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Power law olivine crystal size distributions in lithospheric mantle xenoliths
Lithos 65 (2002) 273 – 285 www.elsevier.com/locate/lithos
Power law olivine crystal size distributions in lithospheric mantle xenoliths P. Armienti *, S. Tarquini Dipartimento di Scienze della Terra, Universita´ di Pisa, via S. Maria 53, Pisa 56100, Italy Received 22 February 2002; accepted 17 June 2002
Abstract Olivine crystal size distributions (CSDs) have been measured in three suites of spinel- and garnet-bearing harzburgites and lherzolites found as xenoliths in alkaline basalts from Canary Islands, Africa; Victoria Land, Antarctica; and Pali Aike, South America. The xenoliths derive from lithospheric mantle, from depths ranging from 80 to 20 km. Their textures vary from coarse to porphyroclastic and mosaic – porphyroclastic up to cataclastic. Data have been collected by processing digital images acquired optically from standard petrographic thin sections. The acquisition method is based on a high-resolution colour scanner that allows image capturing of a whole thin section. Image processing was performed using the VISILOG 5.2 package, resolving crystals larger than about 150 Am and applying stereological corrections based on the Schwartz – Saltykov algorithm. Taking account of truncation effects due to resolution limits and thin section size, all samples show scale invariance of crystal size distributions over almost three orders of magnitude (0.2 – 25 mm). Power law relations show fractal dimensions varying between 2.4 and 3.8, a range of values observed for distributions of fragment sizes in a variety of other geological contexts. A fragmentation model can reproduce the fractal dimensions around 2.6, which correspond to well-equilibrated granoblastic textures. Fractal dimensions >3 are typical of porphyroclastic and cataclastic samples. Slight bends in some linear arrays suggest selective tectonic crushing of crystals with size larger than 1 mm. The scale invariance shown by lithospheric mantle xenoliths in a variety of tectonic settings forms distant geographic regions, which indicate that this is a common characteristic of the upper mantle and should be taken into account in rheological models and evaluation of metasomatic models. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Mantle xenoliths; Olivine size distribution; Power law; Fractals
1. Introduction A wide range of textures characterises mantle xenoliths. According to the classification proposed
*
Corresponding author. Tel.: +39-50-847212; fax: +39-50500675. E-mail address: [email protected] (P. Armienti).
by Harte (1977) and Frey and Prinz (1978), they may vary from coarse to mosaic – porphyroclastic and granoblastic, through a large variety of terms that account for the recrystallization history of the mantle. In this work, we attempt a quantitative evaluation of rock textures of upper mantle Type I xenoliths (Fig. 1) sampled by alkali basalt. Xenoliths were sampled in different geological contexts to better understand the driving forces of textural evolution. We adopted an
0024-4937/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 4 - 4 9 3 7 ( 0 2 ) 0 0 1 9 5 - 0
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Fig. 1. Modal compositions of the populations of Type I refractory mantle xenoliths examined in this work. Most of the samples are spinelbearing harzbugites, with the exception of a few garnet-bearing lherzolites from Pali Aike volcanic field.
automatic image analysis approach in order to systematically identify and measure the size of a large number of mineral grains in each mantle sample. Analysed rocks are unaltered spinel- and garnetbearing harzburgites and lherzolites from three suites of mantle xenoliths found in alkaline basalts from Canary Islands, Africa; Greene Point (Victoria Land, Antarctica); and Pali Aike, South America. Samples were derived from lithospheric mantle, from depths ranging from 80 to 20 km (Perinelli, 2000; Beccaluva et al., 1991; Siena and Coltorti, 1993; Stern et al., 1989). Textures vary from coarse and mosaic – equant to porphyroclastic, and modal abundance of Fo-rich olivine (Fo 89 – 92) is always larger than 60%. Larger olivine crystals (2– 25 mm) exhibit ‘‘kink banding’’ and often show a preferred orientation due to the occurrence of oriented stress. Enstatite (En 83 – 97), the dominant pyroxene, and minor amounts of interstitial diopside (Wo 44.4– 48.2, En 45.6– 52, Fs 3.6 – 6.2) seldom reach a size larger than 2 mm. Spinel and,
when occurring, garnet are minor interstitial Al-bearing components. Table 1 lists the analysed samples and summarises their petrographic features. Olivine is an essential constituent of the upper mantle; thus, the analysis focuses on the size distribution of this mineral that strongly constrains the estimates of rheological parameters and grain boundary areas per unit volume. These parameters are relevant for attenuation mechanisms of seismic wave propagation, the estimate of mantle rheological properties, and reequilibration of fluids percolating in the mantle. A variety of models can account for the evolution of the number of particles populating a system and many of them find a suitable quantitative description by adopting the concept of Particle Size Distribution (PSD). This is defined as the number N(L) of particles of size between L and L + DL per unit volume, normalised to the size interval, DL. N(L) is represented as a discrete function of L, computed by choosing the size of class intervals according to linear
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Table 1 List of samples and summary of petrographic features Olivine (vol.%) Analyses on one thin section LANZ 3 83 LANZ 5 92 LANZ 6a 80 LANZ 6b 84 LANZ 15 #1 96 LANZ 19 90 LANZ 24 #1 96 LANZ 25 87 LANZ 26 83 LANZ 28 97 LANZ 26a 96 LANZ 26b 95 GP 26 96 GP 30 88 GP 32 97 GP 40 96 GP 42 91 GP 43 93 GP 44 95 GP 45 97 GP 46 93 GP 47 91 GP 48 93 GP 50 96 PA 203 #1 72 PA 128 #1 60
Olivine crystals 215 335 662 535 388 466 587 653 675 558 411 522 446 549 499 928 583 804 786 694 872 623 771 523 503 380
Analyses on multiple thin sections (number in parenthesis) LANZ 24 (10) 8424 LANZ 15 (10) 4819 PA 203 (9) 5021 PA 128 (3) 1349
Texture after Harte (1977)
m
r
porphyroclastic porphyroclastic mosaic – porphyroclastic granoblastic porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic granoblastic porphyroclastic porphyroclastic granoblastic coarse porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic porphyroclastic mosaic – porphyroclastic granoblastic
3.81 2.96 3.11 2.59 2.89 2.82 3.34 3.69 3.44 3.14 3.27 2.62 2.65 3.03 2.43 2.89 2.60 2.99 3.27 2.83 2.78 2.56 2.76 2.66 2.48 2.35
0.41 0.38 0.18 0.15 0.17 0.16 0.19 0.21 0.20 0.31 0.40 0.15 0.15 0.18 0.14 0.17 0.15 0.17 0.19 0.16 0.16 0.47 0.24 0.29 0.14 0.14
3.33 2.95 2.43 2.46
0.19 0.17 0.14 0.14
LANZ: Lanzarote (Canary Island); spinel-bearing. GP: Greene Point (Antarctica); spinel-bearing. PA: Pali Aike (Patagonia, South America); garnet-bearing. m: fractal dimension of olivine size distribution. r: standard deviation of m.
or logarithmic scales, and adopting a stereological correction of two-dimensional data (Higgins, 2000; Sahagian and Proussevitch, 1998). L is chosen as a typical linear dimension of the particle (e.g., the equivalent diameter or the length). The physical dimension of N(L) is l 4. In many cases, N(L) can be theoretically derived from physical assumptions, through equations accounting for the evolution of the number of particles as a function of their size; thus, measured PSDs can be explained on the basis of physical models.
Crystal size distribution (CSD) analysis gained widespread attention in petrologic investigation since the early works of Marsh (1988) and Cashman and Marsh (1988), which adopted the logN(L) vs. L plot to describe the evolution of crystal size in open systems according to the equation (Marsh, 1988): L logN ðLÞ ¼ þ logðN0 Þ ð1Þ Gs where G is the crystal growth rate s is the residence time of magma, and N0 is the nucleation density.
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Evolution of crystal sizes can be also described in crystallising closed systems (Armienti et al., 1994) by the equation: N ðLÞ ¼
J G
ð2Þ
where J and G are the crystal nucleation and growth rates at the time in which crystals of size L appeared in the system, respectively. When scale-invariant processes like fragmentation or chaotic phase transition occur (e.g., degassing), PSDs are better described by power laws like: N ðLÞ ¼ CLm
ð3Þ
(Blower et al., 2001; Simakin et al., 1999; Turcotte, 1995), where C is a constant and m is the fractal dimension of the PSD.
2. Analytical method Mineral grain identification is performed by inspecting images of standard petrographic thin sections acquired with a commercially available, low-cost, high-resolution film scanner. The device allows the collection of data from thin sections, with a maximum image resolution of 10 Am/pixel; technical details of this new tool are reported by Tarquini and Armienti (2001). Maximum size of the image is 24 36 mm, practically the whole thin section. The procedure involves the acquisition in transmitted light of a set of four images with crossed polarized light, coupled with one image in plane polarized light. Crystal boundaries are extracted from the images at crossed polarizers, while markers for phase attribution of each grain are obtained from the inplane polarized light image (Fig. 2). This method is
Fig. 2. Set of crossed polarized light images (A – D) and plane polarized light (E) images analysed to obtain mineral identification and delineate grain boundaries (F). Sample 24 H from Lanzarote. The size of the sections is 22.6 32.3 mm. Acquisition resolution is set at 1500 dpi.
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particularly efficient in rocks like harzburgites and lherzolites that contain no more than four different minerals exhibiting distinct optical properties. Image processing for grain identification and detection of grain boundaries has been performed developing procedures with the VISILOG 5.2 package. The software procedure developed relies on the evaluation of absorption colors of each grain and detection of grain boundaries as revealed by changes in interference colours of olivine grains with different orientations (Tarquini and Armienti, 2001). Modal data obtained from thin sections are converted into CSD (the number of crystals per unit size per unit volume: N(D) [mm 4], where D [mm] is the equivalent diameter of a sphere with the same volume of the particle). In order to get the CSDs, two-dimensional size data from binary images have been converted into volume data by applying stereological corrections based on the Schwartz – Saltykov algorithm (De Hoff and Rhines, 1972). This procedure does not cause artifacts such as the pronounced reduction of the scatter of data observed for the method proposed by Peterson (1996) (Pan, 2001; Hammer et al., 1999). Stereological corrections based on the Schwartz – Saltykov algorithm as adopted by Higgins (2000) and Armienti et al. (1994) have been carefully tested against real and theoretical cases and represent a satisfactory solution in most cases encountered in petrologic investigation. The method, in fact, also allows a proper identification of size distribution maxima in uni- and polymodal size distributions obtained from sieved particles, and a complete reconstruction of synthetic size distribution created with suitable algorithms. No correction for the shape factors has been taken into account since elongation of olivine crystals is not such to require this treatment of data (Higgins, 2000): in fact, data of larger samples that have been collected on three orthogonal planes and do not show significant differences. The unfolding procedure adopted includes a systematic check of the closure effect (Higgins, 2002) that is obtained by minimizing the difference between the areal mode (directly obtained from the binary image of the phase of interest) and the mode obtained from the second-order moment of the CSD (see Eq. (6)). The difference is kept within 2% by changing the interval between linearly binned size classes.
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Volumetric data are reported as CSD in bilogarithmic diagrams (Fig. 3a). We assume the size of the particles to be equal to the diameter D of the equivalent sphere, since most olivine grains exhibit almost equant habit. The working resolution for image acquisition was set at 1500 dpi ( c 17 Am/pixel). In order to avoid consideration of particles smaller than the thin section thickness ( c 30 Am) and to exclude ‘‘noise’’ arising from considering as ‘‘grains’’ clusters of a few pixels, a filter was used to cut particles smaller than 50 Am. Crystal size data may show truncation effects arising from resolution (left-hand truncation) of the adopted measurement device and from the size of the specimen (right-hand truncation). Evaluation of left-hand and right-hand truncation effects on the CSD is essential when studying these kinds of data sets before attempting any interpretation (Pickering et al., 1995). Left-hand truncation arises from the resolution of the image analysis procedure as a whole and is revealed by a decrease in the number density of the counted grains. This truncation is reduced when data are acquired with a greater resolution (Armienti et al., 1994). It is important to observe that resolution is controlled both by the resolution of the starting image and by errors introduced by image processing. Thus, due to the heavy image manipulations needed for image segmentation, in spite of the higher resolution of starting images, our device allows the determination of CSDs without truncation effects only for crystal sizes larger than 0.2 mm, with left-hand truncation appearing around 150 Am (Fig. 3a). Any interpretation of a ‘‘true’’ reduction of the number density of crystal smaller than 0.2 mm as due to coarsening processes would require the acquisition of images with a different device (e.g., mineralogy microscope). Right-hand truncation produces the flattening of size distributions on nearly horizontal trends at large size ranges in which not all the classes of size occur in the sections (Fig. 3a and b). This results in a constant value of the number density for classes that contain at most one particle. A logarithmic binning of size classes (i.e., logarithmic variation of the width of the class size) allows to partly avoid this effect by counting a larger number of grains in larger (and wider) classes, but the effect still appears at the limit of the distribution with an increase of the slope at the
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Fig. 3. Effects of left-hand and right-hand truncation shown for samples LANZ 24 and LANZ 15. (a) The typical effect of left-hand truncation is to reduce the number of counted individual crystals near the resolution limit of the acquisition procedure. This can be observed both in the CSDs computed from one thin section and from 10 thin sections. Right-hand truncation arises from the limited size of the explored thin section that allows the detection of only a few (at the limit only one) particles of large size in each size interval: this results in the constant value of N(D) revealed by the flattening of the CSD trend (empty circles). For the same choice of size interval, the effects may be avoided by analyzing a larger population of crystal (small diamonds). A different choice of size intervals would also reduce this effect as is evident in the CSDs obtained from computed logarithmic binning of the size interval (squares). (b) Right-hand truncation may not disappear, even when handling a large population of crystals (small diamonds) if the number of size intervals is too large: this is particularly evident in sample LANZ 15, where CSD is plotted against D. In this case, right-hand truncation is revealed by the constant values of log(N(D)) and by the shift of linearly binned CSD with respect to the logarithmic one.
right hand of the distribution. Right-hand truncation effects can been limited by enlarging the size of the measured sample. This is obtained by measuring CSDs of several (up to 10) thin sections of each mantle xenolith, each cut at a different positions (Fig. 3a and b). The adoption of a logarithmic binning for the calculation of CSD on a single thin section ensures satisfactory results for measurements performed on mantle xenoliths, as shown by the closely comparable results provided by CSDs obtained from cumulative regressions from several sections (up to
10) (Table 1). It is important to state that logarithmic binning tends to smoothen CSDs: in some instances, where necessary, we switched to linear binning of class intervals, considering data in regions far from left- or right-hand truncation effects. No account has been taken of the edge effect due to large grains only partly contained in the thin section, being cut by its borders. This would lead to some underestimates and instability of the value of N(D) for larger grains in samples where CSDs were measured in a single thin section (Fig. 4), but it only affects size classes that are
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Fig. 4. Determination of the reproducibility of the slope determination is performed by repeated acquisition of the CSD on the same thin section. This explains the close similarity of the distinct series. The edge effect (see text) is different for each run of acquisition since different areas were delimited for scanning, but it only affects size classes that are also biased by right-hand truncation. The box contains measured slopes from distinct procedures of acquisition, in the interval not affected by truncations (0.2 – 2.0 mm as shown by the arrows). Lines connect logN(D) values of contiguous size classes for which a nonzero value was obtained. Mean and standard deviation (r) are reported.
also biased by right-hand truncation . Edge effect is compensated by the very large number of total grains counted in samples where CSDs are measured on several sections. In the size range 0.2 – 25 mm, where truncation effects are avoided, all the samples show a linear dependence of log(N(D)) from log(D), with a negative slope, i.e., logN ðDÞ ¼ mlogðDÞ þ logC
ð4Þ
which implies that: N ðDÞ ¼ CDm
ð5Þ
where C is a constant, D is the equivalent diameter of the crystals, N(D) is the number of crystals per unit size per unit volume, and m is the fractal dimension of the size distribution (Turcotte, 1995). In the presentation of results, the fractal dimension of the crystal size distribution set is defined according to Eqs. (4) and (5). Repeated acquisitions of data from the same thin section (Fig. 4) allow an estimate of the error (1r ) of
the procedure of CSD slope determination, which, in most cases, is F 6% of the value of m and is always smaller than 12% (Table 1).
3. Results For all the mantle xenoliths, a relatively constant linear covariance is observed for olivine CSDs, allowing detection of a scale-invariant trend (Simakin et al., 1999; Tarquini and Armienti, 2001). Fig. 5a– c illustrates the measured size distributions for the three suites of mantle xenoliths from Lanzarote, Greene Point, and Pali Aike, while Table 1, for the same data sets, reports the values of the fractal dimensions and the number of counted crystals. As a reference, the size interval in which the scale-invariant relation has been measured is reported in Fig. 5a. CSDs reported in Fig. 5a represent the mean of 10 distinct measurements performed on each thin section. This gives them a ‘‘smooth’’ appearance and reveals fine detail of the CSDs that are not included in measurements reported in Fig. 5b, where results obtained from single
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Fig. 5. CSDs measured on Lanzarote (a), Greene Point (b), and Pali Aike (c) mantle xenoliths. (a) CSDs obtained from linear binning of the size interval, for measurements performed on single thin sections. In the first diagram of the series (top left), small squares show the results of N(D) determination obtained through the logarithmic binning of the size interval, according to the procedure of Higgins (2000). In the size interval without truncation effects (between the arrows), the slopes obtained with the two methods overlap for all samples. Note in the top-left diagram the scale of D in millimeters. (b) CSDs obtained from linear and logarithmic binning for measurements performed in single thin sections. Lines connect logN(D) values of contiguous size classes for which a nonzero value was obtained with the linear binning. (c) CSDs obtained from linear and logarithmic binning, for measurements performed in up to 10 thin sections, of Pali Aike and Lanzarote samples. These are largest xenoliths available in our sampling (Fup to 25 cm); thus, it was possible to cut thin sections in three orthogonal planes at distances larger than about 20 mm (the maximum size detected among olivine grains). In LANZ 24, there is no right-hand truncation in the CSD data set. The horizontal ‘‘tails’’ of the other xenoliths reveal that a larger number of sections should have been analyzed to get linear trends with the linear binning. This requirement is overcome when data are handled with the method of Higgins (2000).
measurements (and linear scaling of class sizes) are compared with CSDs computed with the logarithmically scaled class sizes. Fig. 5c reports measurements obtained from several thin sections of the same xenolith. Care was taken in cutting thin sections on three orthogonal planes and spacing the cuts at a distance larger that the largest detectable crystal. The two samples measured for Pali Aike are given as measurements on multiple sections. In both Lanzarote samples, the larger number of thin sections allows to insert in the data set a greater number of large crystals, thus greatly reducing the right-hand truncation effects for sample LANZ 24. The values obtained for the fractal dimension of LANZ 24 and
LANZ 15 agree within F 1r with measurements done on single sections (Table 1). The presence of horizontal ‘tails’ of CSD at large D values for the other samples reveals that the number of measured thin sections is still insufficient to rely only on linear binning of class sizes for large crystals. In the Lanzarote data set, where porphyroclastic textures are more widespread, the values of the fractal dimension of the CSDs are usually larger than in other areas, and the larger values of m are associated to cataclastic textures akin to mylonites (e.g., LANZ 3 with m = 3.81). On the contrary, granoblastic samples show the lowest values of m (e.g., LANZ 6b with m = 2.59). The accurate measurements performed for
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this data set reveal that the porphyroclastic texture is associated with a small relative maximum of the CSD, usually between 1 and 2 mm. Such detail cannot be detected in the Greene Point and Pali Aike xenoliths that are generally characterised by smaller values of m and by larger maximum sizes of olivine crystals. This is consistent with the less pronounced cataclastic textures observed in these peridotites. Pali Aike xenoliths are the only garnet-bearing rocks of the data set, but this has no influence on the scale-invariant relation between the number of crystals per unit size per unit volume and their size. If we define the ith order moment of the size distribution by the expression: Z l Mi ¼ Di N ðDÞdD; ð6Þ 0
it follows that for spherical particles, M0 is the number of crystals per unit volume, M1 is the total length of
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crystals per unit volume, 2pM2 is the total area of crystals per unit volume, and 23 pM3 is the total volume of crystals per unit volume. These useful relations can be used to asses several textural data of the measured samples. Fig. 6 reports for six samples of Lanzarote the volume fraction of particles of given sizes obtained from the third-order moments of the measured CSDs. It is evident that in porphyroclastic xenoliths, smaller grains represent a larger volume fraction. This is not the case for granoblastic peridotites where the largest volume fractions are accounted for by larger grains. Second-order moment of CSDs allow the computation of the surface fraction of each size class; where a large fraction of grain falls among smaller grains, an even larger fraction of the surface is characteristic of the smaller grain sizes, like in sample LANZ 26.
Fig. 6. Calculated granulometric curve for porphyroclastic and granoblastic samples from Lanzarote. Volume fractions of grains of given size are computed starting from the third-order moment of crystal size distributions (Eq. (6)). It appears that in porphyroclastic samples, the mode at 0.5 mm is more pronounced, accounting for the large number of smaller crystals.
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4. Discussion of data The scale-invariant relation N(D) = CD m in the size range 0.2– 25 mm characterises mantle xenoliths brought to the surface by alkaline basalts in geological contexts ranging from intraplate hot spot magmatism (Lanzarote: Siena and Coltorti, 1993), to rift magmatism (Greene Point, Antarctica: Perinelli, 2000; Beccaluva et al., 1991) and subduction of a triple point (Stern et al., 1989; D’Orazio et al., 2000). All the samples derive from the lithospheric mantle from depths ranging from 20 to 80 km. Though the sampling is necessarily limited to volcanic areas where basaltic magmas rise to the surface, the scale invariance of olivine grain size distributions seems to be characteristic of upper mantle in disparate tectonic settings. In these Type I xenoliths, as in other mantle samples from around the world, olivine exhibits strong kink banding that is considered to be evidence of oriented stress: often kink banding evolves in true cataclastic textures in which olivine grains are disrupted to form a fine-grained matrix of neoblasts. Thus, the fragmentation model may provide an interpretation of the observed scale invariance. Following Turcotte (1995), if grains are ordered in grain size classes and f is the probability that a grain of ith order will fragment to give P particles of (i + 1)th order, the fractal dimension of the size distribution is given by: m ¼ 3*lnðP*f Þ=lnðPÞ:
ð7Þ
If f is allowed to vary between 1/P and 1, then 0 < m < 3. The most recurrent values obtained for fractal dimension of olivine size distributions in our xenoliths are around 2.6. If at each fragmentation eight grains of (i + 1)th order are generated (P = 8) and if only two are preserved at the following fragmentation ( f = 6/8), Eq. (7) gives m = 2.58. This simple model of crushing of a constant fraction of grains illustrates how recursive application of a very simple law can produce scale invariance effects. Different choices of P and f provide different values of m; all are, however, between 0 and 3. Indeed, many fractal dimensions obtained in strongly deformed samples are larger than 3 and do not conform to the simple model of Eq. (7). Values larger than 3 are sometimes reported in the literature for the
fractal dimensions of fragment distributions (see Turcotte, 1995); in our case, the development of a cataclastic texture may be accompanied by recrystallization of smaller grains of olivine. Thus, when larger grains are still crushing (increasing their number density), smaller ones may start to form larger neoblasts, decreasing in number (see the slight flexure in sample LANZ 3, Fig. 5a). This effect is well known in mantle xenoliths, where olivine neoblasts often show compositions different from those of porphyroclasts (e.g., Perinelli, 2000; Siena and Coltorti, 1993), and it may result in a counterclockwise rotation of CSD, whose fractal dimension is thus allowed to become larger than 3.
5. Conclusions Fig. 5c (see sample LANZ 15) shows that, when a logarithmic binning is adopted for regression of data collected on multiple sections, crystals with a maximum size up to 25 mm follow a scale-invariant relation. In general, olivine CSDs show scale invariance astride almost three orders of magnitude in the size interval 0.2 – 25 mm. Fragmentation models offer a possible interpretation for this fractal size distribution, providing a theoretical model that can reasonably predict a value for m c 2.6, as found in most samples. However, selective crushing of larger crystals and recrystallization of smaller neoblasts may provide an interpretation for distributions with m>3. The mean size of olivine grains is a fundamental parameter that is considered in rheological models of the mantle (e.g., Ranalli, 1998), and controls the transition from diffusion to dislocation creep mechanisms. It is evident that, if olivine crystal size distributions exhibit scale invariance, it is not possible to define a mean value for the size of this mineral in rheological maps, and this should be taken into account in modelling of mantle physical behaviour. Our data set (Fig. 6) also shows that the larger surface fraction of mantle grains is associated with smaller grains; thus, a large reaction surface area is available for reequilibration of smaller crystals through a diffusion mechanism. Measurements of CSDs in mantle xenolith can provide useful estimates of the total surface available for reactions of fluids percolating in mantle during metasomatic events.
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Acknowledgements This work was funded by the grant MURST 1998 no. 9804307319002, ‘‘Materiali terrestri ed analoghi sintetici ad alta pressione ed alta temperatura: proprieta’ fisiche, chimiche e reologiche.’’
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Higgins, M.D., 2000. Measurement of crystal size distributions. Am. Mineral. 85, 1105 – 1116. Higgins, M.D., 2002. Closure in crystal size distributions (CSD), verification of CSD calculations, and the significance of CSD fans. Am. Mineral. 87, 171 – 175. Marsh, D.B., 1988. Crystal Size Distribution (CSD) in rocks and the kinetics and dynamics of crystallisation: I. Theory. Contrib. Mineral. Petrol. 99, 277 – 291. Pan, Y., 2001. Inherited correlation in crystal size distribution. Geology 29, 227 – 230. Perinelli, C., 2000. Modificazioni indotte dal metasomatismo di alta pressione ed alta temperatura nelle peridotiti del mantello superiore in zone di rift attivo, di margine passivo e di magmatismo d’intraplacca. PhD thesis, XII ciclo. Universita` di Pisa, 98 pp. Peterson, T.D., 1996. A refined technique for measuring crystal size distributions in thin sections. Contrib. Mineral. Petrol. 124, 395 – 405. Pickering, G., Bull, J.M., Sanderson, D.J., 1995. Sampling powerlaw distributions. Tectonophysics 248, 1 – 20. Ranalli, G., 1998. Inference on Mantle rheology from creep laws. In: Wu, P. (Ed.), Dynamics of the Ice Age Earth, A Modern Perspective. GeoResearch Forum, vols. 3 – 4, pp. 323 – 340. Sahagian, D.L., Proussevitch, A.A., 1998. 3D particle size distributions from 2D observations; stereology for natural applications. J. Volcanol. Geotherm. Res. 84, 173 – 196. Siena, F., Coltorti, M., 1993. Thermobarometric evolution and metasomatic processes of upper mantle in different tectonic settings: evidence from spinel peridotite xenoliths. Eur. J. Mineral. 5, 1073 – 1090. Simakin, A.G., Armienti, P., Epel’baum, M.B., 1999. Coupled degassing and crystallization: experimental study in continuous pressure drop, with application to volcanic bombs. Bull. Volcanol. 61, 275 – 287. Stern, C.R., Saul, S., Skewes, M.A., Futa, K., 1989. Garnet peridotite xenoliths from the Pali-Aike alkali basalts of southernmost South America. Geol. Soc. Aust. Spec. Publ. 14, 735 – 744. Tarquini, S., Armienti, P., 2001. Slide color scanner as a new and cheap tool for image analysis in petrology. Image Analysis and Stereology, Suppl. 1. Proceedings of the 8th European Congress for Stereology and Image Analysis, Bordeaux, France, 2001. vol. 20, pp. 572 – 576. Turcotte, D.L., 1995. Fractals and Chaos in Geology and Geophysics, 2nd ed. Cambridge Univ. Press, Cambridge, 221 pp.
Power law olivine crystal size distributions in lithospheric mantle xenoliths P. Armienti *, S. Tarquini Dipartimento di Scienze della Terra, Universita´ di Pisa, via S. Maria 53, Pisa 56100, Italy Received 22 February 2002; accepted 17 June 2002
Abstract Olivine crystal size distributions (CSDs) have been measured in three suites of spinel- and garnet-bearing harzburgites and lherzolites found as xenoliths in alkaline basalts from Canary Islands, Africa; Victoria Land, Antarctica; and Pali Aike, South America. The xenoliths derive from lithospheric mantle, from depths ranging from 80 to 20 km. Their textures vary from coarse to porphyroclastic and mosaic – porphyroclastic up to cataclastic. Data have been collected by processing digital images acquired optically from standard petrographic thin sections. The acquisition method is based on a high-resolution colour scanner that allows image capturing of a whole thin section. Image processing was performed using the VISILOG 5.2 package, resolving crystals larger than about 150 Am and applying stereological corrections based on the Schwartz – Saltykov algorithm. Taking account of truncation effects due to resolution limits and thin section size, all samples show scale invariance of crystal size distributions over almost three orders of magnitude (0.2 – 25 mm). Power law relations show fractal dimensions varying between 2.4 and 3.8, a range of values observed for distributions of fragment sizes in a variety of other geological contexts. A fragmentation model can reproduce the fractal dimensions around 2.6, which correspond to well-equilibrated granoblastic textures. Fractal dimensions >3 are typical of porphyroclastic and cataclastic samples. Slight bends in some linear arrays suggest selective tectonic crushing of crystals with size larger than 1 mm. The scale invariance shown by lithospheric mantle xenoliths in a variety of tectonic settings forms distant geographic regions, which indicate that this is a common characteristic of the upper mantle and should be taken into account in rheological models and evaluation of metasomatic models. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Mantle xenoliths; Olivine size distribution; Power law; Fractals
1. Introduction A wide range of textures characterises mantle xenoliths. According to the classification proposed
*
Corresponding author. Tel.: +39-50-847212; fax: +39-50500675. E-mail address: [email protected] (P. Armienti).
by Harte (1977) and Frey and Prinz (1978), they may vary from coarse to mosaic – porphyroclastic and granoblastic, through a large variety of terms that account for the recrystallization history of the mantle. In this work, we attempt a quantitative evaluation of rock textures of upper mantle Type I xenoliths (Fig. 1) sampled by alkali basalt. Xenoliths were sampled in different geological contexts to better understand the driving forces of textural evolution. We adopted an
0024-4937/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 4 - 4 9 3 7 ( 0 2 ) 0 0 1 9 5 - 0
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Fig. 1. Modal compositions of the populations of Type I refractory mantle xenoliths examined in this work. Most of the samples are spinelbearing harzbugites, with the exception of a few garnet-bearing lherzolites from Pali Aike volcanic field.
automatic image analysis approach in order to systematically identify and measure the size of a large number of mineral grains in each mantle sample. Analysed rocks are unaltered spinel- and garnetbearing harzburgites and lherzolites from three suites of mantle xenoliths found in alkaline basalts from Canary Islands, Africa; Greene Point (Victoria Land, Antarctica); and Pali Aike, South America. Samples were derived from lithospheric mantle, from depths ranging from 80 to 20 km (Perinelli, 2000; Beccaluva et al., 1991; Siena and Coltorti, 1993; Stern et al., 1989). Textures vary from coarse and mosaic – equant to porphyroclastic, and modal abundance of Fo-rich olivine (Fo 89 – 92) is always larger than 60%. Larger olivine crystals (2– 25 mm) exhibit ‘‘kink banding’’ and often show a preferred orientation due to the occurrence of oriented stress. Enstatite (En 83 – 97), the dominant pyroxene, and minor amounts of interstitial diopside (Wo 44.4– 48.2, En 45.6– 52, Fs 3.6 – 6.2) seldom reach a size larger than 2 mm. Spinel and,
when occurring, garnet are minor interstitial Al-bearing components. Table 1 lists the analysed samples and summarises their petrographic features. Olivine is an essential constituent of the upper mantle; thus, the analysis focuses on the size distribution of this mineral that strongly constrains the estimates of rheological parameters and grain boundary areas per unit volume. These parameters are relevant for attenuation mechanisms of seismic wave propagation, the estimate of mantle rheological properties, and reequilibration of fluids percolating in the mantle. A variety of models can account for the evolution of the number of particles populating a system and many of them find a suitable quantitative description by adopting the concept of Particle Size Distribution (PSD). This is defined as the number N(L) of particles of size between L and L + DL per unit volume, normalised to the size interval, DL. N(L) is represented as a discrete function of L, computed by choosing the size of class intervals according to linear
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Table 1 List of samples and summary of petrographic features Olivine (vol.%) Analyses on one thin section LANZ 3 83 LANZ 5 92 LANZ 6a 80 LANZ 6b 84 LANZ 15 #1 96 LANZ 19 90 LANZ 24 #1 96 LANZ 25 87 LANZ 26 83 LANZ 28 97 LANZ 26a 96 LANZ 26b 95 GP 26 96 GP 30 88 GP 32 97 GP 40 96 GP 42 91 GP 43 93 GP 44 95 GP 45 97 GP 46 93 GP 47 91 GP 48 93 GP 50 96 PA 203 #1 72 PA 128 #1 60
Olivine crystals 215 335 662 535 388 466 587 653 675 558 411 522 446 549 499 928 583 804 786 694 872 623 771 523 503 380
Analyses on multiple thin sections (number in parenthesis) LANZ 24 (10) 8424 LANZ 15 (10) 4819 PA 203 (9) 5021 PA 128 (3) 1349
Texture after Harte (1977)
m
r
porphyroclastic porphyroclastic mosaic – porphyroclastic granoblastic porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic granoblastic porphyroclastic porphyroclastic granoblastic coarse porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic granoblastic porphyroclastic porphyroclastic porphyroclastic porphyroclastic mosaic – porphyroclastic granoblastic
3.81 2.96 3.11 2.59 2.89 2.82 3.34 3.69 3.44 3.14 3.27 2.62 2.65 3.03 2.43 2.89 2.60 2.99 3.27 2.83 2.78 2.56 2.76 2.66 2.48 2.35
0.41 0.38 0.18 0.15 0.17 0.16 0.19 0.21 0.20 0.31 0.40 0.15 0.15 0.18 0.14 0.17 0.15 0.17 0.19 0.16 0.16 0.47 0.24 0.29 0.14 0.14
3.33 2.95 2.43 2.46
0.19 0.17 0.14 0.14
LANZ: Lanzarote (Canary Island); spinel-bearing. GP: Greene Point (Antarctica); spinel-bearing. PA: Pali Aike (Patagonia, South America); garnet-bearing. m: fractal dimension of olivine size distribution. r: standard deviation of m.
or logarithmic scales, and adopting a stereological correction of two-dimensional data (Higgins, 2000; Sahagian and Proussevitch, 1998). L is chosen as a typical linear dimension of the particle (e.g., the equivalent diameter or the length). The physical dimension of N(L) is l 4. In many cases, N(L) can be theoretically derived from physical assumptions, through equations accounting for the evolution of the number of particles as a function of their size; thus, measured PSDs can be explained on the basis of physical models.
Crystal size distribution (CSD) analysis gained widespread attention in petrologic investigation since the early works of Marsh (1988) and Cashman and Marsh (1988), which adopted the logN(L) vs. L plot to describe the evolution of crystal size in open systems according to the equation (Marsh, 1988): L logN ðLÞ ¼ þ logðN0 Þ ð1Þ Gs where G is the crystal growth rate s is the residence time of magma, and N0 is the nucleation density.
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Evolution of crystal sizes can be also described in crystallising closed systems (Armienti et al., 1994) by the equation: N ðLÞ ¼
J G
ð2Þ
where J and G are the crystal nucleation and growth rates at the time in which crystals of size L appeared in the system, respectively. When scale-invariant processes like fragmentation or chaotic phase transition occur (e.g., degassing), PSDs are better described by power laws like: N ðLÞ ¼ CLm
ð3Þ
(Blower et al., 2001; Simakin et al., 1999; Turcotte, 1995), where C is a constant and m is the fractal dimension of the PSD.
2. Analytical method Mineral grain identification is performed by inspecting images of standard petrographic thin sections acquired with a commercially available, low-cost, high-resolution film scanner. The device allows the collection of data from thin sections, with a maximum image resolution of 10 Am/pixel; technical details of this new tool are reported by Tarquini and Armienti (2001). Maximum size of the image is 24 36 mm, practically the whole thin section. The procedure involves the acquisition in transmitted light of a set of four images with crossed polarized light, coupled with one image in plane polarized light. Crystal boundaries are extracted from the images at crossed polarizers, while markers for phase attribution of each grain are obtained from the inplane polarized light image (Fig. 2). This method is
Fig. 2. Set of crossed polarized light images (A – D) and plane polarized light (E) images analysed to obtain mineral identification and delineate grain boundaries (F). Sample 24 H from Lanzarote. The size of the sections is 22.6 32.3 mm. Acquisition resolution is set at 1500 dpi.
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particularly efficient in rocks like harzburgites and lherzolites that contain no more than four different minerals exhibiting distinct optical properties. Image processing for grain identification and detection of grain boundaries has been performed developing procedures with the VISILOG 5.2 package. The software procedure developed relies on the evaluation of absorption colors of each grain and detection of grain boundaries as revealed by changes in interference colours of olivine grains with different orientations (Tarquini and Armienti, 2001). Modal data obtained from thin sections are converted into CSD (the number of crystals per unit size per unit volume: N(D) [mm 4], where D [mm] is the equivalent diameter of a sphere with the same volume of the particle). In order to get the CSDs, two-dimensional size data from binary images have been converted into volume data by applying stereological corrections based on the Schwartz – Saltykov algorithm (De Hoff and Rhines, 1972). This procedure does not cause artifacts such as the pronounced reduction of the scatter of data observed for the method proposed by Peterson (1996) (Pan, 2001; Hammer et al., 1999). Stereological corrections based on the Schwartz – Saltykov algorithm as adopted by Higgins (2000) and Armienti et al. (1994) have been carefully tested against real and theoretical cases and represent a satisfactory solution in most cases encountered in petrologic investigation. The method, in fact, also allows a proper identification of size distribution maxima in uni- and polymodal size distributions obtained from sieved particles, and a complete reconstruction of synthetic size distribution created with suitable algorithms. No correction for the shape factors has been taken into account since elongation of olivine crystals is not such to require this treatment of data (Higgins, 2000): in fact, data of larger samples that have been collected on three orthogonal planes and do not show significant differences. The unfolding procedure adopted includes a systematic check of the closure effect (Higgins, 2002) that is obtained by minimizing the difference between the areal mode (directly obtained from the binary image of the phase of interest) and the mode obtained from the second-order moment of the CSD (see Eq. (6)). The difference is kept within 2% by changing the interval between linearly binned size classes.
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Volumetric data are reported as CSD in bilogarithmic diagrams (Fig. 3a). We assume the size of the particles to be equal to the diameter D of the equivalent sphere, since most olivine grains exhibit almost equant habit. The working resolution for image acquisition was set at 1500 dpi ( c 17 Am/pixel). In order to avoid consideration of particles smaller than the thin section thickness ( c 30 Am) and to exclude ‘‘noise’’ arising from considering as ‘‘grains’’ clusters of a few pixels, a filter was used to cut particles smaller than 50 Am. Crystal size data may show truncation effects arising from resolution (left-hand truncation) of the adopted measurement device and from the size of the specimen (right-hand truncation). Evaluation of left-hand and right-hand truncation effects on the CSD is essential when studying these kinds of data sets before attempting any interpretation (Pickering et al., 1995). Left-hand truncation arises from the resolution of the image analysis procedure as a whole and is revealed by a decrease in the number density of the counted grains. This truncation is reduced when data are acquired with a greater resolution (Armienti et al., 1994). It is important to observe that resolution is controlled both by the resolution of the starting image and by errors introduced by image processing. Thus, due to the heavy image manipulations needed for image segmentation, in spite of the higher resolution of starting images, our device allows the determination of CSDs without truncation effects only for crystal sizes larger than 0.2 mm, with left-hand truncation appearing around 150 Am (Fig. 3a). Any interpretation of a ‘‘true’’ reduction of the number density of crystal smaller than 0.2 mm as due to coarsening processes would require the acquisition of images with a different device (e.g., mineralogy microscope). Right-hand truncation produces the flattening of size distributions on nearly horizontal trends at large size ranges in which not all the classes of size occur in the sections (Fig. 3a and b). This results in a constant value of the number density for classes that contain at most one particle. A logarithmic binning of size classes (i.e., logarithmic variation of the width of the class size) allows to partly avoid this effect by counting a larger number of grains in larger (and wider) classes, but the effect still appears at the limit of the distribution with an increase of the slope at the
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Fig. 3. Effects of left-hand and right-hand truncation shown for samples LANZ 24 and LANZ 15. (a) The typical effect of left-hand truncation is to reduce the number of counted individual crystals near the resolution limit of the acquisition procedure. This can be observed both in the CSDs computed from one thin section and from 10 thin sections. Right-hand truncation arises from the limited size of the explored thin section that allows the detection of only a few (at the limit only one) particles of large size in each size interval: this results in the constant value of N(D) revealed by the flattening of the CSD trend (empty circles). For the same choice of size interval, the effects may be avoided by analyzing a larger population of crystal (small diamonds). A different choice of size intervals would also reduce this effect as is evident in the CSDs obtained from computed logarithmic binning of the size interval (squares). (b) Right-hand truncation may not disappear, even when handling a large population of crystals (small diamonds) if the number of size intervals is too large: this is particularly evident in sample LANZ 15, where CSD is plotted against D. In this case, right-hand truncation is revealed by the constant values of log(N(D)) and by the shift of linearly binned CSD with respect to the logarithmic one.
right hand of the distribution. Right-hand truncation effects can been limited by enlarging the size of the measured sample. This is obtained by measuring CSDs of several (up to 10) thin sections of each mantle xenolith, each cut at a different positions (Fig. 3a and b). The adoption of a logarithmic binning for the calculation of CSD on a single thin section ensures satisfactory results for measurements performed on mantle xenoliths, as shown by the closely comparable results provided by CSDs obtained from cumulative regressions from several sections (up to
10) (Table 1). It is important to state that logarithmic binning tends to smoothen CSDs: in some instances, where necessary, we switched to linear binning of class intervals, considering data in regions far from left- or right-hand truncation effects. No account has been taken of the edge effect due to large grains only partly contained in the thin section, being cut by its borders. This would lead to some underestimates and instability of the value of N(D) for larger grains in samples where CSDs were measured in a single thin section (Fig. 4), but it only affects size classes that are
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Fig. 4. Determination of the reproducibility of the slope determination is performed by repeated acquisition of the CSD on the same thin section. This explains the close similarity of the distinct series. The edge effect (see text) is different for each run of acquisition since different areas were delimited for scanning, but it only affects size classes that are also biased by right-hand truncation. The box contains measured slopes from distinct procedures of acquisition, in the interval not affected by truncations (0.2 – 2.0 mm as shown by the arrows). Lines connect logN(D) values of contiguous size classes for which a nonzero value was obtained. Mean and standard deviation (r) are reported.
also biased by right-hand truncation . Edge effect is compensated by the very large number of total grains counted in samples where CSDs are measured on several sections. In the size range 0.2 – 25 mm, where truncation effects are avoided, all the samples show a linear dependence of log(N(D)) from log(D), with a negative slope, i.e., logN ðDÞ ¼ mlogðDÞ þ logC
ð4Þ
which implies that: N ðDÞ ¼ CDm
ð5Þ
where C is a constant, D is the equivalent diameter of the crystals, N(D) is the number of crystals per unit size per unit volume, and m is the fractal dimension of the size distribution (Turcotte, 1995). In the presentation of results, the fractal dimension of the crystal size distribution set is defined according to Eqs. (4) and (5). Repeated acquisitions of data from the same thin section (Fig. 4) allow an estimate of the error (1r ) of
the procedure of CSD slope determination, which, in most cases, is F 6% of the value of m and is always smaller than 12% (Table 1).
3. Results For all the mantle xenoliths, a relatively constant linear covariance is observed for olivine CSDs, allowing detection of a scale-invariant trend (Simakin et al., 1999; Tarquini and Armienti, 2001). Fig. 5a– c illustrates the measured size distributions for the three suites of mantle xenoliths from Lanzarote, Greene Point, and Pali Aike, while Table 1, for the same data sets, reports the values of the fractal dimensions and the number of counted crystals. As a reference, the size interval in which the scale-invariant relation has been measured is reported in Fig. 5a. CSDs reported in Fig. 5a represent the mean of 10 distinct measurements performed on each thin section. This gives them a ‘‘smooth’’ appearance and reveals fine detail of the CSDs that are not included in measurements reported in Fig. 5b, where results obtained from single
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Fig. 5. CSDs measured on Lanzarote (a), Greene Point (b), and Pali Aike (c) mantle xenoliths. (a) CSDs obtained from linear binning of the size interval, for measurements performed on single thin sections. In the first diagram of the series (top left), small squares show the results of N(D) determination obtained through the logarithmic binning of the size interval, according to the procedure of Higgins (2000). In the size interval without truncation effects (between the arrows), the slopes obtained with the two methods overlap for all samples. Note in the top-left diagram the scale of D in millimeters. (b) CSDs obtained from linear and logarithmic binning for measurements performed in single thin sections. Lines connect logN(D) values of contiguous size classes for which a nonzero value was obtained with the linear binning. (c) CSDs obtained from linear and logarithmic binning, for measurements performed in up to 10 thin sections, of Pali Aike and Lanzarote samples. These are largest xenoliths available in our sampling (Fup to 25 cm); thus, it was possible to cut thin sections in three orthogonal planes at distances larger than about 20 mm (the maximum size detected among olivine grains). In LANZ 24, there is no right-hand truncation in the CSD data set. The horizontal ‘‘tails’’ of the other xenoliths reveal that a larger number of sections should have been analyzed to get linear trends with the linear binning. This requirement is overcome when data are handled with the method of Higgins (2000).
measurements (and linear scaling of class sizes) are compared with CSDs computed with the logarithmically scaled class sizes. Fig. 5c reports measurements obtained from several thin sections of the same xenolith. Care was taken in cutting thin sections on three orthogonal planes and spacing the cuts at a distance larger that the largest detectable crystal. The two samples measured for Pali Aike are given as measurements on multiple sections. In both Lanzarote samples, the larger number of thin sections allows to insert in the data set a greater number of large crystals, thus greatly reducing the right-hand truncation effects for sample LANZ 24. The values obtained for the fractal dimension of LANZ 24 and
LANZ 15 agree within F 1r with measurements done on single sections (Table 1). The presence of horizontal ‘tails’ of CSD at large D values for the other samples reveals that the number of measured thin sections is still insufficient to rely only on linear binning of class sizes for large crystals. In the Lanzarote data set, where porphyroclastic textures are more widespread, the values of the fractal dimension of the CSDs are usually larger than in other areas, and the larger values of m are associated to cataclastic textures akin to mylonites (e.g., LANZ 3 with m = 3.81). On the contrary, granoblastic samples show the lowest values of m (e.g., LANZ 6b with m = 2.59). The accurate measurements performed for
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this data set reveal that the porphyroclastic texture is associated with a small relative maximum of the CSD, usually between 1 and 2 mm. Such detail cannot be detected in the Greene Point and Pali Aike xenoliths that are generally characterised by smaller values of m and by larger maximum sizes of olivine crystals. This is consistent with the less pronounced cataclastic textures observed in these peridotites. Pali Aike xenoliths are the only garnet-bearing rocks of the data set, but this has no influence on the scale-invariant relation between the number of crystals per unit size per unit volume and their size. If we define the ith order moment of the size distribution by the expression: Z l Mi ¼ Di N ðDÞdD; ð6Þ 0
it follows that for spherical particles, M0 is the number of crystals per unit volume, M1 is the total length of
283
crystals per unit volume, 2pM2 is the total area of crystals per unit volume, and 23 pM3 is the total volume of crystals per unit volume. These useful relations can be used to asses several textural data of the measured samples. Fig. 6 reports for six samples of Lanzarote the volume fraction of particles of given sizes obtained from the third-order moments of the measured CSDs. It is evident that in porphyroclastic xenoliths, smaller grains represent a larger volume fraction. This is not the case for granoblastic peridotites where the largest volume fractions are accounted for by larger grains. Second-order moment of CSDs allow the computation of the surface fraction of each size class; where a large fraction of grain falls among smaller grains, an even larger fraction of the surface is characteristic of the smaller grain sizes, like in sample LANZ 26.
Fig. 6. Calculated granulometric curve for porphyroclastic and granoblastic samples from Lanzarote. Volume fractions of grains of given size are computed starting from the third-order moment of crystal size distributions (Eq. (6)). It appears that in porphyroclastic samples, the mode at 0.5 mm is more pronounced, accounting for the large number of smaller crystals.
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4. Discussion of data The scale-invariant relation N(D) = CD m in the size range 0.2– 25 mm characterises mantle xenoliths brought to the surface by alkaline basalts in geological contexts ranging from intraplate hot spot magmatism (Lanzarote: Siena and Coltorti, 1993), to rift magmatism (Greene Point, Antarctica: Perinelli, 2000; Beccaluva et al., 1991) and subduction of a triple point (Stern et al., 1989; D’Orazio et al., 2000). All the samples derive from the lithospheric mantle from depths ranging from 20 to 80 km. Though the sampling is necessarily limited to volcanic areas where basaltic magmas rise to the surface, the scale invariance of olivine grain size distributions seems to be characteristic of upper mantle in disparate tectonic settings. In these Type I xenoliths, as in other mantle samples from around the world, olivine exhibits strong kink banding that is considered to be evidence of oriented stress: often kink banding evolves in true cataclastic textures in which olivine grains are disrupted to form a fine-grained matrix of neoblasts. Thus, the fragmentation model may provide an interpretation of the observed scale invariance. Following Turcotte (1995), if grains are ordered in grain size classes and f is the probability that a grain of ith order will fragment to give P particles of (i + 1)th order, the fractal dimension of the size distribution is given by: m ¼ 3*lnðP*f Þ=lnðPÞ:
ð7Þ
If f is allowed to vary between 1/P and 1, then 0 < m < 3. The most recurrent values obtained for fractal dimension of olivine size distributions in our xenoliths are around 2.6. If at each fragmentation eight grains of (i + 1)th order are generated (P = 8) and if only two are preserved at the following fragmentation ( f = 6/8), Eq. (7) gives m = 2.58. This simple model of crushing of a constant fraction of grains illustrates how recursive application of a very simple law can produce scale invariance effects. Different choices of P and f provide different values of m; all are, however, between 0 and 3. Indeed, many fractal dimensions obtained in strongly deformed samples are larger than 3 and do not conform to the simple model of Eq. (7). Values larger than 3 are sometimes reported in the literature for the
fractal dimensions of fragment distributions (see Turcotte, 1995); in our case, the development of a cataclastic texture may be accompanied by recrystallization of smaller grains of olivine. Thus, when larger grains are still crushing (increasing their number density), smaller ones may start to form larger neoblasts, decreasing in number (see the slight flexure in sample LANZ 3, Fig. 5a). This effect is well known in mantle xenoliths, where olivine neoblasts often show compositions different from those of porphyroclasts (e.g., Perinelli, 2000; Siena and Coltorti, 1993), and it may result in a counterclockwise rotation of CSD, whose fractal dimension is thus allowed to become larger than 3.
5. Conclusions Fig. 5c (see sample LANZ 15) shows that, when a logarithmic binning is adopted for regression of data collected on multiple sections, crystals with a maximum size up to 25 mm follow a scale-invariant relation. In general, olivine CSDs show scale invariance astride almost three orders of magnitude in the size interval 0.2 – 25 mm. Fragmentation models offer a possible interpretation for this fractal size distribution, providing a theoretical model that can reasonably predict a value for m c 2.6, as found in most samples. However, selective crushing of larger crystals and recrystallization of smaller neoblasts may provide an interpretation for distributions with m>3. The mean size of olivine grains is a fundamental parameter that is considered in rheological models of the mantle (e.g., Ranalli, 1998), and controls the transition from diffusion to dislocation creep mechanisms. It is evident that, if olivine crystal size distributions exhibit scale invariance, it is not possible to define a mean value for the size of this mineral in rheological maps, and this should be taken into account in modelling of mantle physical behaviour. Our data set (Fig. 6) also shows that the larger surface fraction of mantle grains is associated with smaller grains; thus, a large reaction surface area is available for reequilibration of smaller crystals through a diffusion mechanism. Measurements of CSDs in mantle xenolith can provide useful estimates of the total surface available for reactions of fluids percolating in mantle during metasomatic events.
P. Armienti, S. Tarquini / Lithos 65 (2002) 273–285
Acknowledgements This work was funded by the grant MURST 1998 no. 9804307319002, ‘‘Materiali terrestri ed analoghi sintetici ad alta pressione ed alta temperatura: proprieta’ fisiche, chimiche e reologiche.’’
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