## Abstract

^{40}Ar/^{39}Ar apparent age spectra have been measured for unusually retentive potassium feldspars (K-feldspar) from the South Cyclades Shear Zone, Ios, Greece. Our results imply that the Argon Partial Retention Zone (Ar PRZ) for the most retentive domains in potassium K-feldspar can expand into the ductile regime, even when temperatures in excess of about 400–450 °C apply. In such cases K-feldspar could be used as a geochronometer to estimate the timing and duration of deformation and metamorphism events. Therefore, we have reassessed traditional methods used to analyse Arrhenius plots by simulating the effect of step-heating experiments on argon loss. Fractal multidomain diffusion models were used, with theoretical distributions of diffusion domain size and volume. We discovered a Fundamental Asymmetry Principle that offers objective constraints on slope fitting to allow an analysis to be consistent with the multidomain diffusion hypothesis, and which consistently leads to the estimation of higher activation energies. Reanalysis of existing datasets is encouraged to allow reassessment of the significance of the average values reported. Retentive diffusion parameters for K-feldspar might prove to be commonplace.

The South Cyclades Shear Zone (SCSZ) is an approximately 1 km-thick bowed-up extensional ductile shear zone that defines the carapace of a deeply eroded gneiss dome that outcrops on the island of Ios in the Cycladic archipelago, Aegean Sea, Greece. The SCSZ is of geological importance as it is a key structural element in the tectonic evolution of the Cycladic Massif. It defines the carapace of the dome defining the first-recognized Aegean metamorphic core complex (Lister *et al.* 1984, 2007; Lister & Forster 1996). More significantly, Forster & Lister (2009) showed that the SCSZ operated from about 35 Ma, for about 5 Ma, and it therefore links the Eocene high-pressure stage of the evolution of the Cycladic eclogite–blueschist belt with the later lithosphere-scale stretching that took place during rollback of the Hellenic slab.

There have been several studies using ^{40}Ar/^{39}Ar geochronology on Ios (Lister & Baldwin 1996; Baldwin 1996; Baldwin & Lister 1998; Forster & Lister 2009). The results are often difficult to interpret because deformation occurred at relatively low temperatures, and was partitioned in discrete, often anastomosing zones (from the metre-scale to the <µm-scale). This geometry allowed preservation of lenses of relict material, on all scales, while the relatively low temperatures involved allowed the preservation of relatively old apparent ages. This is also true in zones where younger deformation has produced intense overprints of earlier fabrics, with minimum recrystallization. Baldwin & Lister (1998) suggested that the observed heterogeneity of apparent ages obtained using ^{40}Ar/^{39}Ar geochronology was made possible because the SCSZ operated in the Argon Partial Retention Zone (Ar PRZ), and under such conditions the argon system is only partially reset during overprinting events. By analysing the degree of resetting, or more particularly the degree of preservation of older gas reservoirs, they provided data that allowed modelling of the duration of specific events (using the ‘MacArgon’ software: Lister & Baldwin 1996). Baldwin & Lister (1998) suggested that ambient temperatures prior to deformation pulses were lower than about 280 °C, and that during its operation the SCSZ had been subject to short thermal pulses.

It was decided to investigate the question of thermal pulses during shear zone operation in more detail, and therefore argon geochronology was conducted along a traverse through the SCSZ (Forster & Lister 2009). The duration of a thermal pulse can be estimated if the following conditions apply: (1) relatively old apparent ages are preserved through a younger thermal event; (2) the peak temperature during the younger thermal event can be estimated using independent means, for example using metamorphic petrology; (3) loss of argon from relict fabrics is related to diffusion; and (4) accurate estimates are available as to the diffusion parameters that applied. At first sight it appeared that these conditions could be met. Relatively old apparent ages are preserved in K-feldspar. The SCSZ operated at temperatures sufficiently high as to allow growth of biotite in microdilation sites (i.e. >400 °C). Locally this growth appears to have taken place immediately adjacent to what appears to be new grown garnet, allowing estimates of the temperatures that applied locally in the SCSZ when it was operating to be in excess of about 450 °C (Spear 1993). Forster & Lister (2009) ascertained the timing of operation of the SCSZ by analysing grains that grew during specific deformation events during the life of the shear zone. By inference, the cores of relict Hercynian K-feldspar phenocrysts lost argon only by diffusion. The remaining issue is to provide accurate estimates of the diffusion parameters that applied, and this is the subject of this paper.

It is evident that the SCSZ operated at temperatures above what are loosely referred to as diffusional closure temperatures for the minerals in question. The operation of the shear zone may thus have been associated with short-lived thermal pulses, as suggested by Baldwin & Lister (1998). Since it is possible to eliminate potential heat sources such as advection, either due to the emplacement of magma or due to fluid percolation, the origin of these temperature excursions could be associated with the conversion of mechanical work to heat during plastic deformation. Baldwin & Lister (1998) inferred Arrhenius parameters close to the UCLA average reported by Lovera *et al.* (1997). If these parameters apply, the duration of the inferred thermal pulses would have to have been extremely short, as reported by Baldwin & Lister (1998). This conclusion is valid, however, if and only if K-feldspar is as unretentive as implied by these diffusion parameters. Our data potentially force us to conclude that this may not be the case. Forster & Lister (2009) were able to use cataclased and/or recrystallized K-feldspar to directly date the timing of different movements in the shear zone, and to show that partial resetting of the argon system only occurred when renewed motion in the shear zone led to deformation and/or recrystallization. These circumstances are possible only if K-feldspar in the SCSZ is far more retentive than these earlier estimates would allow.

K-feldspar can only be used as a geospeedometer to constrain the duration of these different events if we are able to accurately estimate the diffusion parameters. Traditionally, such estimates are provided by the analysis of Arrhenius plots. There is considerable latitude in respect to how such Arrhenius data can be analysed, however, as illustrated in Figure 1. Different strategies can be applied, either to reduce the activation energy obtained (Fig. 1a) or to maximize the result (Fig. 1b, c). Significant variation in the inferred activation results, depending on the strategy adopted. Without objective guidelines as to how to proceed, one might be tempted to err in the direction that provides the closest answer to the values reported for argon@UCLA. We can minimize the inferred activation energy if we do not take into account the results obtained from the first Arrhenius points (e.g. Fig. 1a: Baldwin & Lister 1998). This may be a valid approach as these steps are obtained from the first points in the heating sequence and, supposedly, pertain to degassing of the minute, least retentive, diffusion domains. They may be more susceptible to experimental error. Conversely, we can maximize the inferred activation energy if we ignore the overall trend, and select two or three points from the start of the sequence (Fig. 1c: Mahon *et al.* 1998).

## Fractals and multidomain diffusion models

In an effort to provide objective guidelines in respect to the analysis of Arrhenius plots we began to explore ramifications of multidomain diffusion (MDD) models based on fractal distributions of size and volume, using a modelling and simulation approach. As we know the values of the diffusion parameters we actually input, we are able to investigate why variation is obtained in estimates obtained by numerical analysis of the results. To achieve this result ‘Program eAr’ (available for download from http://rses.anu.edu.au/tectonics/programs/) was modified to allow simulation of argon release from arbitrary MDD models, and thereafter to allow the resultant Arrhenius plot to be analysed in the traditional fashion. The data required to be input to generate an arbitrary MDD model includes the diffusion parameters: including *E*, the activation energy, and *D*_{0}, the frequency factor; and a list of diffusion domains, specifying for each, its overall volume and a characteristic diffusion radius (in this case the half-thickness of a slab). The temperature history used in the simulated step-heating experiment must also be specified.

The temperature histories that we use below consist of sequences of one or more isothermal steps, with the temperature incremented by (for example) 50 °C between each step in the isothermal sequence. The temperature reached during an individual step can be the same as that achieved during the preceding heating step, or it can increase. The sequence of temperatures applied is thus monotonically upward. The step-heating schedules for the modelling below have a starting temperature of 400 °C, with the first step in an isothermal sequence set at 5 min. If the simulation uses several isothermal steps, subsequent steps double in their duration (e.g. heating takes place over 5, 10 or 20 min if three isothermal steps are applied in sequence). To simplify matters, if we utilize isothermal steps, the same number of isothermal steps will be applied for each temperature in the sequence.

In all simulations reported in this paper a constant value for the activation energy (*E*=75.0 kcal mol^{−1}) has been applied to all diffusion domains. Based on the trend line describing the best fit to the UCLA data reported by Lovera *et al.* (1997) the frequency factor is determined by:

We were drawn to self-similarity in terms of an explanation for the peculiar distributions of volume and diffusion domain radius in published data (Lovera *et al.* 1989, 1997). Therefore we used two fractal distributions of volume and diffusion domain size for comparative purposes, the fractal cube (Fig. 2a) and the Menger sponge (Fig. 2b). The distribution of volume and diffusion domain radius is shown in Table 1. It should be noted that the choice of fractal is not of particular significance, however; although in many ways the fractal cube allows simulation of the effect of a cubic diffusion domain with a rough surface. In contrast, the Menger Sponge can simulate some of the effects of recrystallization and grain growth or, conversely, the effect of increasing breakdown of the lattice structure in K-feldspar grains in a weathering environment, or when stewing slowly in a sedimentary basin (e.g. Mahon *et al.* 1998; cf. Lee 1995 and Parsons *et al.* 1999).

It should also be noted that use of fractal MDD models offers a numerical approximation to quantify the effects of roughness, for as temperatures increase so does the characteristic distance associated with diffusion. In effect the domains become smoother. This means that it is potentially futile to search for a one to one correspondence between elements in the microstructure and individual diffusion domain sizes. Diffusion from lamellae surrounded by what is effectively a network of pipe-like fast escape pathways for argon expelled from the surrounding lattice (Fitz Gerald *et al.* 2006) may be approximated by a MDD model, for example.

The choice of fractal is not particularly significant, although interesting analogies can be drawn in consequence of any particular choice (Ben-Avraham & Havlin 2000; Landrenau 2000). In the fractal cube (http://txspace.tamu.edu/handle/1969.1/3776?show= full) the largest diffusion domain is a cube, with six cubic diffusion domains of dimension half of that cube edge set upon each face. Upon each of the five free faces of those cubes we then attach five cubes of dimension half of that cube edge, and so on. Eventually an octahedron with holes will be outlined, defining a fractal with dimension [(log 5)/(log 2)=2.32]. The first iteration generates six cubes with the edge of each cube half that of the original cube. The second iteration generates five cubes for each of the cubes generated in the first iteration, with the edge of each cube half that of the first iteration cubes (a total of 30 cubes in total). Subsequent iterations continue in the same fashion, halving the cube edge each time. For the Menger Sponge (http://en.wikipedia.org/wiki/Menger_sponge) the holes are assumed to define ‘intact’ diffusion domains, with the sponge considered to be a mineral lattice that is highly diffusive because of the presence of defects or artifacts (defining a fractal with dimension 2.73). It should be noted that other geometries could also be considered if one attempts to more exactly simulate the effect of surface roughness for a 3D diffusion domain geometry.

## The Fundamental Asymmetry Principle

An important result emerged, which we have termed the Fundamental Asymmetry Principle (or FAP), since it is implicit in (and required by) any MDD model. The FAP applies to any line fitted to data from a sequence of step-heating experiments in which the temperature applied from step to step may be the same, or higher, but may not decrease. Lovera *et al.* (1997) set out the basis of the MDD method, and shows how a straight line can be fitted to a set of Arrhenius data points, under specific circumstances. The fitted line will faithfully replicate the essential Arrhenius parameters only as long as the data points are derived from a set of partially degassed diffusion domains. In part, this is because the estimated diffusivity from any MDD model will progressively decrease as the less retentive domains are degassed. Mathematically, it follows that, to be consistent with a MDD model, a line fitted to any sequence of Arrhenius points must divide the population by rank order. Points from data obtained earlier in the sequence of step-heating experiments must lie on the fitted line, or to the right of it. Points from data obtained later in the sequence must lie on the fitted line, or to the left of it. Violations of the FAP will lead to consistently underestimated values for the activation energy (*E*) and for the frequency factor (*D*_{0}).

## Variation owing to domain size range

Figure 3 shows modelling for an Arrhenius plot derived from a single diffusion domain, and then the variation that would be obtained if eight then 16 iterations of a fractal cube were utilized to determine the volume–size distribution. This volume–size distribution is shown in Table 1. It is evident that smaller domain sizes define an increasing percentage of the total volume as the depth of the fractal size–volume relationship increases. The simulations show that these smaller grains have a significant impact on the Arrhenius plots that will be obtained.

As described by Lovera *et al.* (1997) for a single diffusion domain (Fig. 3a), we obtain a single straight line and we are able to accurately determine the activation energy used in the simulation (*E*=75 kcal mol^{−1}). This is also true for a volume–size distribution defined by eight iterations of the fractal cube. However, at higher temperatures the simulation shows a deviation from this straight-line trend. As noted by Lovera *et al.* (1997), this deviation begins to occur as degassing of smaller domains is completed. Nevertheless, applying the Fundamental Asymmetry Principle we were still able to obtain an accurate estimate of the activation energy (Fig. 3b).

In the third simulation of this type we utilize a volume–size distribution defined by 16 iterations of the fractal cube (Fig. 3c). The result is that the initial slope is no longer discernible: a direct consequence of this expanded volume–size distribution. Again, as noted by Lovera *et al.* (1997), the deviation is what is to be expected once smaller grains have completely degassed. The initial slope is no longer visible because, even at the lowest temperatures used in the step-heating experiment, there are smaller grains that have completely degassed. In these circumstances, any attempt at line-fitting will result in a significant underestimation of the activation energy. Using a standard statistical approach to determine the line of best fit (e.g. least squares, allowing a few outliers to be rejected), a lesser value for the activation energy is obtained. Coincidentally, the value obtained (*E*=46.5 kcal mol^{−1}) is close to the UCLA average (*E*=46 kcal mol^{−1}). We discover that application of the Fundamental Asymmetry Principle allows by far the most accurate estimates, although only two points could be utilized.

Although two points are all that is required to define a maximum slope (thereby here predicting *E*=72.0 kcal mol^{−1}) this approach would be questionable on numerical grounds if real data were being utilized, since errors may be introduced in many ways. Errors that originate from the actual measurements need to be taken into account, which may require some FAP violations to be tolerated. Application of standard statistical methods (e.g. least squares, allowing a few outliers to be rejected) will produce results that consistently underestimate the Arrhenius parameters. Although strict adherence to the Fundamental Asymmetry Principle sometimes requires a choice of only two points to define the fitted line, perhaps this practice should be avoided. Use of too few points may lead measurement inaccuracies to cause the use of the FAP to somewhat overestimate the retentivity. Even so, leaving the topic of error in real measurements for later discussion, theoretical simulation of the effect of step-heating experiments on MDD models show that application of the Fundamental Asymmetry Principle provides the minimum underestimate of the actual value of the activation energy used in these simulations, whereas allowing FAP violations significantly increases the error in any particular determination.

## The effect of isothermal duplicates

Lovera *et al.* (1997) pointed out that the use of sequences of isothermal steps in a step-heating experiment provides a strategy that allows discrimination of Arrhenius data that has been affected by degassing of less retentive domains. The experiments shown in Figure 3 were, therefore, repeated using sequences of isothermal steps (see Fig. 4). There are effects that relate to the number of isothermal steps, but in this paper we report only on the consequence of two steps at each temperature. It is evident that once degassing of less retentive domains begins to occur, the estimates of *E* and *D*_{0}/*r*^{2} in these two steps begin to differ. The effect is evident only on the last step of the simulation for a single diffusion domain (Fig. 4a). Simulations using isothermal steps on the volume–size distribution for eight iterations of the fractal cube show that the effect is evident immediately the change of slope begins to take place (Fig. 4b). Simulations using isothermal steps on the volume–size distribution for 16 iterations of the fractal cube show that the feathering effect can be evident from the outset (e.g. Fig. 4c). Interestingly, the variation obtained by including two isothermal steps allows recognition that the feathering is taking place. A fitted line obeying the Fundamental Asymmetry Principle allows a far more accurate estimate of the activation energy, *E*=69.5 kcal mol^{−1} (i.e. with an error of *c.* 7%, Fig. 4c), than the line fitted ignoring the initial steps, and which yielded *E*≈46.5 kcal mol^{−1} (i.e. with an error of *c.* 40%, Fig. 3c).

## Variation owing to discontinuous domain size ranges

The final type of simulation explored in this paper considers the effect of variation in fractal dimension, and the effects of non-overlapping but limited domain size ranges.

Figure 5a shows an Arrhenius plot derived for a MDD model based on four iterations of a Menger Sponge. This should be compared with Figure 3b, which shows an Arrhenius plot derived for a MDD model based on eight iterations of a fractal cube. A fractal cube volume–size relation has a fractal dimension of 2.32, whereas, with a fractal dimension of 2.73, the Menger Sponge will always have a larger volume fraction of small grains (Table 1). An increase in fractal dimension leads to higher values of *D*_{0}/*r*^{2} being estimated.

The effect of non-overlapping but limited domain size ranges similarly causes larger offsets between the value of *D*_{0}/*r*^{2} estimated for the largest domains and the average values determined by the fitted line. Figure 5b shows the effect of a population of large domains co-existing with a population of much smaller domains, with a volume–size distribution defined by a Menger Sponge. Four iterations of the Menger Sponge were allowed to take place before new diffusion domains were added to the volume–size distribution. The smaller diffusion domains are approximately 0.05–1% of the size of the large domain, but they account for around 22% of the total volume. The result is a marked offset in the Arrhenius plot.

A feathering effect is observed when the smaller diffusion domains are very much smaller than the larger domains. In Figure 5c, 16 iterations of the Menger Sponge are allowed to take place before new diffusion domains are added to the volume–size distribution. The smaller diffusion domains are now 4–5 orders of magnitude smaller than the smaller domains in the volume–size distribution used in the previous simulation, and the smaller domains now account for a mere approximately 3% of the total volume. The result is a marked deviation in slope, and a trend away from the maximum slope that represents an accurate estimation of the actual activation energy (*E*=75 kcal mol^{−1}). The inflection in the gradient inferred from the Arrhenius plot results from the population of minute diffusion domains defined by the Menger Sponge. This is interesting because, as we will show later in this paper, actual data from real K-feldspar often reflects similar inflections, and such effects have not been taken into account previously in descriptions of variations expected as the result of multidomain diffusion theory (Lovera *et al.* 1997). Irrespective of the above considerations, application of the Fundamental Asymmetry Principle to the Arrhenius data obtained in these simulations allows more accurate estimates of the activation energy.

The effect of ‘iteration level’ may be important since the effect of diffusion in a deformed crystal might be simulated by switching the fractal dimension early in the iteration process, and allowing a greater number of iterations to define the population of smaller grains in the Menger Sponge.

## Application of the Fundamental Asymmetry Principle to the analysis of real data

The Fundamental Asymmetry Principle can be readily applied to the analysis of real data. Here we compare and contrast the results of Arrhenius data analysed so that the fitted line does not violate the FAP, with the results of the application of statistical methods requiring a minimum number of three or more data points. From the theoretical simulations it is evident that effects related to fractal feathering lead to consistent underestimates of the actual activation energy. The same might be expected in real data.

The data from the step-heating experiments performed on K-feldspar from the South Cyclades Shear Zone were therefore re-evaluated, using ‘Program eAr’ to implement methods as described in this paper. Figure 6a, c, e show three examples for which the data are least scattered, and the result obtained using a line fitted using least-squares regression, but allowing the rejection of outliers. Figure 6b, d, f illustrate the reanalysis of these data in a way that is compliant with the Fundamental Asymmetry Principle (i.e. no FAP violations allowed). There is a systematic increase in the magnitude of the activation energy estimated. Similarly, Figure 7 shows more scattered Arrhenius data, with lines fitted using least-squares regression, allowing the rejection of outliers. Figure 8 shows the effect of reanalysis of these data with no FAP violations allowed. There are quite significant increases in the magnitude of the activation energy estimated. It is also evident that these data provide examples of inflections of the type illustrated in Figure 5c. By analogy, this feathering might be taken to imply a fractal MDD model simulating a high degree of roughness and/or heterogeneity in the domain size distribution.

A comparison of the apparent retentivity of the samples determined utilizing these two contrasting strategies is provided in Table 2. This shows that application of the Fundamental Asymmetry Principle causes the average of the activation energies determined to increase from *c*. 60 kcal mol^{−1} to a total of *c.* 73 kcal mol^{−1}. This is *c*. 1.6 times the UCLA average reported by Lovera *et al.* (1997).

Figure 9a, b shows the inferred diffusion parameters plotted on a graph modified from Lovera *et al.* (1997). Figure 9a shows data derived by fitting a line through the Arrhenius points based on the methods developed at UCLA, as described by Lovera *et al.* (1997). The values obtained are significantly more retentive than the UCLA average (activation energy, *E*=46 kcal mol^{−1}), although they fall approximately on the same trend line and within the limits of the spread of data reported by Lovera *et al.* (1997). Figure 9b shows data derived by fitting a line through the Arrhenius points based on the methods developed in this paper, requiring compliance with the Fundamental Asymmetry Principle.

## Discussion

This study was conducted in order to help us analyse Arrhenius data from a sequence of step-heating experiments applied to K-feldspar from the South Cyclades Shear Zone. We obtained ‘suspiciously high’ values for the activation energy, and therefore put some effort into ensuring that there were no systematic measurement errors (e.g. as would result if temperature was consistently overestimated as a result of calibration errors). Finally, examining the results of Baldwin & Lister (1998) from the same rocks (Fig. 1a), we realized that there were numerical issues in respect to the way different authors analysed their data. Baldwin & Lister (1998) reported a value for activation energy (*E*=46.5 kcal mol^{−1}) and this is a value that is close to the UCLA average (Lovera *et al.* 1997). Figure 1a shows, however, that the line of best fit was obtained for five points, rejecting outliers defined by the first two steps. If we used the initial steps, however, we obtained a higher estimate for the activation energy (Fig. 1b). Is one estimate better than the other? The work conducted in this paper was an attempt to resolve this question.

In one aspect we obtained a clear answer. Mathematically, there is a fundamental asymmetry required by the set of diffusion equations in an MDD experiment. Arrhenius data points for domains less retentive than those used to determine the line of best fit must fall on, or to the right of, the line, while data points for domains more retentive than those used to determine the line of best fit must fall on, or to the left of, that same line. Baldwin & Lister (1998) were able to obtain a value close to the 46 kcal mol^{−1} UCLA average, but only by (inadvertent) violation of the Fundamental Asymmetry Principle (FAP). Otherwise they would have obtained higher values, but, again, not as high as reported in this study.

The activation energy estimated in Figure 1b is *E*=60.2 kcal mol^{−1}. This is lower than the values obtained in our analysis of K-feldspar from the SCSZ, but it is nevertheless close to the minimum values we obtained. Note, however, that Baldwin & Lister (1998) did not use a step-heating schedule that involved isothermal duplicates, whereas our simulations show that this is a key aspect if one wants to avoid underestimates of the actual activation energy. In any analysis of Arrhenius data, systematic application of the FAP leads consistently to higher estimates of activation energy.

The Arrhenius data obtained by applying the FAP (Fig. 9b) can be seen, in part, to plot outside of the range of values recorded by Lovera *et al.* (1997) from the extensive UCLA dataset. Are these values realistic? We cannot say. Step-heating experiments with real K-feldspar are as well behaved as theoretical considerations allow.

We have shown that there are issues with the different ways that numerical methods can be applied to the Arrhenius data. Described simply, the problem is how to select which points to include in a numerical analysis of a line of best fit when attempting to determine values for activation energy and frequency factor on an Arrhenius plot. We illustrated this point in Figure 1, first using data from Baldwin & Lister (1998). These authors chose Arrhenius points (Fig. 1a) that took no account of outliers defined by the data from the least retentive domains, inadvertently violating the FAP, but nevertheless thereby producing a lower estimate for the activation energy. Coincidentally, this value was close to the UCLA average. In Figure 1c we illustrate another dataset, this time from Mahon *et al.* (1998) who appear to have used as few as two or three data points to determine the slope. This strategy will always maximize the estimate obtained for the activation energy, which otherwise in this case would have been far less than the UCLA average. Interestingly, this analysis appears consistent with the FAP, and is therefore consistent with the principles we advocate in this paper.

It is beyond the scope of this paper to reanalyse data from the entire UCLA dataset. We would have to begin by eliminating all experiments that did not perform isothermal duplicates, for example, as such experiments lose vital information and invariably allow lower estimates of the activation energy to be obtained. Perhaps, indeed, systematic application of the FAP would lead to a different distribution of Arrhenius data.

The SCSZ K-feldspar does appear to be unusually retentive. The ‘MacArgon’ program (adapted for use with the MacOSX operating systems) was used to compute closure temperatures for cooling through the Argon Partial Retention Zone (Table 3), for a range of cooling rates, and for zero pressure as well as for a pressure of 10 kbar (100 MPa). Calculations for a single sample (AG03-07) are tabulated. A fit of the Arrhenius data that produces values close to the UCLA average can be obtained if the first isothermal duplicate is discounted by discarding the first two Arrhenius points as outliers, and using a mean square regression on the next four isothermal duplicates (i.e. allowing FAP violations). The K-feldspar is relatively retentive compared to the Lovera *et al.* (1997) analysis of the UCLA archive [*E*=46.9 kcal mol^{−1}, log(*D*_{0}/*r*^{2})=5.37] because there is a 4 orders of magnitude difference in *D*_{0}/*r*^{2} values for the most retentive domains compared with data for the fitted line. A fit of the Arrhenius data produces values close to the ANU average (*E*≈73 kcal mol^{−1}, see Table 2) if no significant FAP violations are allowed, and a minimum of three Arrhenius points included. There is now a 7 orders of magnitude difference in *D*_{0}/*r*^{2} values for the most retentive domains compared with data for the fitted line, and this leads to a considerable increase in their estimated closure temperatures. We thus suggest that it is not reasonable to expect that all K-feldspar should display Arrhenius parameters that compare with the UCLA average (Lovera *et al.*, 1997) or with the gem-quality Madagascar K-feldspar studied by Arnaud & Kelley (1997).

We note that there are no particular issues with the computation of lines of best fit in many experiments, but for some experiments there can be significant outliers. These same experiments commonly show inflection points in the variation of gradient. We have shown that such inflection points can be readily explained by effects related to fractal feathering and limits to the size ranges encompassed by coexisting fractal populations. In terms of numerical analysis, the MDD method allows the effect of the roughness of a diffusion domain to be simulated, and/or the effect of variations in the spacing of fast-diffusion escape pathways for the argon released from a diffusion domain. Effects as reported in this paper could be obtained by allowing fractal variation in the geometry of fast-diffusion pathways that allow escape of argon; for example, a chicken-wire mesh of irregularly spaced pipes acting as fast escape pathways (cf. Fitz Gerald *et al.* 2006). We suggest that the MDD model merely approximates the effect of such non-ideal geometries; for example, the effect of a rough-edged diffusion domain. We suggest fractal feathering in Arrhenius plots is commonplace, especially in samples altered by weathering or samples that have been stewed at low temperature in a sedimentary basin before they have been subject to step-heating experiments (e.g. Mahon *et al.* 1998). Such effects introduce uncertainty that so far has not been taken into account in attempts to extract cooling histories from K-feldspar. The time dependence of microstructural changes during deformation and metamorphism (or while stewing in a sedimentary basin) might be approximated by increasing depth in the fractal distribution of size and volume. Extraction of useful geospeedometry data (e.g. with respect to the cooling path) from a microstructure undergoing such a progressive time-dependent evolution would become increasingly problematic.

## Conclusion

The Argon Partial Retention Zone using the UCLA average value of *E* of 46 kcal mol^{−1} would restrict the resting state of the SCSZ so that it could not lie within the ductile field (*T*>350 °C) if such values applied. If the ANU average is applied then argon enters the retentive zone: (1) the resting state of the SCSZ can lie within the ductile field; (2) the duration of the thermal pulse associated with the operation of the SCSZ can be as long as the 5 Ma we estimate that this shear zone was active; (3) a far smaller difference in temperature may apply between the shear zone resting state and its operating state. In other words, the more retentive values for the Arrhenius parameters allow a far more realistic geodynamic scenario to be modelled. Therefore, we conclude that the retentive values we have estimated for the Arrhenius parameters are not unreasonable given the constraints offered by the geological environment in which they are found. As noted by Baldwin & Lister (1998), thermal pulses associated with the operation of the SCSZ have to be extremely (and unrealistically) short if Arrhenius values as defined by the UCLA average apply.

There are issues to be resolved in respect to the analysis of errors, but these are minimized if step-heating experiments are utilized that involve sequences of isothermal duplicates. Fractal feathering does appear to be commonplace, and this, in part, could explain the differences in the average values obtained for the Arrhenius parameters for argon@ANU compared with argon@UCLA. Nevertheless, any analysis of Arrhenius data from step-heating experiments using K-feldspar should take account of the Fundamental Asymmetry Principle as this is an inherent part of any multidomain diffusion model. If the Fundamental Asymmetry Principle is not applied then numerical analysis will invariably underestimate the actual value of activation energy used in simulating the effect of step-heating experiments on fractal volume–size distributions.

## Acknowledgments

M. A. Forster acknowledges the support of an Australian Research Fellowship provided by the Australian Research Council (ARC). Research support provided by ARC Discovery Grants DP0449975 ‘Revisiting the Alpine Paradigm: The Role of Inversion Cycles in the Evolution of the European Alps’ and DP0343646 ‘Tectonic Reconstruction of the Evolution of the Alpine–Himalayan Orogenic Chain’. Irradiations were funded by the Australian Institute of Nuclear Science and Engineering (AINSE Award Grants) and facilitated by the Australian Nuclear Science and Technology Organization (ANSTO) at Lucas Heights, New South Wales, Australia. Argon analyses and microprobe analyses were carried out at the Research School of Earth Sciences Laboratories at the Australian National University. Spectra analysis was performed with ‘Program eAr’ written by G. S. Lister. The Institute of Geological and Mining Exploration (IGME) provided permission for fieldwork and sample collection in Greece. O. Lovera and an anonymous reviewer are thanked for their contribution to the final paper.

- © The Geological Society of London 2010