Organization of oscillatory zoning in zircon: Analysis, scaling, geochemistry, and model ofa zircon from Kipawa, Quebec, Canada
ANTHONY FOWLER,1,* A NDREAS PROKOPH,1 RICHARD STERN,2 and CELINE DUPUIS3
1Ottawa-Carleton Geoscience Centre and Department of Earth Sciences, University of Ottawa, 140 Louis Pasteur,Ottawa, Ontario K1N 6N5, Canada
2Geological Survey of Canada, 601 Booth Street, Ottawa, Ontario K1A OE8, Canada3Department of Earth Sciences, Biological and Geological Building, University of Western Ontario, London, Ontario N6A 5B7, Canada
(Received January 29, 2001;accepted in revised form June 29, 2001)
Zircon is a widespread accessory mineral of evolved igne-ous, sedimentary, and metamorphic rocks. It is an excellentgeochronometer because during crystallization, U readily sub-stitutes into its lattice whereas Pb is excluded, and becausediffusivities of U and Pb within it are generally small. Zircon isoften oscillatory zoned, characterized by fluctuations in submi-cron to millimeters wide optically and chemically distinctbands parallel to crystal faces. Largely because of its use ingeochronology, the formation of oscillatory zoning (OZ) ofzircon has been the subject of several recent studies (Haldenand Hawthorne, 1993; Vavra, 1994; Mattinson et al., 1996;Pidgeon et al., 1998; Hoskin, 2000).
A theme of several articles has been the characterization ofthe zoning and its interpretation. A central idea is that undercertain circumstances, crystal growth modifies its local envi-ronment and hence its own growth conditions. Such feedbackproduces nonlinearities that may cause chemical oscillatorypatterns to spontaneously emerge (e.g., zoning). These patternsare said to be self-organized—that is, they arise from intrinsiccrystal growth processes rather than extrinsic or bulk system-scale fluctuations. The idea behind characterization of the zon-ing is to search for organization within the OZ, as one expectsa measure of the zoning (e.g., back-scattered electron [BSE]signal intensity, zone thickness) to be related to the processesthat led to zone formation. Therefore, a central objective of this
work is to examine the data of zircon oscillatory zonation for anunderlying organization or pattern.
The quantification of the nature of the zoning (e.g., nonlin-ear, chaotic, stochastic, or harmonic) allows one, with geolog-ical or geochemical information, to better understand the dy-namical conditions that gave rise to the zoning. Although OZhas been known for well over a century and occurs in everymajor mineral class (Shore and Fowler, 1996), there is noagreement on its mode of origin. This stems in part from thefact that there are likely numerous related modes of origin andcrystals often have only a few zones (�50) that can be verynarrow, making quantitative analysis difficult. Hence, an im-portant aspect of this work on zircon was the acquisition oflarge crystals with abundant and wide zones. Here, we build onexisting work through the use of new ion microprobe analyseson an abundantly zoned zircon. In addition, we employ newtechniques for the data analysis and construct a semiquantita-tive data–driven model of the zircon OZ.
2. ANALYTICAL METHODS
2.1. General Approach
We proceeded by examining numerous zircon crystals in orientedpolished thin sections by use of transmitted light and Normarski phasecontrast illumination (Fig. 1). These procedures enabled the selectionof crystals with abundant zones and minimal amounts of metamictmaterial. Data were acquired via optical microscopy, scanning electronmicroscopy (SEM), electron microprobe (EMP), and ion microprobeanalyses. Spatial series of zone thickness vs. crystal core-to-rim dis-tance were obtained from SEM traverses on oriented sections. Theintensity of the SEM back-scatter signal was correlated with chemicalcomposition of the crystal zones determined from the ion microprobe
* Author to whom correspondence should be addressed ([email protected]).
and EMP data. The spatial series were analyzed by signal analysistechniques, notably wavelet and Fourier transforms, to characterize theseries and to search for periodicities, singularities, persistence, andstationarity. The correlation function method, a series analysis tech-nique from nonlinear dynamics, was used to investigate the organiza-tion of the series (i.e., random or deterministic behavior). Hurst expo-nents were also determined by using the Fourier power spectra (e.g.,Halden and Hawthorne, 1993; Hoskin, 2000) and a rescaled rangeanalysis (Turcotte, 1997). The chemical data were analyzed and werealso correlated with the intensity of the SEM back-scatter signal.
2.2. Geochemical Methods
Abundances of trace elements in zircon were determined by second-ary ion mass spectrometry (SIMS) by use of SHRIMP II (sensitivehigh-resolution ion microprobe) following the method described byStern (1999); Appendix 1. An O primary ion beam was used to sputter15- to 20-�m-wide, 1-�m-deep pits in selected areas of the zirconmegacryst. Sequential analyses of selected isotopes of P, Y, Ba, rareearth elements (REEs), Hf, Th, and U were carried out at a massresolution of �7000, and count rates were referenced to an isotope ofZr. Sensitivity factors for the isotope ratios were determined by refer-ence to an in-house zircon standard (BR266) that is chemically homo-geneous at the micron scale (Stern, 2001), and for which abundances oftrace, minor, and major elements have been determined by independentmethods, including inductively coupled plasma mass spectrometry,thermal ionization mass spectrometry (TIMS), and EMP (Stern, 2001;unpublished data). Analytical uncertainties (1�) are generally in therange �1 to 5% for P, Y, Hf, Ce, Gd-Lu, U, and Th, �5 to 10% for Nd,Sm, and Eu, and �15 to 25% for Ba, La, and Pr. These errors includecounting statistical errors and uncertainties in the relative sensitivityfactors. Additional external uncertainties of 2 to 5% relating to thestandard composition have not been included in the error estimatesbecause they do not affect the intercomparison of zones. Abundances ofSi in zircon were determined with an EMP calibrated with natural andsynthetic zircon standards and have an uncertainty of about �2%.
A Cambridge Instruments S360 scanning electron microscope wasused to obtain BSE images of the zircon. Although cathodolumines-cence imaging could also have been used in the current context, thespatial sensitivity of BSE imaging was found to be superior—that is,considerably narrower oscillatory zones could be resolved. Further-more, our initial work also showed that the relative BSE intensity wasdirectly proportional to trace element contents (e.g., U, Th), thusenabling the use of BSE gray levels as a trace element proxy.
Samples were also imaged by use of Nomarski differential interfer-ence phase contrast microscopy (NIC), a method used for observingsurfaces of metals and alloys (Nomarski and Weill, 1954). It overcomesthe resolution degradation inherent in transmitting light through a�30-�m-thick section associated with petrographic microscopy(Anderson, 1984; Clark, 1986). For NIC, the introduction of a Nomar-
ski prism into the ray path of the objective in reflected light mode splitsthe primary light beam and recombines it after reflection. As a result,subtle surface features cause phase contrast interference of the reflectedlight, and features as small as 2 to 10 nm can be resolved. The zirconsections were first highly polished and then etched with hydrofluoric(37%) acid so as to produce surface relief dependent on the chemistryof adjacent zones. The technique is limited in its horizontal resolutionto that of the optical microscope, �0.5 �m.
2.3. Mathematical-Statistical Methods
Continuous wavelet transform produces a space of wavelet coeffi-cients W�(a,b), of scale (e.g., wavelength) a and position b (e.g., time,zone number or distance from crystal core) from a time (or length)series f(t), with f representing the signal (e.g., gray value) amplitudesand t representing the time or depth axis, depending on the shape of themother wavelet �:
W��a,b� � �1
a�� f�t��� t � b
a �dt. (1)
We used the sinusoidal-shaped Morlet wavelet (Grossman and Morlet,1984) as mother wavelet
�a,bl �t� � ��
14�al ��
12e�i2�
1a�t�b�e�
12�t�b
al �2
(2)
with l representing the scaling ratio of the analyzing window of theanalyzing Morlet wavelet vs. scale b and is set to l� 10. This providesparticularly good resolution in periodicity (Prokoph and Barthelmes,1996). Edge effects of W were reduced by using “zero padding” on thebeginning and the end of the data series. The “scalogram” is thetwo-dimensional graphic representation of the wavelet coefficients inlogarithmic spaced a vs. b. We used a four-step (0 to 25%, 25 to 50%,50 to 75%, 75 to 100%) gray-level scale for coding W�(a, b).
Fourier analysis is a type of spectral analysis for which significancetests are well established. It is used to transform the signal from timeto frequency domain (e.g., Davis, 1986). Its output is spectral power Pf
for frequencies f (or periods) that is the fraction of variance that theamplitude has at a specific period. However, time information is lost inFourier analysis. Moreover, wavelet analysis permits distinction be-tween stationary and nonstationary signals, such as gradual and abruptchanges in signal frequency, phase, and amplitude. Figure 2 is a simpleexample constructed using sine waves. It illustrates the use and termi-nology of both the Fourier and wavelet techniques.
We determined the 95% confidence levels for nonrandomness of thespectral power by calculating the power spectrum from the originaldata though shuffled (i.e., randomized) in time. The 95% level of thespectral power data from the shuffled data represents the 95% confi-dence level for the original data. This randomization method has theadvantage over other methods of white noise and confidence levelcalculation (e.g., Bartlett, 1966; Mann and Lees, 1996) in that it isdistribution independent (i.e., the statistical distribution of the originalset is retained in the shuffling).
For testing determinism and nonlinearity, we utilized the correlationfunction technique (Grassberger and Procaccia, 1983) and calculatedthe slope of the power spectrum (� value), fractal dimension, and Hurstexponent from the power spectrum (e.g., Halden and Hawthorne, 1993;Turcotte, 1997).
The correlation dimension, determined from a correlation functionplot, is commonly used to analyze nonlinear deterministic systems(Grassberger and Procaccia, 1983). Its limitations are discussed inFowler and Roach (1993) and Essex (1991). The method requiresevenly spaced data, such as zone numbers. The correlation dimensiond depends on the time (or data distance) lag �, the range between thescaling amplitudes rmax and rmin, and the size of the embeddingdimension E. The algorithm searches for correlations within the seriesthat has been transformed into an array of E-dimensional vectors bycombining observations Xo(ti) � Xo(t1)…Xo(tN), in the followingmanner:
Fig. 1. Nomarski microscopy of zircon zoning, section parallelc-axis. Note the variability in zone thickness, the sharp contrast be-tween successive zones, and the domain of bright and dark zones.
312 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
The correlation function C(r) can then be calculated as
C�r� �1
N2�i, j�1
N
�r � abs�Xi � Xj��, (4)
with ( x) � 0 if x � 0, ( x) � 1 if x 0.In a deterministic time-series pattern, the correlation function for
small values of C(r) fulfills the relationship C(r) � k*rd, with d represent-ing the correlation dimension (comparable to fractal dimension). If dconverges and saturates below the size of the embedding dimension, d can
give the dimension of a strange attractor or the number of independentparameters involved in the dynamics, giving rise to the formation of theseries (Grassberger and Procaccia, 1983). No saturation of d at lowembedding dimension represents randomness. In the algorithm, 18 valuesof radius r are used in the range from rmin � a single pixel to rmax � 2*maximum zircon zone thickness for each data set.
3. KIPAWA ZIRCON
The study zircon occurs at a rare mineral site within theKipawa Syenite Complex (Currie and van Breemen, 1996),
Fig. 2. (A) Model time (length) series with three abrupt changes (V1, V2, V3) in simulated crystal zoning growth. (B)Wavelet (Eqn. 1) scalogram. The vertical axis shows continuous spectra of possible cycle periods on a logarithmic scale.The horizontal axis is distance, as in (A). The dark gray and black areas in the scalogram mark well-pronounced cyclicityfor specific time intervals of the 1000-�m-long record. In other words, the 40-�m period portion of the signal occupies �2/3of the scalogram, whereas the 80-�m period portion occupies �1/3, consistent with their distribution in the original signal(A). Note that there is almost no appearance of edge effects, and the low-frequency harmonics of the periodic signals areonly marginally present. (C) Morlet wavelet with scale a, location b, and length of analyzing wavelet l � 10 � a (see text).(D) Spectral analysis (Fourier transform) with periodogram of the model time series. Note that the abrupt changes V1 andV2 are not detectable in the Power spectra, and period jump V3 results in two different peaks in the spectrum, which couldbe incorrectly interpreted as a synchronization of both periods. Wavelet analysis has the advantage in that one has scale anddistance information preserved.
313Organization and model of zircon oscillary zoning
located in the western part of the Mesoproterozoic GrenvilleOrogen, Quebec, Canada. The following brief description isderived from Currie and van Breemen (1996). The syenitecomplex comprises peralkaline igneous rocks emplaced at�1240 Ma ago into host granitic gneisses, marbles, and calc-silicate rocks that were together deformed and metamorphosedto amphibolite facies during the terminal phases of the Gren-ville orogen (�1000 Ma). The marbles and calc-silicate rocksat the margins of the complex were imbricated with the syeniteduring deformation and metamorphism, remobilizing alkalis,halogens, REEs, Zr, and Nb and generating anatectic melts.The action of fluids, melts, or both within the marbles and hostsyenite and granitic rocks resulted in the generation of a skarn-like unit containing many rare minerals, including megacrysticzircons. The zircon crystals reach several centimeters in diam-eter in pegmatitic patches within amphibolitic units of theskarn. Thus, the zircon megacrysts appear to have formed fromF-rich brines, anatectic melts, or both. These zircons have beendated by TIMS methods at 993 Ma (Currie and van Breemen,1996; Stern, 1997). Zircons from the locality are typicallybrown to red, millimeter- to centimeter-sized, doubly termi-nated tetragonal prisms. In large crystals, zoning is observablewith the naked eye.
After we examined numerous crystals and sections, we chosea crystal section perpendicular to the c-axis for detailed anal-ysis. The section was chosen because of its large size andnumerous zones, and because it has a core-to-rim area withrelatively few fractures and domains of metamictization. Zon-ing is clearly visible as layers of varying interference colors.
Large-scale zoning can be identified as alternating bands show-ing well-developed polygonal fractures with yellow third-orderinterference colors, and having bands much less fractured, withgreen-blue to pink fourth-order interference colors. Figure 1 isa NIC image of a portion of the crystal etched for 20 s. Thezoning is very sharply defined and fine-scaled, down to a widthof �5 �m, and is not uniform. The apparent pinching out ofzoning is due to the orientation of the section with respect to thepyramidal crystal faces
A mosaic of BSE SEM images perpendicular to c-axis cov-ering the whole zircon surface was taken at an acceleratingvoltage of 20.0 kV and a magnification of 150�. The imageshows homogeneous unzoned cores surrounded by bands ofoscillating intensity. Two cores representing two separate crys-tals with parallel c-axes, one bright and one dark in BSEintensity, are visible (Fig. 3). Although their cores and broad-scale zoning patterns appear similar, the two crystals likelygrew independently because their fine-scale zoning does notcorrelate. This particular feature provides good empirical evi-dence that the fine-scale zoning is not strictly due to large-scalechanges in the environment of growth. In general, the darkbands are more fractured, whereas the light bands are thinner,ranging from approximately 0.15 to 1.8 mm (Fig. 3).
Smaller scale zoning is visible inside these broad zones andhas been explored in more detail on SEM images of highermagnification (Fig. 4). Small-scale OZ is found everywhereexternal to the crystal core. In almost all regions imaged, thezoning is still clearly visible at magnifications higher than1000�. The smallest bands observable were at a magnification
Fig. 3. Scanning electron microscope back-scatter photomicrograph of zircon crystals cut normal to c-plane (resolution,1 pixel � 20.8 �m), with location of ion-probe samples 9 to 15 (Fig. 4 shows the remaining analysis areas). Box showsthe location of Figure 4A. Note that the two crystals have contrasting cores, one light- and the other dark-colored, in SEMback-scatter mode. Both are unzoned. Close to the core, the broad-scale zoning appears correlative. The bright area in thetop left of the image is interpreted to be the mineral thorite.
314 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
of 4.36 � 103 (Fig. 4D) and have a width of approximately 0.3�m. The latter observation was made on the half-section adja-cent to the dark core where zoning is well developed andclearly visible from the core to the margin of the zircon (Fig.4A).
Fine, bright zones are best distinguished from thicker grayzones at higher resolution; they merge to fuzzy medium gray
bands at low resolution (Figs. 4B–D). This pattern of nestedzonation occurs with successive higher resolution (Figs. 4C,D)and is good empirical evidence of scale invariant or fractalpatterning within the zonation.
The effect of zircon metamictization and expansion of thecrystal lattice is clearly shown on Figure 3. Microfractures tendto develop perpendicular to the crystal faces in bands of high
Fig. 4. Four levels of magnification of zircon zoning at resolutions back-scatter SEM. Windows show magnified area foreach successive scale. Resolution: (A) 1 pixel � 2.43 �m with ion-probe samples; (B) 0.93 �m; (C) 0.195 �m; and (D)0.03 �m. Dotted line shows traverse for wavelet zone thickness analysis of Figure 9.
315Organization and model of zircon oscillary zoning
Fig. 5. Wavelet analysis (Eqn. 1) periodogram constructed by program CWTA.F (Prokoph and Barthelmes, 1996). Dataanalyzed are the SEM back-scatter gray level values (white � 255) for the half-section normal to zircon zones (Fig. 4A)at four different magnifications. All scales are in microns. The bar at the top of the figure gives the gray-scale coding forthe magnitude of the wavelet coefficients expressed as a percentage (black, 75 to 100% of maximum coefficient; dark gray,50 to 75%, light gray, 25 to 50%, and white �25%). Magnitude is plotted as a function of scale (wavelength) a, vs. positionalong the crystal b. Analyses and their data are plotted as pairs of panels {(A, B) (C, D) (E, F) (G, H)}, with the scale ofdata collection decreasing in each pair of panels. Rectangles relate the panels through changes in scale of data collection.
316 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
birefringence but do not propagate across metamict layers. Thisfeature is clearly visible on the SEM images, where the fracturesare strongly concentrated in the darker bands, which are relativelydepleted in U and Th and consequently are less metamict.
3.1. Mathematical Analyses of Zoning Patterns
To decipher the oscillatory-zoned patterns in terms of theirunderlying dynamics, quantitative techniques were employed(wavelet, Fourier, rescaled range, and correlation dimension)on two separate types of data sets: series consisting of lineararrays of crystal core-to-rim gray-scale data, and series con-sisting of the thickness of the zones as a function of crystalcore-to-rim distance.
3.2. Gray-Scale Data Analysis
Digitized gray-value (0, black; 255, white) back-scatter SEMback-scatter data and zone thickness data of the zircon crystalwere taken from tiff images at four different magnifications(Figs. 4A–D) with 1 pixel � 2.43, 0.94, 0.2, and 0.033 �m,respectively. Time series of gray values and thickness wereconstructed perpendicular to the zoning in the half-sectionnormal to the c-crystallographic axis (Fig. 4A).
3.2.1. Wavelet Analysis
Wavelet analysis (continuous wavelet transform) (Eqn. 1) ofthe gray-value time series reveals different periodicity in dif-ferent parts of the crystal and at different microscopic resolu-tion (Fig. 5). The most pronounced and stationary periodicity ata resolution of 1 pixel � 2.43 �m has a wavelength of � 1200�m (black band in Fig. 5G), which is equivalent to � 4.5 cyclesand ranges across the crystal half-section (� 5000 �m). Peri-odicities of 60, 250, 280, and 600 �m at this resolution are lessstationary (noncontinuous black and dark gray patches in Fig.5G) or less intensive (i.e., light gray band at wavelength of�2500 �m). Zooming into higher resolutions (Figs. 5A–F)shows some periodicities occur in a nonstationary manner (i.e.,sporadic) at distinct bandwidths of 1.8 to 2, 6 to 9, 13 to 15, 20to 26, �60, �80, and �250 �m. The major periodicities andtheir locations are summarized in Table 1.
3.2.2. Fourier Analysis
The power spectra of gray values also show significantperiodicities, which are dependent on the microscopic magni-fication (Figs. 6A–D). The periodicities above the 95% confi-dence level of nonrandomness are close to or the same as foundby wavelet analysis, with addition of periods of �5.2 �m (Fig.6C) and � 51 �m (Fig. 6B). It is possible that some periodi-cities are not independent but are multiples. For example, the2500- and 630-�m periods may be 2f and f/2 multiples of the1200-�m periodicity (Fig. 6A). The smallest significant peri-odicity (�2 �m) detected by continuous wavelet transform(black band in Fig. 5A) and Fourier analysis occurs as oscilla-tions of bright zoning (Fig. 4D) with gray values fluctuatingfrom �140 to 200 (Fig. 5B).
The � value is defined by � � ��log(Pf)/�log(f), with f �number of repetitions that ranges from 1 to the number ofequidistant data entries/2, and Pf � spectral power density at f.
A flat slope (i.e., � � 0) defines white noise, whereas a steepslope may indicate self-similarity in the analyzed pattern (Man-delbrot, 1982). The fractal dimension D is defined by � � 5 �2D (Halden and Hawthorne 1993, Turcotte 1997). In our study,� is relatively constant for all resolutions and ranges from 0.97to 1.26 with absolute errors from 0.38 to 0.42, respectively.Consequently, D for our gray value data of zircon zoningranges from 1.87 to 2.02 � ��0.2, which is also characteristicfor fractional Brownian motion (1 � D � 2, Turcotte, 1997).The Hurst exponent H � 2 � D defines self-affinity in timeseries and ranges from �0 to 0.11 (��0.2) in our study. Thisvalue is much smaller than previously estimated for the Hurstexponent for zircon zoning of H � 0.34 to 0.55 (Halden andHawthorne, 1993; Hoskin, 2000). The zircon time series usedby Halden and Hawthorne (1993) and Hoskin (2000) are muchsmaller (�280 data) than ours (780 to 2100 data per timeseries) and restricted previous spectral analysis to � 140 fre-quencies. For comparison, we calculated the � value for onlyour first 100 frequencies simulating less zones and smallercrystals (Fig. 7). For this case, the fractal dimension D forresolutions of 1 pixel � 2.43 �m and 1 pixel � 0.03 �m are1.74 (� �0.19) and 1.69 (��0.18), and Hurst exponents H are0.26 � �0.19 and 0.31 ��0.18, respectively. Low Hurstexponents H � 0.32 indicate sharp peaks and antipersistence inthe time series (Korvin, 1992; Hoskin, 2000). The difference of thefractal dimensions between different resolutions and differentparts of the power spectra suggests that the gray values fromzircon zoning are not purely a fractal distribution, and that self-organization is unlikely the sole mechanism of crystal growth.
3.2.3. Rescaled Range Analysis
An alternative method of measuring H is derived from (RN/SN)Average� (N/2)H (e.g., Turcotte, 1997) with range R andstandard deviation S in parts of the time series containingblocks of N data points each. The block �R/S values areaveraged; H is given by the slope of the line ln(RN/SN)Average)/ln(N/2). The rescaled range analysis shows that the gray scale
Table 1. Major periodicities in the zoning pattern.a
0.2 (Fig. 2C) Rim �80Rim 23–26Part of rim 13–15Part of rim 7–9 10Part of rim 1.8–2 2.5
0.033 (Fig. 2D) Rim 15Rim �6.5Rim 2
a Italic indicates 75% of maximum wavelet coefficient.
317Organization and model of zircon oscillary zoning
data have H � 0.09 for both the raw data and the data aftershuffling (Fig. 8).
3.3. Zone Thickness Data Analysis
To produce a series of crystal core-to-rim zone thicknessdata, we initially devised an algorithm to search for disconti-
nuities in traverses along the two-dimensional SEM gray-scaleback-scatter files. Because of perturbations such as fracturesand areas of metamictization, this was not successful. Accord-ingly, the series was constructed by hand through careful mea-surement of the zone thicknesses. The fractures tend to benormal to the zoning; consequently, we were able to choose a
Fig. 6. Logarithmic scaled power spectra of Fourier analysis for gray value time series (Fig. 5) from four resolutions ofzircon zoning. X-axes: logarithmic scaled frequency f in number of repetitions. Dotted lines mark 95% levels fornonrandomness of power (see text for explanation). Solid lines mark �log(Pf)/�log(f) trend with slope ��. Significantperiodicity are indicated on top of each power spectra. Resolution: (A) 1 pixel � 2.43 �m; (B) 1 pixel � 0.93 �m; (C) 1pixel � 0.195 �m; and (D) 1 pixel � 0.03 �m.
318 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
section line so as to minimize their intersection. Where frac-tures were encountered, the section was constructed from alocation immediately adjacent to the fracture having correlativezones on either side of the fracture. Thus, we are confident thatthe section is a faithful representation of the data.
3.3.1. Wavelet Analysis
Wavelet analysis of the zoning thickness of the whole sec-tion shows stationary (i.e., stable in time) �28 zone periodicity(Fig. 9A), which may be detected by eye directly from the data(Fig. 9B). Small sets or single thick zones are followed by 20to 25 thinner zones. In addition, a small interval close to thecore shows a 4.5-zone periodicity, and close to the rim, a�3-zone periodicity (black patches in Fig. 9A). The periodicityin the high-resolution zoning pattern is less stationary (Fig.
9C). A 38-zone periodicity (four cycles) is transferred to a65-zone wavelength (only one cycle) closer to the rim. Peri-odicities of 10 and 20 zones are nonstationary (Fig. 9C).
3.3.2. Fourier Analysis
Fourier analysis reveals similar or the same zoning periodi-cities as detected by wavelet analysis (Figs. 10A,B). The lowperiodicities of 2 to 3.7 zones signify intense high-frequencyoscillations, which are shown to be nonstationary by waveletanalysis. As demonstrated above, the intense high frequenciescause the slope of the log/log-regression line (� value) to bevery flat (� � 0.3 ��0.24) both for whole and detailedsections, similar to a white noise dominated zone thicknessdata.
3.3.3. Rescaled Range Analysis
For the zone-thickness time series of resolution of 1 pixel �2.43 �m (Fig. 9D), we calculated H� 0.22 for N/2 � 3…72.According to Turcotte (1997), the corresponding � value wouldbe �fBw � 5 � 2(2 � H) � 1.43 for fractional Brownian walkrepresenting antipersistence, and �fGn � 2H � 1 � �0.57 forfractional gaussian noise. Both results are in contrast to theprevious results (Halden and Hawthorne, 1993; Hoskin, 2000)for � values from Fourier analysis (� � 0.2) and likely due tothe influence of long periods in the power spectrum. Conse-quently, the power spectra of the zoning pattern cannot solelybe interpreted as a long-term memory process (e.g., fractionalBrownian walk model), but by antipersistence (e.g., stronghigh-frequencies) superimposed by long-term periodic forcing,such as repetition frequency of 29 zones).
3.3.4. Correlation Dimension
The technique was not used for the gray-scale data becauseit requires a sequence of event data, for instance zone numbers.The correlation dimension analysis shows a significant deter-ministic behavior of the zone thickness data because the slopeof correlation dimension d vs. embedding dimension E satu-rates at a low value of E � 6 at d � 4.2 for both detail andwhole section data (Fig. 11). On the contrary, the same data,although shuffled, do not show any saturation of d. We interpretthis, consistent with the wavelet and Fourier analysis, to meanthat there is significant organization within the data set. How-ever, on the basis of the Hurst exponents determined, we cannotrule out a stochastic element within the dynamics that gave riseto the OZ. In contrast, the shuffled zone thickness data arerandomly distributed consistent with expectation. The period-icity in the zoning pattern may account for most of the deter-ministic pattern. Nonstationary periodicity may also be deter-ministic (e.g., nonlinear) rather than random.
In conclusion, the data analysis (wavelet and Fourier) dem-onstrate that there are significant periodicities within the OZrecord and the correlation function analysis shows that the dataare deterministic, and not randomly distributed. Accordinglywe propose a simple nonlinear deterministic model for thezonation.
Fig. 7. First 100 logarithmic scaled power spectra values of Fourieranalysis for two gray value time series (Figs. 5B,H) from two resolu-tions of zircon zoning.
Fig. 8. Hurst exponent analysis by rescaled range technique: timeseries of 2096 original gray values (white � 255) and the same data,although shuffled; resolution of 1 pixel � 2.43 �m. Note that bothHurst exponents are close to zero.
319Organization and model of zircon oscillary zoning
4. PERIODICALLY FORCED NONLINEAR OSCILLATIONMODEL
We expect the chemical composition of the zircon tochange most sharply at the strong dark-bright transitions(Figs. 4A–D), although higher resolution ion-probe analysesbeyond the limits of current technology are necessary toverify this assumption. The high-resolution cyclicity occursmostly at relatively bright zones (i.e., with high REE � P �U � Th concentration and low Hf concentration).
The objective of our data-driven model is to preserve thestatistical properties of the original signal by implementing alimited number of periodic and nonlinear parameters. For thezircon-growth model, we respect three major characteristics:
(1) Periodicity and nonlinearity (i.e., determinism) in thezone thickness.
(2) Periodicity in the gray level value (i.e., geochemistry ofthe zones).
(3) Zoning pattern of zircon is at least partially self-orga-
Fig. 9. Wavelet analysis of two zone-thickness data series. (A) Wavelet scalogram of (B) 145 zones from resolution 1pixel � 2.43 �m (Fig. 4A); (C) wavelet scalogram of (D) 225 zones from resolution 1 pixel � 0.195 �m (for sectionlocation and length, see dotted line in Fig. 4A). For explanation of scalogram shading, see Figure 5.
320 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
nized (e.g., these results [Fig. 11] and previous work: Haldenand Hawthorne, 1993; Hoskin, 2000).
Our self-organization model (SO model) for the simulationof a nonlinear pattern in zone thickness makes use of thelogistic map (May, 1976), which has been used as a simpleanalog of the behavior of competitive populations. The logisticmap is defined by
xn�1 � �xn�1 � xn�, (5)
with a range of the control parameter � from [0,4], and nrepresenting the zone number and is iterated using a seed valueof X [0,1]. One can think of this map as a simple means ofexpressing the competition between various elements for oc-cupancy within the zircon. For example, Xn could represent theconcentration of a trace element in the melt adjacent to thecrystal at a particular time, whereas Xn�1 represents the con-centration at a later time calculated from the nonlinear expres-sion (Eqn. 5). This feedback is caused by the concentration ofXn itself in conjunction with a system control parameter �,
which in this data-driven model is represented, at a conceptuallevel only, by crystal growth velocity.
For values of � [1,3], output series are regular in nature andcharacterized by convergence to single values, and for �[3,3.45], output series x form periodic cycles, until � 3.45,where “chaotic regions” with infinite, although bound, unstablevalues for x begin (May, 1976).
Consequently, values of � (�3.45) can be related to linear orcyclic (i.e., predictable) behavior in output of Eqn. 5, whereasvalues of � 3.45 can be used to indicate “deterministic”chaotic behavior (e.g., long-term unpredictable outcomes andsensitivity to initial conditions). In the SO model, we replacethe fixed parameter � by the variable periodic forcing param-eter . We chose � ranging from [2.8,4] compatible with thedetermination that thickness fluctuations of zircon zones arecyclic and might often be chaotic and antipersistent (see aboveand Hoskin, 2000). Oscillations of � between 2.8 and 4 and arerealized by substitution:
� 3 1.2 n 2.8 (6)
with
n � �cos�2�n
28 � 1� / 2 (7)
The wavelength � � 28 is determined from the stationary28-zone periodicity in thickness pattern at resolution 1 pixel �2.43 �m (Fig. 9A). According to the results, the zone numberin our model is restricted to 145. The initial value x1 is set to0.1. Realizations of xn depict thick (x below average) andrelatively thin zones (above average). The output for x isbetween [0,1]. For adjustment to the real zoning with thickzones (�150 �m), we tuned the output series xn (see Eqn. 1) to[0,150] by
yn � 150�1 � xn0.5�. (8)
Our gray-value model links the zone thickness model with thegray-value periodicity �� of � 1200 �m (Figs. 5G,H), follow-ing the periodically reoccurrence of dark (gray value �120) tobright (gray values of � 200) (Fig. 5H). The average gray-value change between two successive zones is 25.6, with arange of 17 to 66. Consequently, we transform the thicknessdata into length data and include the statistical characteristicsand periodicity from wavelet transform to give the model grayvalue zn,
zn � 180 50� sin�2�n
35 � ��
2� �yn
3, (9)
which includes the parameter for mean gray value (180), meangray-value amplitude caused by 1200-�m periodicity (50), thezone number n, a negative phase shift �/2 corresponding to thedark core at the beginning of the data set, and the zone-thickness–related gray value fluctuation on zone contacts ofy/3.
The model zone thickness data sum up to a total st � 5073�m at nt � 145 zones, which is close to the original sectionlength of 5075 �m. The 1200 �m gray-value periodicity �� istransformed into zone number periodicity �� by
�v � ��*nt/st. (10)
Fig. 10. Logarithmic scaled power spectra of Fourier analysis forzone-thickness time series (Fig. 9) from 2 resolutions of zircon zoning.X-axes: logarithmic scaled frequency f in number of repetitions. Dottedlines mark levels for 95% nonrandomness of power. Significant peri-odicities are indicated on top of each periodogram. Solid lines mark�log(Pf)/�log(f) trend with slope ��. Resolution, (A) 1 pixel � 0.195�m; and (B) 1 pixel � 2.43 �m.
321Organization and model of zircon oscillary zoning
The model output from the thickness SO model (Fig. 12A)shows the 28 zone cyclicity in the zone thickness according tothe external forcing. The most intense jumps in zone thicknessare symbolized by black spots at low wavelengths in thescalograms. Note that a possible scale-invariant higher-fre-quency zoning occurs at higher resolutions (e.g., 1 pixel � 0.03�m, Fig. 4D) at these transitions. In addition, the externalperiodicity is transformed in amplitude and number of succeed-ing thick zones due to the nonlinearity in the model. Conse-quently, random noise is not necessary to explain variations inzone thickness and high frequency periodicity (Fig. 9). Spectralanalysis (Fig. 13) also indicates that this model provides alog(Pf)/log(f)slope comparable to the original data, in contrastto a largely nonlinear model for the zircons of Hoskin (2000).Consequently, external periodic forcing is likely involved in theforming the OZ at least in the crystal studied.
The gray-value model (Fig. 12B) shows altered periodicityaround �1200 �m, including a low-contrast zoning at relativehigh gray level (bright REE � P rich zones) corresponding tothe crystal data (Figs. 5G,H). Wavelet analysis reveals that theperiodic forcing is altered to � 900 to 1300 �m periodicity dueto the influence of the nonlinear feedback mechanism. Note that�60 and 200 �m periodicities occur in the gray-level modeldue to nonlinearity, despite the fact that these periodicities arenot included in the external forcing. In addition, the gray levelcontrast between thick zones is higher than between thin zones,also correlative to the crystal data. The relationship betweenzone thickness and gray level is also similar between data andmodel, as shown from the trend line and similar data scatteringin Figure 12.
Rescaled-range analysis determines H � 0.28 for measure-ment data, randomized (“shuffled” ) data and the periodic forcednonlinear zone-thickness model (Fig. 14). Consequently, theresults from the two methods for determining the Hurst expo-
nent (i.e., log/log-slope from Power spectra and rescaling anal-ysis) may be inconsistent, and these techniques are not unam-biguous or in our opinion not particularly useful to distinguishthe persistence, nonlinearity or randomness in time series.
We think that the data driven model is realistic because thereis strong evidence that there is a periodic component to the OZ.Also, our analysis, consistent with that of others for differentzircon crystals (Halden and Hawthorne, 1993; Mattinson et al.,1996; Hoskin, 2000) shows that determinism is present withinthe thickness series. From observation we noted that the ana-lyzed crystal (Fig. 3) is part of two adjoined crystals having acommon orientation. The fine-scale zones of the two crystalscannot be correlated by visual inspection. This is strong evi-dence that the OZ is not the simple one to one reflection ofexternal system changes upon the growing crystals. On theother hand, one might expect that, if periodic forcing was animportant factor, it would be recorded, and visually correlativebetween adjacent crystals. There is some indication of this inFigure 3, especially close to the two cores; however, the non-linearities in a periodically forced chaotic system mask theperiodicity. In geological terms, we expect that the periodicforcing could have arisen during the late pegmatitic stage ofcrystallization for instance, as fluctuations in pressure or influxof new fluid. Such changes could drive the system far fromequilibrium conditions providing the feedback for a nonlineargrowth regime. Under far from equilibrium conditions crystalgrowth is less controlled by surface kinetics. Diffusion ofgrowth constituents plays an important role. Close to the crystalface, fast crystal growth may result in the depletion of Zr andthe enrichment in components (e.g., U, Th, REE) relative to thebulk melt. Thus crystal growth eventually switches to a U-, Th-,and REE-rich composition, depleting their concentration in themelt as Zr concentration is reestablished by diffusive resupply,and so on. A hallmark of some systems driven far from equi-
Fig. 11. Correlation function analysis. Plot of correlation dimension d vs. embedding dimension E for two originalzone-thickness time series (Fig. 9B � whole section, Fig. 9D � detail) and shuffled time series (see text for explanation).Solid line marks 100% white noise; dotted line marks approached saturation of d � 4.2 for detail and whole-section timeseries. Note that the shuffled time series does not approach saturation levels.
322 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
Fig. 12. (A) Periodic forced nonlinear time-series model of zone thickness of zircon (bottom) with wavelet scalogram(top). For explanation of parameters and scalogram, see text. (B) Periodic gray-level model of imaging of zircon crystal(bottom). Length is transformation of zone thickness model, wavelet scalogram (top); for explanation, see text. Note thesimilarities in the occurrence of nonstationary high-frequency cyclicity to the wavelet scalograms of the original data.
323Organization and model of zircon oscillary zoning
librium is that they fluctuate from one composition to another.These systems are termed chemical oscillators (Gray and Scott,1990).
5. GEOCHEMISTRY
Analyses of individual growth zones were undertaken todocument the trace element chemistry of the zones and theirintra and interzone chemical variations. The emphasis was onunderstanding the main compositional controls on BSE inten-sity (gray level), which was used for time series analysis.Twenty-nine spot analyses were performed by both EMP andSIMS (SHRIMP) and EMP at the same locations on the crystal(Figs. 3, 4). For analysis of zone homogeneity, seven spotswere analyzed within the same zone (Fig. 3).
The chondrite-normalized log REE plots (Fig. 15) are char-acterized by steep positive slopes as a result of significant
HREE enrichment that is typical of most zircons. The HREEshave a rather prominent convex-up curvature. The chondrite-normalized La/Lu values range from 1.5 � 10�3 to 7.5 � 10�6
and are correlated with fairly large chondrite-normalized Ceanomalies (Ce/Ce*) ranging from 9 to 650, respectively (i.e.,the steeper patterns have the larger anomalies). The REE pat-terns are also characterized by slight moderately negative Euanomalies, Eu/Eu* � 0.32 to 0.45, possibly due to prior pla-gioclase crystallization. The core of the zircon is characterizedby high Th/U (0.8 to 1.5), whereas the remainder has lowervalues of 0.16 to 0.47. In chondrite normalized terms the Laconcentration varies from �3 � 10�2 to �101, whereas Luranges from 8 � 102 to 104. Abundances of trace elements varywidely: Lan (0.03 to 9.7), Y (246 to 5214 ppm), Th (17 to 876ppm), and U (21 to 2245 ppm), whereas Hf concentrations arerelatively homogenous (0.74 to 1.05 wt%). Although not con-sidered here, we note that the molar (REE � Y)/P values are�1, indicating that the xenotime substitution scheme (REE �p � Zr � Si; Speer, 1980) by itself is insufficient to explain thesubstitution of REEs in this particular zircon (cf. Hoskin et al.,2000).
Linear correlation and regression analysis shows that BSEgray-level measured at a resolution of 1 pixel � 2.43 �mcorrelates significantly positively with REE � Y and U � Thbut negatively with Hf (Fig. 16). The bright BSE zones areassociated with relatively high abundances of HREEs � Y �U � Th and lower abundances of Hf. Substitution of the formergroup of elements appears to be take place, in part, at theexpense of Hf and Zr. In contrast, there is no correlationbetween any chemical element or group and zone thickness,indicating that at the �20-�m resolution the SIMS analyses,the deterministic zoning thickness pattern is independent of thezone chemistry.
The gray levels and the geochemistry within a single growthzone show slight correlated heterogeneities. For example, anal-yses 9 to 15 were (as best as could be estimated from the SEMimages) taken from within the same relatively bright BSEgrowth zone. The variations in the sum of REEs � Y (2546 to5821 ppm) and U � Th (1255 to 2910 ppm) correlate positivelywith gray-level values of 193 to 255 (measured at a resolutionof 1 pixel � 20.8 �m). Despite the chemical heterogeneity of�2.3 times observed for these chemical parameters within thisparticular zone, such variations are considerably smaller thanthose observed across the entire crystal (18 to 75 times).
6. DISCUSSION AND CONCLUSIONS
Wavelet and other time series analysis of zircon from theKipawa syenite demonstrate that the pattern of OZ has featuresconsistent with periodic and deterministic precipitation mech-anisms. We suggest that the low Fourier power spectrum slopethat we determined, compared with previous studies of othercrystals, is a result of the existence of relatively high spectralpower for high frequencies (repetition numbers). This may bedue to local high-frequency zoning—in particular, brightlyzoned bands (Figs. 4A–D). As shown, these can be missed inshort sequences.
The number of zones and their thicknesses were measuredfor the whole core-rim section (145 zones, 5130 �m) at reso-lution of 1 pixel � 2.43 �m, and for a detailed section on the
Fig. 13. Power spectra of Fourier analysis for periodic forced zone-thickness model (Fig. 12). X-axes: logarithmic scaled frequency f innumber of repetitions. Dotted lines mark 99.5 and 95% probability fornonrandomness. Note that the slope of the �log(Pf)/�log(f)–trend(�0.55) is similar to the original data (�0.34) for the resolution 1pixel � 2.43 �m.
Fig. 14. Hurst exponent analysis by rescaled range technique. Orig-inal and shuffled zone-thickness time-series data (Figs. 9B,D) and zonethickness model data (Fig. 12A) are shown.
324 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
Fig. 15. (A) Chondrite-normalized REE plots samples 1 to 15. (B) Samples 16 to 29.
325Organization and model of zircon oscillary zoning
crystal rim (225 zones over 796 �m) at a resolution of 1pixel � 0.195 �m (Figs. 9B,D). The mean zoning thickness forthe whole section is 35.38 �m and for the detailed section 3.54�m or 0.029 zones/�m and 0.28 zones/�m, respectively. Con-sequently, a � 13� zoom results in �10� more detected zonesper interval. An extrapolation of the high-resolution image tothe whole crystal could reveal �1420 zones/�5080 �m with-out consideration that this high-frequency zoning is most pro-nounced in the strong gray-value-contrast parts of the crystal.This resolution dependency of the zoning data means that evendetailed studies such as this have not fully resolved the zoning.Clearly, understanding the scale and resolution of zircon OZ isimportant. For instance, recrystallization and metamictizationare related to trace element distribution and reverse discordancein zircon (Mattinson et al., 1996; Nemchin and Pidgeon, 1997).Because the trace element distribution is strongly related to theOZ, the redistribution of elements between zones of differentconcentrations has the potential to perturb age dates.
The SEM back-scatter gray-level data reflect the geochem-istry—that is, bright zones are REE, Y, U, and Th rich. Spotanalyses show that the REE profiles are strongly HREE en-riched with slight negative Eu anomalies and prominent posi-tive Ce anomalies. The fact that there is no correlation betweenany chemical element or group and zone thickness suggests thateither the ion microprobe beam is large with respect to zonethicknesses, or the deterministic zoning thickness pattern isindependent of the zone chemistry. Thus chemical variationsmay be a reflection of oscillations in the mode of growth of thecrystal in response to the local environment.
Hoskin (2000) has calculated both H and the Lyapanovexponent � from data of oscillatory zoned zircons from acompositionally zoned pluton in Australia. The Lyapanov ex-ponent measures the extent to which a series is characterized bydeterministic chaos. For chaotic systems the evolution of thesystem is sensitive on initial conditions, and with time, twoarbitrarily close values of dynamical system variables willdiverge exponentially. In essence, � measures the evolution of
dynamical variables with time in the series. Hoskin’s (2000)results are consistent with antipersistent chaotic behavior forzircons from rocks of relatively low silica content, and non-chaotic persistent behavior for zircons from rocks of relativelyhigh silica content. Hoskin (2000) proposes a model of the OZthat uses a self-organization mechanism wherein an interplaybetween crystal growth velocity and P diffusion controls thexenotime substitution scheme and hence the trace elementabundances. Because under the conditions cited the diffusivityof Zr is greater than that of P, relatively rapid growth of zircondepletes the crystal boundary zone in P, damping the xenotimesubstitution mechanism. This results in a build-up of traceelements around the crystal, causing a poisoning and a reduc-tion of zircon growth velocity. The Kipawa zircons are differ-ent than those of Hoskin (2000) in that they were formed inpegmatite of a skarn environment. The Hurst exponents wemeasure are higher than those of Hoskin (2000); however, wedemonstrate that the determination of H is not particularlyuseful for determining persistence, noise, or nonlinearities inthe OZ of crystals. Zoning within the Kipawa zircon is alsodifferent to that reported by Hoskin (2000) because it incorpo-rates a harmonic component for several thick zones. Theseharmonic zones are roughly correlative between the two crys-tals of Figure 3, suggesting to us that they are due to changesto the system from which the crystals grew.
The model of the zoning is data driven and was created basedon a simple nonlinear equation modified for periodic forcing.The model is rudimentary. Lacking relevant data for zircongrowth rates (e.g., the rates for zircon growth in a SiO2-richsilicate melt vary by four orders of magnitude), precise diffu-sivities of actinides and lanthanides, and other data for pegma-tite growth, a more robust model is not warranted at this time.It replicates the actual data and demonstrates that although theactual patterns are noisy, the dynamics of the pattern-formingprocess can be explained by a simple nonlinear model. Thehomogeneous and unzoned core of the crystal is evidence thatit initially grew under close to equilibrium conditions, possiblyfrom the syenite melt. The subsequent zonation likely aroseduring the later pegmatite/hydrothermal phase, possibly in re-sponse to an influx of new material or a change in intensiveparameters. As a result, crystal growth was fast relative todiffusive resupply such that a feedback was established be-tween growth, supply, and composition.
We postulate that the melt layer close to the crystal becamedepleted in Zr and enriched in U, Th, and REEs relative to thebulk melt thus driving the crystal to a new composition and anoscillatory mode of growth. We speculate that periodic influxesof new material or other large-scale changes in the systemcaused the thick harmonic zoning but also provided the far-from-equilibrium conditions for the OZ. The fact that adjacentand interfering crystals have different zonation patterns is goodevidence that the OZ is not the simple one-to-one reflection ofchanges within the bulk system. Our model of growth is per-missive and data driven, and it is only a first attempt. Althoughthe data have a deterministic component, they are somewhatnoisy. Therefore, we cannot rule out noise or noise-inducedtransitions as being an important factor in the zircon OZ.
Acknowledgments—We thank the Natural Science and EngineeringResearch Council for their continued support. Jeanne Paquette is
Fig. 16. Gray level values (resolution of 1 pixel � 2.43 �m) vs. sumof REEs � Y, U � Th, and Hf (�g/g). Linear regression and correla-tion coefficients are indicated. For sample locations, see Figures 3 and4.
326 A. Fowler, A. Prokoph, R. Stern, and C. Dupuis
thanked for having helped with choosing a section for study anddiscussing crystal-growth mechanisms. We appreciate the detailed re-views and editorial comments made by Paul Hoskin and an anonymousreviewer. We gratefully acknowledge the Canadian Museum of Naturefor providing samples of the Kipawa zircon.
Associate editor: F. J. Ryerson
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327Organization and model of zircon oscillary zoning
Appendix 1. Spot chemical compositions of an oscillatory-zoned Kipaw a zircon megacryst.
SpotName
EMPA SIMS
Si Zr Ba � La � Ce � Pr � Nd � Sm � Eu � Gd � Tb � Dy
