0.8/mm) and intercept (9.79 ± 0.18) of the CSD obtained from 2D length measurements corrected using the CSDCorrections program (Higgins 2000) are equal within error to the values obtained directly for the maximal length (–10.4 ± 2.6/mm for the slope, 9.12 ± 0.77 for the intercept), confirming the validity of the corrections proposed by Higgins (2000). In the case of tabletlike plagioclase crystals, the CSD parameters are also very similar to those of the long axis. On the other hand, it is of note that the program of Higgins (2000) may also be employed using 2D crystal widths as input data. In this case, we find that the proposed corrections give inconsistent results compared to the observed 3D distributions, a fact that we attribute to shape variability in our crystal population (i.e., variable aspect ratios between crystals). We therefore suggest that the use of width as input data in the CSDCorrections program must be limited to cases where it can be demonstrated that crystal shape is constant.
The CSD obtained for the short axis in 3D can be compared with CSDs obtained from 2D measurements for crystal width (Fig. 8b⇑), corrected for the probability of intersection effect using a method based on that of Higgins (1994). Although this method fidoes not explicitly take into account the “cutting effect,” this comparison is justified by the fact that on randomly cut tablets, the probability of intersection parallel to the short axis is large, so that in 2D measurements, the modal width is equal to the short dimension (Higgins 1994). Both measured and derived CSDs are dominated by a linear segment and a downturn toward small sizes in the semilogarithmic plot. The slope of the linear part of the CSD derived from 2D data (–85 ± 13/mm) is equal within error to that obtained from direct 3D measurements (–77 ± 12/mm). The fact that the cutting effect is not taken into account would therefore appear to be of negligible importance. However, the intercept (13.15 ± 0.3) is slightly higher for the CSD derived from 2D images relative to that measured from the 3D reconstruction (11.4 ± 1.0). The observed variation in the intercept can be explained by (1) incorrect estimates of the observed volume (or surface), and (2) shape variability that induces an incorrect estimate of the shape factors used to correct for the probability of intersection effects. The problem of the volume estimate is particularly crucial in the 3D reconstruction because, at the center of the experimental charge (where olivine crystals were present) and near the uppermost and lowermost sections, crystals have not been drawn. The number of crystals per unit volume is therefore most probably underestimated in the 3Dreconstruction, leading to an underestimate of the intercept by up to 0.5 ln unit. In the CSDs obtained from 2D sections, the incorrect estimate of the shape factor may cause a vertical shift of ca. 1 ln unit (maximum 1.3 units for crystal length if the S/I and S/L factor both vary in the extreme range 0.1–0.6).
To illustrate further the influence of 3D crystal shapes on corrected CSDs derived from 2D data, we have applied the CSDCorrections program to our 2D data assuming extreme values of the aspect ratio of 3D grains. For example, if the shape factor of the most extreme acicular crystals is assumed, the slope of the CSD derived from 2D measurements of crystal length is −3.15/mm (compared to −11.7 when the modal values of shape parameters are used). At the other extreme, assuming shape parameters that correspond to the most sheetlike crystals leads to a calculated slope of −18.6/mm. However, we note that as long as the shape remains close to tabletlike, the calculated slopes remain constant within errors. As an aside, we also note that shape variability does not affect the calculation of the slope of the CSD when using the simplified method of stereological correction.
To summarize, Castro et al. (2003) and Mock and Jerram (2005) have demonstrated that recent methods (Higgins 2000; Peterson 1996) may satisfactorily provide sterological corrections for CSDs of tabletshaped crystals. The results presented above confirm that currently available correction methods are sufficient to calculate the parameters of the linear part of 3D crystalsize distributions for short and maximal crystal dimensions. In detail, the slope and intercept for maximal length are found to be correctly reproduced by the CSDCorrections of Higgins (2000) when 2D crystal length is used as input data, but we caution against the use of 2D crystal width as input data, unless the population of grains can be demonstrated to be of constant shape factor. The slope of the CSD for the short axis S may be retrieved from the correction method proposed by Higgins (1994), used by Pupier et al. (2008), which only takes account of the probability of intersection effect. On the other hand, the intercept is slightly underestimated.
Acknowledgments
We thank Laurent Tissandier and Cédric Deumeurie for their assistance with the experiments and sample preparation. Many thanks also to the students of the Ecole Nationale Supérieure de Géologie, Florence Bonnet and Gaëlle Maury. This work benefited from financial support from the French Ministère de l’Education Nationale et de la Recherche and the Institut National des Sciences de la Terre. Many thanks as well to Julia Hammer, Don Baker, and Michael Higgins for their careful and constructive reviews.
Footnotes

↵1 Deposit item AM08025, data set of 2D measurements. Deposit items are available two ways: For a paper copy contact the Business Office of the Mineralogical Society of America (see inside front cover of recent issue) for price information. For an electronic copy visit the MSA web site at http://www.minsocam.org, go to the American Mineralogist Contents, find the table of contents for the specific volume/issue wanted, and then click on the deposit link there.

Manuscript handled by Don Baker
 Manuscript Received April 23, 2007.
 Manuscript Accepted December 30, 2007.