# European Journal of Mineralogy

- E. Schweizerbart'sche Verlagsbuchhandlung, D-70176 Stuttgart

## Abstract

We study oscillatory banding in plagioclase and pyroxene crystals from different volcanic environments. We use digitized images from an optical microscope. Assuming that oscillatory banding is selfaffine, we calculate the Hurst coefficient *H*, for each crystal, used here as a phenomenological characterization of banding. We find that calculated values of *H* are reproducible, not only in crystals from the same thin section, but also in rocks from the same environment. Plagioclase crystals from different volcanoes exhibit different values of the Hurst coefficient *H.*

## Introduction

Chemical banding in minerals is long known in nearly all geological environments (Rutley, 1875; Shore & Fowler, 1996); in some minerals (*e.g.*, plagioclase) it can easily be observed under an optical microscope. It has been studied extensively, but a clear understanding of this common phenomenon has not yet been reached.

Attempts to understand the origin of oscillatory zoning can broadly be classified into two categories: In the first one, banding is thought to reflect deterministic dynamics intrinsic of the crystal in relation with its immediate environment (*e.g*, Haase *et al.*, 1980; Allègre *et al.*, 1981; Higman & Pearce, 1993). Here, physico-chemical variations of the crystallizing system are not considered to influence banding significantly, possibly because they take place on time scales very different from those of crystal growth. Methods based on non-linear analysis and deterministic chaos would seem fit to recognize deterministic structures in band growth (see later).

Studies falling in the second category are those that consider chemical banding as caused by variations of intensive variables, and/or in the chemistry of the crystallizing system (*e.g.*, Jamtveit *et al.*, 1993; Holten *et al.*, 1997). Environment changes, in this case, are implicitly considered to take place on time scales of the same order of that of crystal growth.

Chemical banding has been studied in a variety of different ways. Optical and electronic microscopes can detect banding in a number of minerals, whereas ion microprobes can yield quantitative information in any chemically zoned mineral. Cathodoluminescence and Nomarski interferometry have also been used. In this work we use digitized images of banded minerals obtained by optical microscopy. We use fractals to obtain a phenomenological characterization of oscillatory banding in various minerals. We then show that similar minerals from different volcanic materials are characterized by different scaling properties.

## Experimental methods

Thin-section samples were inspected under a Zeiss optical microscope, mounted with a B/W TV camera (Sony SSC-C370P, 752X582 square pixels). Camera images are sent to a personal computer, where they are captured with a Flashpoint digitizing board and saved in one of the standard formats. Grey-scale profiles were obtained by means of an image analysis computer program, the University of Texas Health Science Center in San Antonio UTHSCA *Image-Tool*, freely available on the site ftp://maxrad6.uthscsa.edu. Information about this software can be found at http://ddsdx.uthscsa.edu/. Fractal and non-linear analysis calculations were made with the Chaos Data Analyzer (CDA) package, produced by Physics Academic Software. Control calculations of the Hurst exponent were also made with different PC programs.

## Non-linear analysis

Crystals grow in concentric shells, thus a compositional profile going from the center outwards represents a sequence ordered in time. Equally spaced points of the sequence, however, would be also equally spaced in time only if the crystal growth velocity had always been constant, a very unrealistic assumption. This prevents a rigorous application of non-linear methods, which always require the points in the time series to be equally spaced in time. In spite of this very serious limitation, we have applied some of these methods to our longest profiles, looking for any hint of determinism, to be studied eventually with some other methods. Phase portraits, correlation dimension, Fourier spectra and Non-linear Forecasting all fail to suggest the existence of orderly structures, periodicities or low dimensional chaos. Thus we agree with Holten *et al.* (1997) that empirical evidence does not support the idea that oscillatory banding is due to self-organization and local interface kinetics.

## Fractal analysis

Fractals are becoming increasingly popular in earth sciences. In fact they were introduced with the very purpose of describing the irregular geometry of natural shapes, plenty of which are encountered in geology, geomorphology, mineralogy and in relates sciences, including geophysics (Mandelbrot, 1983). It has been suggested (Halden & Hawthorne, 1993; Holten *et al.*, 1997) that oscillatory zoning of minerals can be described by fractal geometry, and can be characterized by the Hurst coefficient, *H*, that can vary between 0 and 1. A random walk function *X*(w) has *H* = 0.5 if at each step the function has the same probability of increasing or decreasing. If the probability that an “up step” is followed by another “up step” is larger than 0.5, then the function is said to be persistent, and 0.5 < *H* ≤ 1. If an “up step” is more likely to be followed by a “down step” then the function is called antipersistent, and 0 ≤ *H* < 0.5. If the profile really is selfaffine (fractal), then a plot of log (*RMS*) *vs.* log (w) yields a linear trend with slope *H* when w varies a few orders of magnitude. For large values of w the trend rolls over due to saturation; thus the slope has to be calculated in the first part of the trend, typically not exceeding the square root of w. For a more detailed introduction to the Hurst coefficient, to the use of fractals in the earth sciences and practical algorithms, see Barton & La Pointe (1995), Halden & Hawthorne (1993) and Holten *et al.* (1997).

## Results and discussion

Image analysis cannot yield quantitative compositional data about chemically banded crystals, and optical microscopy is actually a rather crude method of investigation, but it has some advantages: it is fast, cheap and non-destructive.

Five to ten profiles were studied for each crystal (totalling over 1,000 profiles). For all samples, plots of log (*RMS*) *vs.* log (w) are reasonably linear for length scales spanning 1.5–2 orders of magnitude (*e.g.*, see Fig. 1c and 2c); we *assume* that oscillatory banding in the examined samples is self-affine (fractal). As explained earlier, self-affinity can be characterized by the Hurst coefficient *H.* Table 1 shows the average values of *H* for each volcano, with their statistical errors (1 σ level). It shows that the overall geological error is so small that differences between different groups of samples can be considered significant. The calculated values of *H* are quite robust. With many test runs, we have verified that they are relatively independent from magnification and from the orientation of the crystal; this is what should be expected if the profiles really are self-affine.

The values of *H* for pyroxene are always smaller than 0.5, and show antipersistence. They are grouped around 0.4, (in agreement with previous studies) and their distribution is indistinguishable from a Gaussian one (see Fig. 3a). The *H* values for plagioclase are much more variable, and range from 0.4 (mild antipersistence) to 0.72 (persistence). Their distribution can be suspected to be bimodal or multimodal (Fig. 3b), clearly strengthened by the observation that *H* values for plagioclase samples are well grouped for each volcano. Thus one may hope that plagioclase can be more useful in discriminating among different geological environments. In each single rock the values of *H* for plagioclase are always higher than those for pyroxene.

Fig. 4 summarizes a very crude attempt to characterize each lava by its values of *H*. Here *H* values for pyroxene crystals are plotted against those for plagioclase crystals. It must be said that pyroxene and plagioclase crystals for each region are from the same lavas, but not always from the same thin section, since suitably banded crystals are not so easily found. Different samples are well distinguished in Fig. 4.

The most interesting finding is the large difference between lavas and pyroxenites of Vesuvius. The bulk chemistries of the magmas from which the lavas and pyroxenites crystallized were roughly similar (Hermes & Cornell, 1981). Both the lavas and the pyroxenites crystallized in a range of pressures, but presumably the pyroxenites spent longer times at larger depths. We do not know what caused the difference in banding, which is reflected in the difference in the values of *H*. As explained in a previous section we find no empirical evidence for periodicities or self-organization of the oscillatory banding, so we cannot look for explanations of this sort. The plagioclase in the pyroxenites has higher values of *H*, showing persistence, that is a more regularly varying chemical composition than in the lavas. It can be imagined that a deeper environment may be more protected from geological noise than a shallower one. This is our best suggestion for the differences in the values of *H* in the plagioclase crystals, which, we must confess, leaves us very unsatisfied. On the other hand, pyroxenes have similar values of *H* in the lavas and the pyroxenites, and seem insensitive to the different environment.

The meaning of the term “geological noise” is open to discussion. If crystal-growth rates in amagma must be in the order of 10–11 to 10–13 m/s in order to produce oscillatory banding (Shore & Fowler, 1996), then geological disturbances taking place on time scales of years or tens of years are instantaneous from the point of view of crystal growth, thus can be considered as noise.

Clearly much more work is needed in order to clarify these problems.

## Appendix: Samples of study

**Roccamonfina** (Appleton, 1972). Studied samples are leucitic tephrites with porphyric texture. Phenocrysts are clinopyroxene, plagioclase and leucite.

**Somma-Vesuvius**- Lavas from the 1944 eruption are leucitic phonolite-tephrites. Phenocrysts are clinopyroxene, plagioclasí and abundant leucite. Pyroxenites (Hermes & Cornell, 1981) are medium-grained cumulite rocks, mainly composed by clinopyroxene, with subordinate biotite, olivine and plagioclase.

**Ernici** (Civetta *et al.*, 1979). Studied samples are leucitic tephrites with porphyric texture. Phenocrysts are clinopyroxene and leucite.

**Stromboli** (Horning-Kjarsgaard *et al.*, 1993). Zero-age shoshonitis basalt samples have porphyric texture. Phenocrysts are plagioclase, clinopyroxene and rare olivine.

**Pantelleria** (Civetta *et al.*, 1984). Studied samples are porphyric basalts, with phenocrysts of plagioclase, clinopyroxene and rare olivine.

**Capraia** (Di Girolamo, 1978). Studied samples are porphyric latites. Phenocrysts are plagioclase, clinopyroxene and biotite, with relics of olivine.

## Acknowledgments

We thank an anonymous referee for careful reading of this manuscript and useful comments.

- Received 14 March 2000.
- Modified version received 14 November 2000.
- Accepted 11 December 2000.