Systematic analysis of K-feldspar 40Ar39Ar step heating results: I. Significance of activation energy determinations

Geochimica et Cosmochimica Acta, Vol. 61, No. 15, pp. 3171-3192, 1997 Copyright 0 1997 Elsevier Science Ltd Printed in the USA. All rights reserved CO16-7037/97$17.00 + .OO

Pergamon

PI1 SOO16-7037(97)00147-6

Systematic analysis of K-feldspar 40Ar/39Ar step heating results: I. Significance of activation energy determinations OSCAR M. LOVERA, MARTY GROVE, T. MARK HARRISON,

and K. I. MAHON

artment of Earth and Space Sciences and IGPP, University of California, Los Angeles, Los Angeles, California 90024, USA (Received

November

12, 1996; accepted

in revised form March 25, 1997)

Abstract-To better understand the argon retention properties of basement K-feldspars, 40Ar/39Ar stepheating results for 115 specimens representing a wide range of temperature-time evolution from a diverse array of geologic environments have been systematically evaluated. In carrying out the measurements, we instituted nonconventional analysis routines including: ( 1) duplicate isothermal steps; (2) multiple isothermal steps at 1100°C to extract as much gas as possible prior to melting; and (3) temperature cycling. To maintain a self-consistent approach, we systematically applied a weighted least square regression to determine the activation energies (E) and log(&/ri) values. Activation energies were found to define a normal distribution (46 ? 6 kcallmol) spanning over 30 kcal/mol. Corresponding log(D,Jr i) values (5 t- 3 s-‘) were highly correlated with E, yielding a slope that reproduces the previously documented feldspar compensation relationship. Numerical analysis of this correlation permits us to rule out large systematic laboratory errors in our data. In applying the multiple diffusion domain (MDD) model, we observed general tendencies in the domain distribution of basement K-feldspars. For the majority of samples, most 3gAr resides in the larger domains. The smallest domains generally constitute <5% volume fraction of the sample and tend to plot -2 orders of magnitude above log (Da/r i). Alternatively, the largest domains constrained by 39Ar released below melting have log( D/r*) values that are typically -3 orders of magnitude smaller than log(D,lr i). Systematic analysis of the database demonstrates that the diffusion behavior predicted by the MDD model prevails for virtually all samples. However, two kinds of anomalous degassing behavior were observed. The first appears to be due to our inability to isolate effects resulting from distributions characterized by very small domains while the second may result from inhomogeneous K-distributions. Although these phenomena are capable of producing a wide range of calculated diffusion parameters, close inspection reveals that important systematics of the K-feldspar results cannot be accounted for these factors and remain best explained by real intrasample differences in diffusion properties. Finally, while uncertainties in E determined from 40Ar/3gAr step-heating experiments can produce significant disnersion in calculated temueratures ( - lO”C/kcal/mol), the overall form of the cooling history, usually the most important result for tectonic applications. is preserved. Copyright 0 1997Elsevie-r Science I

Ltd

1. INTRODUCTION

In the MDD model, domains are represented by activation energy (E), frequency factor (Do), domain size (p), and volume fraction (4). Because, K-feldspar thermochronologic studies require extrapolation of laboratory determined 39Ar diffusion properties to crustal conditions, the magnitude of the diffusion parameters (E, Do) are of particular significance in determining the temperature dependence of argon release. Accordingly, determining whether or not intrasample variation of these macroscopic parameters exists in basement K-feldspars is of fundamental importance. A related issue is whether it is possible to accurately determine their values in a 4oAr/3gAr step-heating experiment (e.g., York and Hall, 1990). In systems described by a single diffusion length scale, determination of E is obtained in a straightforward manner from the slope of the linear array of log(Dlr*) values produced in an Arrhenius plot. In materials characterized by multiple diffusion domains however, nonlinear Arrhenius arrays result from differences in diffusive length scale (p) and argon concentrations (4) within the individual domains. Application of the MDD model to such materials, therefore, raises the following issues: can stepheating experiments provide the basis to distinguish between

Argon diffusion studies (Sardarov, 1957; Foland, 1974; Harrison and McDougall, 1982; Zeitler, 1987) have long indicated that low-temperature alkali feldspars are characterized by variable argon retention properties. In order to utilize 40Ar/39Ar step-heating measurements to recover the thermal history information recorded in these materials, we require an internally consistent and geologically meaningful interpretive framework. Several models have been proposed to explain the relationship between the laboratory degassing kinetics and the natural loss mechanisms of argon (Foland, 1974; Zeitler, 1987; Parsons et al., 1988; Lovera et al., 1989; Lee and Aldama, 1992; Foland, 1994). In attributing the variability in K-feldspar argon retentivity to the presence of a discrete distribution of noninteracting domains, the multiple diffusion domain (MDD) model (Lovera et al., 1989) reconciles the two distinct sources of information available from K-feldspar 40Ar/39Ar step-heating experiments: the age spectrum reflecting radiogenic 4oAr (@Ar*) loss over geologic timescales, and the Arrhenius plot determined from the degassing systematics of reactor-produced argon (principally potassium-derived 39Ar or 3gArK) during laboratory heating. 3171

0. M. Lovera et al

3172

the influence of E and the domain distribution parameters upon log(Dlr*) values? Moreover, how do other factors such as the effect of nonuniform K distributions within domains affect the determination of E and DO? In an attempt to address these issues, we have analyzed the results of 115 individual ““Ar/“Ar step-heating experiments performed using low-temperature K-feldspars at UCLA over the last several years (Table I; the complete contents of the database including individual data tables and derivative plots are available from the following WWW site: http://oro.ess. ucla.edu/argon.html). The samples (see Table 1) represent a wide range of feldspar textural varieties collected from numerous distinct geologic environments. To maintain a self-consistent approach, we determine E and log( Do/r (?,) values by systematically applying a weighted least square regression to the initial heating steps of each sample. Because this focus is geared towards addressing sample-tosample variation in diffusion parameters, the issue of whether individual samples can possess multiple activation energies (e.g., Harrison et al., 1991) is deferred to the discussion. Our systematic analysis of these data permits examination of key aspects of the MDD model, clearly defines normal and aberrant kinetic behaviors of K-feldspars, permits comparison of results with alternative models (e.g., Lee and Aldama, 1992), and provides a basis for improvements to the application of MDD theory to recover geological thermal histories. 2. EXPERIMENTAL The 115 K-feldspar samples used in this investigation are listed in Table 1 along with relevant information. In performing the stepheating experiments on these samples, we have employed different types of heating schedules in an attempt to determine and assess the quality of “ArK diffusion parameters. In nearly half of these samples, we adopted the practice of performing isothermal duplicates (obtaining two or more successive measurements at the same temperature before proceeding to the next higher temperature step) during the initial stages of gas release (-450-750°C). Such a heating schedule allows both an indication of reliability of the estimate of activation energy (Lovera et al., 1993) and provides the basis for correcting ages for Cl-correlated, excess 4”Ar (Harrison et al., 1994 ) Another approach to step-heating involved cycling back to lower temperatures. The principal advantage of this method is to permit evaluation of the activation energy at later stages of the degassing (Lovera et al., 1991). An important consideration in all heating schedules is that the volume fraction of 39Ar released prior to melting (above - 1150°C) be maximized. We achieve this by performing progressively longer heating steps at 1100°C. Finally, although the rates of .‘YAr release measured as the K-feldspar melts possess no geologic significance, all “Ar that remains within the sample above I 100°C must be measured in order to estimate accurate fractional loss at each step from which to calculate diffusion coefficients. Our ability to accurately monitor temperature and heating duration and “Ar release represent fundamental concerns in comparing results between K-feldspar 39Ar kinetic experiments. Considerations of secondary importance in the calculation of diffusion parameters including line blanks, irradiation parameters, and mass spectrometer calibration procedures that may vary between samples. In most cases, appropriate experimental details for individual samples are provided in references listed in Table I. 2.1. Temperature-Time

Monitoring

The step-heating experiments were all performed at UCLA using double vacuum, resistance McDougall and Harrison,

furnaces (Harrison and Fitz Gerald, 1986; 1988). Temperature control in all experi-

ments was achieved with a feedback loop between a W-Re thermocouple inserted upwards into a 2 mm diameter well bored into the I cm thick base plug of a vertical Ta crucible (- I -2 mm beneath the sample) and a thyristor unit that regulates power supplied to a split-ring Ta heating element. The duration of heating steps in our experiments varied from 10 min to over 24 h depending the prior heating schedule and the desired fractional loss of “Ar. A run was considered to have begun when the temperature recorded by the thermocouple positioned at the base of the crucible matched the set temperature (normally -2 min after the furnace is powered). The run was considered to be over upon power shutoff to the heating element. Although time can be measured with high precision, uncertainty results from the thermal lag prior to achieving steady-state. Using this system, the precision of the temperature measurement IS nominally limited by the stability of the thyristor (better than 0.0 I mV or 2 1°C). Such precision applies only when heat-flow within the crucible has achieved steady-state. Because this condition must be established each time the furnace is powered, the uncertainty in the temperature measurement is considerably greater for short duration runs. In general, the measurement method appears to result in an overestimate of temperature. For example, tests performed with an independently-calibrated, chrome]-alumel thermocouple inserted against the base of the crucible in the same position that the sample occupies indicate that more than 20 min may be required to achieve steady-state when the crucible is heated from room temperature. The dual thermocouple experiments indicate that reported temperatures are accurate to within ? 10°C at the position of the sample between 500- 1100°C provided that the furnace has been previously equilibrated at -3OO-350°C for -30 min prior to beginning heating steps. These conclusions are supported by melting experiments performed with pure tin, aluminum, and copper foils. In general, we have attempted to begin all heating steps from a preheated condition (- l50-200°C below the target temperature). Despite this, between 1 and 5 min elapse between the time the external thermocouple indicates the set temperature has been reached and the time the internal thermocouple attains steady-state. During this interval, temperature is typically indicated to be within 10°C of the set point. The decline in crucible temperature after cutting power to the furnace is typically between 50 and 100”C/min, depending upon initial temperature. This effect partially offsets the thermal lag during power up. In light of these uncertainties in temperature and the thermal lag time we estimate that relative uncertainties of 25°C and ? I .5 min are representative for the great majority of our experiments. Although the absolute uncertainty in temperatures characterizing different experiments could be greater, we have routinely reproduced diffusion parameters in replicate analyses of the same material. For example, the spread in values of E obtained from six separate diffusion experiments performed with untreated MH- 10 K-feldspar ( -46-49 kcallmol; (Lovera et al., 1991, 1993) essentially agree with an experimental uncertainty of -C1.5 kcal/mol calculated for the individual step-heating experiments. For comparison, a systematic temperature offset of 50°C between different experiments performed with the same material would result in a difference in activation energy of 6 kcal/mol. 2.2 Argon Analysis Three separate mass spectrometers were employed in the stepheating experiments reported in Table 1. Calibration of argon sensitivity and mass discrimination in all three instruments was performed with air pipette systems containing splits of a common air argon ampoule. Most analyses were performed with either a VG 3600 or VGl2OOS mass spectrometer. Samples sizes were generally lo-30 mg, and gas was transferred by expansion. The VG 3600 utilizes a Nier-Bright source yielding an Ar sensitivity of 3 x 10M4 A/torr at a mass resolving power (MRP) of 450. The VG12OOS utilizes the Baur-Signer GS-98 ion source (Baur, 1980) which is characterized by a high argon sensitivity (3 X lo-” Altorr) and very low pressure dependence. Typical operating MRP is 90. The reproducibility of manometric argon measurements in either instrument is typically better than 2%. For example, reproducibility of 4DAr intensities for the VG 3600 determined from eighteen blocks of three to five replicate atmospheric argon aliquots measured over three months ranged

Significance of activation energy determinations

3173

Table 1. K-Feldspar database. Sample

Type of

name

analysis

621 927 940 26-l/90 37190 54190 59190 639-KRX 67190 93-NG-17 93-NG-37 93-NG-6 AC BW-2 CARARA CB2493 CG194 CL-BB-19 CM-l CP-128 cv DC-1OC DC-2A DL-3 DSC ET194 F-185 F-223 FA-1K FA21K FA4lK FASK FA7K FAIK FA9K GR-04 GR-11 GR-12 GR-13 GR- 14 GR-15 GR-16 GR-18 GR- 19 GR-20 JU KAIL2K MCA994 MH8911 MH8913K MH8914K MH8915K MH8916 MH8926 MH8932K MH897K MN-l MN-7

D ED D’ D D D D D D D D M D D : D : M C, D C, D D D D D D c” C M C M M : D D D D D D D D CM D D C M D M M M : D D

Type of analysis (Heating schedules) M = monotonic increments D = isothermal duplicates C = temperature cycling Rock Types Gr = granites Gr. Peg = granite pegmatite Gd = Granodiorite Tn = Tonalite Sy = Syenite Gn = Gneiss Sch = Schist

Rock type Gn Gr Gn Gr Gr Gr Gr Gr. Gr Gr Gr Gr Tn Gr. Gr. Tn Tn Gr Gr. Gd Gd Gn Gn Gd Gd Tn Gr Gr Gn Gn Gn Gn Gn Gfl Gfl Gd Gd Gd Gd Gd Gd :: Gd Gd Tn Gr Tn Gn Gn Gn Gn Gn Gn Gn Gn

Locality

-

Ref.

-

EG EG EG EG

10 10 10 10

EG VT VT VT PRB PRB PRB PRB SEA PRB PRB PRB RRSZ RRSZ PR PRB PRB SET SET RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RZT RZT RZT RZT

10 5 5 5 12 12 -

Peg

Peg Peg

Peg

g5 RZT %:: RZT PRB SWT PRB SH SH SH SH SH SH SH SH GT GT

:: 16 12 12 12 1 1 11 12 13

2 2 2 : 2 2 14 14 14 14 14 14 14 14 14 14 12 3 13 4 t 4 t 4 4

Sample name MP MP-17 N-12 N-13 N-14 N-15A N-16 N-17A N-19 N-21 N-25 N-5 N-6 N-8 PN794 PP- 1 PRP86-89 QH-1 SA409 SEM394 SEM694 SFH SJ394 SL-1 SM-11 SP SY TR-210 TR-83 TR-BB-86 TS-1 v-103KF V-12KF v-1KF v-4KF XHlOB2K YA-24 YANS-1 YANS-11 YN-27 YU-36K YU-38K YU-39K YU-40K YU-56K YU-57K YU-62K YU-94K YU-97K YX-16K YX-29K YX-41CK YX-42BK ZD-41 ZD-50 ZH-2B ZH-3

Type of analysis M C D D D

C, D D C, D D ; D D :: D D D D D D C, D D D ; M D D D : D D D ; D D : : C C : D D ; D D D D D D

Rock type Tn 2 Gd Gd Gr Gd Gr SY Gd Gd Gd Gn Gd Tn Gr Gd Gd Tn Tn Tn Tn Tn Gd Gd Tn Gd Gr Gr Gr Gn Gn Gn Gn Gn Gd Gd Gn Gn Gn Gn Gn Gn Gn Gn Gn Gn Gn Gn Gn Gn Gr :: Gd

Locality GT PRP GT GT GT GT GT GT GT GT

PR PR GT

PRB PRB PRB PR PR PRB SET SET SEA RRSZ RRSZ RRSZ RRSZ GT RRSZ GT GT RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ RRSZ HIM HIM GT GT

Ref. 12 12 8 8 8 8 a : 8 : 13 11 11 8 13 13 13 12 13 11 11 12 12 16 2 6 6 7 6 8 8 2 : : 6 6 6 6 6 2 : 9 7 7

:

Locality EG = Eldiurta Granite, Greater Caucasus, Russia PRB = P&insular Ranges Batholith, southern & Baja CIalifomia RRSZ = Red River Shear Zone. Yunnan and Vietnam SH = Shandong Province, PRC’ GT = Gandese Thrust, southeastern Tibet PR = Puerto Rico VT = Vincent Thrust, southern California, USA HIM = High Himalaya RZT = Renbu Zedong Thrust SWT = Southwest Tibet SET = Southeast Tibet SEA = Southeast Australia

References (1) Leloup et al. (1993) (2) Harrison et al. (1992) (31 Rverson et al. (1995) <4j &en et al. (1942) ’ (5) Grove and Lovera (1996) (6) Harrison et al. (1996) (7) Yin et al. (1994) (8) Harrison et al. (1995) (9) Dransfield et al. (1997) (10) Gazis et al. (1995) (11) Smith and Schellekens ( 996) (12) Grove (1993) ( 13) Rothstein ( 1997) (14) Quidelleur et al. (1997)

0. M. Lovera et al

3174

from 0.4 to I .7% (mean of 1.1% ). Similar analysis over an equivalent time period performed with the VG 1200s yielded a range of 0. I -2.9% (mean of 1.7%). Procedures implemented to regulate the amount of gas delivered to either mass spectrometer limit the dynamic range of pressures applicable to this study to a factor of about 20. Tests performed using atmospheric argon aliquots do not detect pressure nonlinearity effects at the percent level when the reproducibility of individual ion intensities are taken into account. About 10% of the samples were analyzed with a Nuclide 4.560.RSS mass spectrometer operated in the Faraday mode with an ““Ar sensitivity of 1.5 X 1Om’5 mol/mV. Sample sizes were generally -200 mg, and gas was transferred quantitatively by trapping it on activated charcoal maintained at liquid nitrogen temperatures. Because the total quantity of gas admitted to the mass spectrometer was governed using a leak valve, orifice corrections were required. Reproducibility of manometric argon measurements for this instrument were typically l-2%.

ues of log(Dl?) is determined largely by the analytical uncertainties discussed in the previous section. Propagation of these uncertainties into the calculation of 1og(Dlr2) is summarized in Appendix A. Based upon these calculations we estimate that errors in our log( D/r’) values are typically 10.05 log units (in s ~’ ) A composite Arrhenius plot displaying representative experimental results from fifty-seven of our samples is shown in Fig. 1. The samples selected were chosen because all were processed using duplicate isothermal steps from 4.50 to 750°C. Note that despite the variety of materials examined (Table 2)) log( D/r’) values from all samples define a spread of less than 2 orders of magnitude at any given temperature. This variability is significantly less than that obtained when step-heating results are normalized according to an estimated diffusion radius (Mussett, 1969). This observation is maintained when samples degassed with somewhat different heating schedules are also considered. In order to facilitate comparison of our results with those from other laboratories, we have also performed step-heating experiments with a sanidine glass synthesized at 1300°C (Fig. I ). The sanidine glass results are calculated in terms

3. RESULTS

3.1. Arrhenius Parameters Low-temperature tained

within

using

a uniform

Arrhenius

the

database

data for the K-feldspars (Table

1) has been

Our ability to determine

set of criteria.

900 0

I

I

val-

‘U”C)

800 I

con-

examined

600

700 ,

I

m

l 0

500

I I K-feldspars from Database - Step heating Sanidiie Glass (r = 0.04 cm) Madagascar Orthoclase - Step heating Benson Mines Orthoclase - Step heating I

I

I

.

. ..._

-6

10000/T(K) Fig. 1. K-feldspar “Ar kinetic results from UCLA ( 1993- 1995): Arrhenius results obtained from step-heating of a single 40 mg fragment of sanidine glass (filled squares). Note that log(D/r*) values have been calculated assuming plane slab diffusion geometry and normalized to r = 0.04 cm to facilitate comparison with the K-feldspar results. Arrhenius results for fifty-seven K-feldspar samples analyzed following similar heating schedules (isothermal duplicates performed from 450 to 750°C). Shown for comparison are log(Dlr’) results from two gem quality specimens, Benson Mines orthoclase and Madagascar orthoclase. The bulk loss (bold line) and step-heating results (open squares) for 127 pm diameter Benson Mines orthoclase grains were recalculated from Foland ( 1974) and Foland ( 1994). respectively, using plane slab geometry. Madagascar orthoclase results (filled circles) are from UCLA.

Significance of activation energy determinations

3175

Table 2. Results of activation energy determinations. Sample

logW4

name 627 921 940 26-l/90 37l90 54190 59190 639-KRX 67190 93-NG-17 93-NG-37 93-NG-6 AC BW-2 CARARA CB2493 CG194 CL-BB-19 CM- 1 CP- 128 cv DC-1OC DC-2A DL-3 DSC ET194 F-185 F-223 FA-IK FA2lK FA4lK FASK FA7K FAIK FA9K GR-04 GR-I 1 GR-12 GR-13 GR-14 GR-1.5 GR-16 GR-18 GR-19 GR-20 JU KAIL2K MCA994 MH8911 MH8913K MH8914K MH8915K MH8916 MH8926 MH8932K MH897K MN- 1 MN-7

(s-‘) 37.2 38.3 43.5 47.1 35.8 52.6 40.0 41.2 36.9 41.6 44.3 40.8 41.7 32.0 45.0 53.6 44.0 42.7 51.5 43.1 36.8 48.8 46.4 44.0 59.2 63.6 58.7 60.6 45.3 42.5 51.3 51.1 44.4 50.0 50.8 38.9 51.8 42.1 47.0 44.7 50.0 47.6 46.0 43.7 39.0 37.2 43.8 62.5 50.7 55.3 46.1 56.7 53.8 43.3 45.5 47.7 45.6 42.1

? 0.9 2 1.7 t 2.1 + 3.8 + 3.1 rf- 4.1 ? 0.9 + 1.1 + 3.0 2 1.1 t 1.0 + 0.9 2 1.2 2 1.3 2 1.3 ? 3.0 + 1.3 t 2.8 -t 2.1 + 0.9 + 2.0 + 1.2 ? 1.2 ? 0.8 2 3.2 + 3.4 % 1.6 + 3.2 t 1.5 2 1.9 + 1.1 c 2.1 2 1.5 2 1.7 + 1.7 + 2.7 + 2.1 f 1.3 t 1.4 + 1.3 + 2.1 + 1.3 % 1.0 t 1.3 * 1.2 2 2.1 + 0.8 + 4.0 ? 1.7 % 1.5 t 1.1 + 3.0 t 2.7 2 1.2 * 1.5 & 1.5 t 1.3 +- 0.8

3.0 3.4 5.1 5.3 2.5 6.8 3.1 2.9 3.3 3.7 4.4 3.6 4.9 0.3 4.5 7.5 4.2 4.8 7.0 4.0 2.3 6.4 6.2 3.7 8.4 9.0 8.3 9.7 4.8 4.1 6.2 6.7 4.9 6.4 6.3 4.0 6.8 4.4 5.3 4.8 6.3 5.3 4.7 4.3 3.0 3.6 4.1 10.1 5.8 6.8 5.0 7.9 7.2 4.1 5.0 5.8 4.4 3.2

r 0.2 t 0.5 ? 0.6 + 1.1 2 0.9 t 1.2 + 0.2 -+ 0.3 + 0.9 + 0.3 + 0.3 f 0.2 2 0.3 + 0.4 + 0.3 t 0.8 ? 0.3 ? 0.8 t 0.6 + 0.2 t 0.6 ? 0.3 + 0.3 + 0.2 + 0.9 + 1.0 ? 0.4 + 0.9 ? 0.4 * 0.5 % 0.3 2 0.6 t 0.4 c 0.5 5 0.4 t 0.8 + 0.6 + 0.3 + 0.4 + 0.4 + 0.6 * 0.4 ? 0.3 + 0.3 % 0.3 2 0.6 2 0.2 t 1.1 ? 0.4 c 0.4 f 0.3 2 0.8 + 0.7 t- 0.3 + 0.4 % 0.4 + 0.4 t 0.2

Goodness of fit 0.38 0.11 0.94 0.03 0.01 0.24 0.19 0.47 0.01 0.85 0.18 0.90 0.95 0.69 0.97 0.85 0.69 0.61 0.96 0.32 0.2 0.71 0.17 0.89 0.66 0.47 0.94 0.53 0.49 0.80 0.58 0.40 0.69 0.75 0.01 0.69 0.42 0.92 0.82 0.78 0.90 0.95 0.98 0.95 0.83 0.4 1 0.86 0.80 0.35 0.68 0.06 0.41 0.88 0.20 0.67 0.5 1 0.94

Sample N

f

13 7

0.145 0.034 0.043 0.012 0.016 0.011 0.059 0.020 0.013 0.051 0.083 0.08 0.057 0.009 0.041 0.023 0.070 0.030 0.046 0.061 0.009 0.091 0.166 0.055 0.011 0.006 0.068 0.052 0.060 0.040 0.088 0.060 0.082 0.040 0.075 0.029 0.032 0.073 0.055 0.055 0.032 0.050 0.068 0.044 0.038 0.056 0.097 0.020 0.041 0.068 0.055 0.027 0.03 1 0.089 0.085 0.096 0.032 0.060

I 4 4 4 11 10 4 10 11 11 8 8 z 9 5 7 10 4 12 10 14 6 6 9 6 6 : 2 5 5 5 7 9 9 9 I 9 11 9 9 5 14 5 6 6 11 4 4 7 z 9 12

MP MP-17 N-12 N-13 N-14 N-15A N-16 N-17A N-19 N-21 N-25 N-5 N-6 N-8 PN794 PP- 1 PRP86-89 OH-l iA SEM394 SEM694

SFH SJ394 SL-1 SM-11 SP SY TR-210 TR-83 TR-BB-86 TS-1 v-103KF v-12KF V-IKF V4KF XHlOB2K YA-24 YANS-1 YANS-11 YN-27 YU-36K YU-38K YU-39K YU-40K YU-56K YU-57K YU-62K YU-94k YU-97K YX-IhK YX-29K YX-41CK YX-42BK ZD-41 ZD-50 ZH-2B ZH-3

Goodness of fit

l0gWd

name

(s-9

38.8 37.1 45.6 46.8 55.1 58.0 49.3 46.4 54.6 67.5 57.3 46.3 43.4 45.6 58.4 45.2 45.7 50.0 53.1 63.5 62.2 39.8 32.5 44.0 39.2 48.0 44.2 42.5 52.6 42.1 45.5 51.2 59.6 59.1 56.1 39.9 45.2 48.2 55.2 52.2 48.0 46.9 42.4 44.7 49.6 45.9 47.7 42.9 47.7 45.4 49.4 40.5 42.1 61.4 46.3 39.1 31.1

t 1.2 rt 2.0 + 1.9 2 3.1 + 3.5 -c 2.9 -c 3.2 2 1.4 ? 3.5 2 5.2 i- 2.3 + 2.0 t 1.8 ? 1.0 + 3.8 r 3.2 ? 1.6 * 1.5 2 3.6 -c 4.2 2 3.4 If: 1.7 & 2.4 * 1.2 2 1.9 + 1.4 % 1.0 % 1.3 + 3.3 + 2.7 2 1.0 2 1.3 ? 3.3 t 2.2 + 2.9 t 2.0 ? 3.1 ? 1.4 + 3.4 2 2.9 2 1.3 ? 1.2 IT 1.7 % 2.5 & 1.0 + 1.3 * 1.0 + 1.0 ? 0.8 * 0.9 -t 1.0 ? 2.5 I 1.5 % 3.3 % 0.8 ? 3.4 2 2.6

3.4 1.9 5.6 5.8 8.1 8.4 6.7 5.0 8.3 12.2 8.2 5.0 4.2 5.0 7.9 4.6 4.8 5.7 7.0 8.9 9.0 3.7 0.4 4.1 3.4 5.6 4.1 4.6 8.2 4.0 5.0 6.8 9.2 8.7 8.0 5.2 5.0 5.4 7.6 7.1 5.4 4.6 3.5 4.8 5.5 4.7 4.9 4.4 4.1 4.5 5.2 3.4 3.1 9.3 4.4 2.3 0.1

* ? 2 t 2 + r ? t * t 2 ? * + c 2 + * + -c 2 + rt ? t % % 2 + f % t ? + + -c % t ? + 2 * t 2 + r ? ? & 2 * % ? + ? 2

0.3 0.6 0.5 0.9 1.0 0.8 0.9 0.4 1.0 1.5 0.6 0.5 0.5 0.3 1.1 1.0 0.4 0.4 1.0 1.2 0.9 0.5 0.7 0.3 0.5 0.4 0.3 0.3 0.9 0.8 0.3 0.3 0.9 0.6 0.8 0.6 0.9 0.4 1.0 0.8 0.3 0.3 0.5 0.7 0.3 0.3 0.3 0.3 0.2 0.2 0.3 0.7 0.4 0.9 0.2 1.0 0.8

0.42 0.49 0.85 0.97 0.82 0.06 0.84 0.94 0.23 0.01 0.24 0.65 0.44 0.97 0.22 0.49 0.83 0.41 0.36 0.48 0.77 0.24 0.81 0.17 0.96 0.67 0.25 0.41 0.60 0.89 0.35 0.98 0.09 0.70 0.96 0.41 0.74 0.55 0.15 0.13 0.60 0.35 0.70 0.69 0.54 0.87 0.62 0.47 0.99 0.37 0.32 0.25 0.96 0.92 0.93 0.41

N 8 4 7 5 5 7 1; 5 4 7 7 7 11 5 4 8 9 5 5 6 5 5 10 : 9 9 5 5 12 9 6 7” 7 ; 5 : 8 6 4 10 8 9 12 15 14 11 6 8 6 13 4 5

f 0.052 0.005 0.053 0.022 0.023 0.017 0.026 0.046 0.032 0.049 0.032 0.022 0.021 0.103 0.006 0.003 0.02 1 0.036 0.029 o.c05 0.010 0.016 0.004 0.027 0.026 0.070 0.056 0.088 0.077 0.018 0.097 0.099 0.025 0.049 0.019 0.098 0.012 0.045 0.016 0.024 0.071 0.041 0.011 0.010 0.098 0.05 1 0.083 0.096 0.116 0.096 0.063 0.011 0.007 0.017 0.115 0.001 0.002

N = Number of heating stegs regressed Regressed fraction of 3 AI released

f=

of log(Dlr’) using planar geometry and a length scale (r = 0.04 cm) to facilitate comparison with the K-feldspar data. Arrhenius results for two gem-quality orthoclases (Benson Mines orthoclase and Madagascar orthoclase) are provided for additional reference. To ensure direct comparison with the database samples, the Arrhenius plots for Benson Mines were calculated from the Foland ( 1974, 1994) results using plane slab geometry. The measured K-feldspar diffusion coefficients plot intermediate between the

log(Dlr2) values of the two reference materials. The data systematically converge with increasing temperature (to a spread of less than 1 order of magnitude) and become closer to the values obtained from Madagascar orthoclase. According to the MDD model, a linear Arrhenius array of log(Dlr’) values with a slope proportional to the activation energy of the sample is expected as long as the K-derived 39Ar from the smallest diffusion domains present within the K-feldspar are degassed with the appropriate resolution. For

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a plane slab diffusion geometry, this linearity is maintained as long as the fractional loss of ‘9Ar release from the smallest domain does not exceed about 60% (see Appendix C of Lovera et al. ( 1989), Eqns. Cl-g). The likelihood of maintaining this linearity is generally improved by minimizing the fraction of 39Ar released during the initial steps. This is generally accomplished by beginning the step-heating experiment at temperatures below 500°C for heating durations of - 10 min. As j9Ar is exhausted in the smaller domains during progressive heating, log( D/r*) values of successive steps decline causing convex up curvature in the Arrhenius plot (Fig. 1). Our approach to estimating K-feldspar values of E and log (DO/r;) for ‘“Ar diffusion in step-heating experiments is illustrated in Fig. 2 and described in detail in Appendix B. In calculating the best-fit solution, we increase the number of points used in the regression until the goodness of fit weighted by the number of data points achieves a maximum. Because the first data point is likely to misrepresent the diffusion properties of the sample due to surface anisotropy, recoil and/or small scale compositional variations, we considered it as an outlier if it lies more than 2 standard deviations from the resulting line. We emphasize that only the first datum is dealt with in this manner and that less than 10% of our samples were affected. Values of E and log(Dlr*) calculated for our K-feldspars are displayed in Fig. 3a and 3b, respectively. One sigma uncertainties in E reported in Table 1 are typically + l-2 kcal/mol. A modified Kolmogorov-Smirnov goodness-of-fit test using the Kuiper statistic (Press et al., 1988) indicates that the distribution of E and log(Dlr’) values both define crudely normal distributions (Fig. 3a,b). The overall dispersion in E shown in Fig. 3a is about 30 kcal/mol. A plot of E vs. fraction of “Ar used in the regression (Fig. 3c) shows a sharp drop in intrasample variation of activation energy as the regressed gas fraction increases. At the same time it is important to point out the permissible fraction of “Ar regressed by the method outlined in Appendix B is an intrinsic property of each sample. In other words, increasing the number of steps (and hence the “Ar fraction) used in calculating activation energies for low-gas fraction samples in Fig. 3c would shift all the E values to lower values maintaining the observed high dispersion but reducing the overall quality of fit. A final observation is that calculated E and log(DOlri) values exhibit a high degree of correlation (correlation coefficient = 0.97; Fig. 3d). Analysis of the mathematical formulation of the MDD model (Lovera et al., 1989: Eqns. Cl -8) shows that negligible difference in log(Dlr’) between successive isothermal steps ( Alog(Dlr’)) are expected when the smallest domains retains at least 40% of their initial “Ar concentration, Values uncertainty, therefore, of Alog( D/r*) within experimental are expected to represent a sufficient condition to obtain an accurate value of E. Accordingly, we have examined Alog(Dlr’) values to evaluate both the capability of the isothermal duplicates method to identify this condition (smallest domain releasing less than 60% of its “Ar) and the ability of the approach summarized in Fig. 2 to yield the activation energy of the sample.

Plots of Alog(Dlr*) vs. E for the first three pairs of isothermal duplicates (450, 500, and 550°C) are shown in Fig. 4a-c. Note that while Alog(Dlr*) values for the first duplicates (450°C) exhibit more dispersion than those obtained for the 500°C duplicates, a mean value of 0.04 was obtained in each case. The larger dispersion of the first set of duplicates (Fig. 4a) may be due to the second order effects mentioned above that primarily affect the first step. The distribution of Alog( D/r*) values for the third duplicate pair (550°C; Fig. 4c) show a marked shift toward positive values (mean value of 0.11). This shift towards positive values characterizes all subsequent duplicate pairs measured at higher temperatures than 550°C and appears to reflect the progressive depletion of the smallest domains in most of our samples. A plot of weighted mean Alog(Dlr*) values vs. E calculated from results measured at 450 and 500°C is shown in Fig. 4d. The sign of the weighted mean corresponds to that obtained from the sum of both Alog( D/r*) values. A histogram of the weighted mean of the 450 and 500°C Alog(Dlr*) values reveal a Gaussian distribution with a mean of 0.01 and a standard deviation of 0.13 (Fig. 5a). Samples lying more than one standard deviation from the mean tend to exhibit the systematic behavior shown in either Fig. 5b or Fig. 5c. Most of the deviant samples yield positive Alog(Dlr*) values (see 37/90 K-feldspar in Fig. 5b). A more unusual phenomenon involves systematic increase in the log(Dlr*) value for the second of two isothermal steps (see YN-27 K-feldspar in Fig. 5~). Numerical analysis described in Section 4.2. indicate that the strongly positive Alog( D/r*) values obtained from samples similar to 37/90 K-feldspar (Fig. 5b) could lead to underestimation of their activation energies. Alternatively, similar calculations suggest that samples resembling YN-27 K-feldspar (Fig. 5c) would lead to overestimation of activation energies when our calculation scheme is applied. In spite of these expectations, we note that no correlation exists between mean Alog(Dlr*) values and calculated activation energies. 3.2. Domain Distribution Parameters The form of the Arrhenius plot (log( D/r’) vs. 1lT or its associated log( r/r”) plot (Richter et al., 1991) is a function of the parameters that characterize the individual diffusion domains (activation energy (E), frequency factor (DO), domain sizes (p), and volume fractions (4) ; see Fig. 6a). We have recently incorporated routines (Quidelleur et al., 1997) that employ Levenberg-Marquardt methods (a generalizalion of least square routines to nonlinear cases; see Press et al.. 1988 ) to find the maximum likelihood estimate of the domain distribution parameters (p, 4) by minimizing the chi-square quantity

where (T, = standard deviation, y, , x, = data points, u, = model parameters, N is the number of laboratory steps available, and M is the number of independent parameters of the model. For example, when the Arrhenius results are modeled using S diffusion domains, M = 2. S - 2, due to the following normalizations: pN = 1 and I: o = I. For each

Significance of activation energy determinations

3177

-4 E = 41.6 f 1.1 kcal/mol

pWW+@ @@tB@ @@

:

log(DO/r02)= 3.7 f 0.3 s-l

-6

-8 0

Measured below melting

H Measured above melting Values used for least square fit

l

-10.

.

6

.

8

.

12

10

14

10000/T(K) 12L

4 10 46

9

8

42

* x b

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6 4 2 0.

0 34

2

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2

I.

1.

I.

4

6

8

I.

I.

I

00-o *

I

’

10 12 14 16 18 N

Fig. 2. Method employed to calculate Arrhenius parameters (E and log(&/r~) from K-feldspar samples. (a) Arrhenius plot for 93-NG-17 K-feldspar. Uncertainties (k la) shown for individual diffusion coefficients were calculated following the procedure outlined in Appendix A. Diffusion coefficients used to calculate the indicated values for E and log(Q,/r:) are indicated by filled symbols. The solid line represents a weighted, least squares fit to these data (see Appendix B for details); (b) Variation in E resulting from least squares fitting of successive data points (N) starting from the initial measured value; (c) Variation of the goodness of fit (q) as a function of N. As discussed in Appendix B, the set of the first N diffusion coefficients corresponding to the maximum value of q X N was regressed to obtain the diffusion parameters. The filled symbols in (b) and (c) represent this condition.

sample there is both a minimum number of domains required to adequately fit the experimental results and an upper limit to the number of domains beyond which the overall fit is not appreciably improved (Lovera et al., 1991) . To facilitate

intersample comparison of domain distribution parameters we have used eight domains for all samples which results in acceptable distributions. For most K-feldspars, this number of domains lies between the above-mentioned limits.

0. M. Lovera et al

3178

25

-46

+ 6-

25

30

40

50

60

0

;

2

4

6

8

10 12 14

E (kcal/mol)

60a 2 50$ ti ;;; 40intercept = 29.4 f 0.5 slope = 3.33 * 0.08

30-

.210 Cumulative

fraction 3YArreleased

-4

0

4 log(DpJ

8

12

16

s-l

Fig. 3. Arrhenius parameters calculated for K-feldspars contained in the database: (a) Histogram of activation energies for all 115 samples. A modified Kolmogorov-Smirnov goodness-of-fit test using the Kuiper statistic (Press et al., 1988) indicates that the data satisfy criteria for a normal distribution; (b) Histogram of log(&lr~) values for all samples. Again the data define a normal distribution; (c) E vs. Cumulative fraction of “Ar released for the steps used to calculate the diffusion parameters for each of the samples. Variability in calculated E decreases sharply as the gas fraction defining the reference line (log(Dlr$)) increases: (d) E vs. log(D,Jri) for all 115 K-feldspars. The plot exhibits a high degree of correlation.

In order to illustrate the approach, the model domain distribution parameters determined for 93-NG- 17 K-feldspar are shown in Fig. 6a. The diffusion properties of each of the eight domains are represented by the dotted lines. Note that the log( D/r*) values of the smallest domain plot -2 orders of magnitude above the reference line (log(D,Jr;)) while those of the largest domains occur -3 orders of magnitude beneath log( DO/r :) The associated log( T/T~) plot (Richter et al., 1991) for this sample is shown in Fig. 6b. Note that only 68% of the total 39Ar released was degassed at tempera-

tures below melting. Since the domain distribution can only be constrained by “Ar release below melting, there is a large uncertainty in the size and concentration of the largest domain (4 = 33%, Fig. 7a). Provided that only ““Ar/3’Ar ages calculated from gas extracted below melting are used to constrain the possible T - t paths experienced by the sample, this ambiguity does not have an effect on the determination of the thermal history. Figure 7a displays the log(D,,lp’) values vs. their corresponding “Ar concentrations (4) for the diffusion domain

Significance

of activation

energy

determinations

I

70

14 weighted rn#tn = 0.04

.,

.,

*

.,

weighted IX&I = 0.04

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.,

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-

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Alog(D/r*) Weighted Mean

Fig. 4. Difference in log( D/r’) values between successive isothermal steps ( Alog( D/P-~)) vs. calculated activation energy for fifty-seven samples for which appropriate data are available: Errors for Alog(D/r’) represent kla. and were calculated assuming no correlation. Results from isothermal duplicates performed at 450, 500, and 550°C are shown in (a), (b), and (c). respectively. The proportion of Alog( D/r’) values that are zero within uncertainty are 60% and 80% for 450°C and 500°C. respectively. As indicated in (c), Alog(D/r’) values for isothermal duplicates at 550°C and higher temperatures tend not to be zero within uncertainty. (d) Weighted mean between 450 and 500°C results used to discuss results presented later in the text.

0. M. Lovera et al

3180

0 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Weighted Mean Aiog(D/r*)

-4

0P) 37/90 K-feldspar m

1s I*,

E = 36 f 3 kcal/mol log(D,[r,,‘)

= 2.4 ? 0.8 s

’

-10

CC :I f 7,,, n$ 9 & 0

-6

Yn-27 K-feldspar :1:

+

r.4 fl

from the database are presented in Fig. 7c. Because values of log( DO/p’) are strongly correlated with activation energy ( Fig. 3d). we have restricted our analysis to the seventy-six samples whose activation energies lie within t- I CJof the mean (Fig. 3a). The data define a broad peak centered on log(D,,lp’) = 2.5 s-’ with an asymmetric tail extending to higher values (i.e., less retentive domains). Note that the spread in the largest domain sizes of Fig. 7c is almost twice the spread obtained for 93-NG- I7 (Fig. 7b). A larger contrast appears at the other extreme of the spectrum, where the spread of the distribution of smallest domains is now several orders of magnitude. The composite domain size distribution of Fig. 7c reflects the typical domain properties of lowtemperature potassium feldspars: (I ) the majority of ‘“Ar resides in the largest domains; (2) the smallest domains typically constitute <5% volume fraction of the samples (black columns) and tend to plot -2 orders of magnitude above log( Do/r:) (Fig. 3~); and (3) the largest domains constrained by the samples (-20% by volume; gray columns) have log( Do/p’) values that are typically -3 orders of magnitude below log( Do/r i) (Fig. 7~). The log( D/r’ )

*.

-8 E = 52 f 3 kcahol bg(L)/r,?) -lu

It .

= 7.1 *0.x \ 1300

,

10

11

12

13

1100

900

T(“C)

700

500

14

lllOOO/T(K)

Fig. 5. Examples of different types of systematic Arrhenius behavior exhibited by K-feldspar in low-temperature step-heating experiments. (a) Weighted Mean of Alog(Dlr’) values calculated from 450°C and 500°C isothermal duplicates. Nearly 80% of the samples yield values within uncertainty of zero (Type I behavior: see 93. NG-I 7 in Fig. 2a); (b) Type II behavior refers to samples in which log( D/r’) drop by an amount greater than experimental uncertainty in successive isothermal duplicate pairs; (c) Type III behavior refers to the opposite tendency (see text for details).

-10

Domains 8

6

lOOOUiT(K)

1.8

used to model “Ar data of 93-NG- I7 K-feldspar as shown in Fig. 6a. The ambiguity in the determination of the largest domain and the uncertainties in measured “Ar values leads to a multitude of solutions that fit the experimental data in an equivalent manner. For example, one hundred equivalent domain distributions calculated from the “Ar results of 93-NG- 17 were used to construct the weighted histogram in Fig. 7b. In producing this plot, mean ‘“Ar concentrations for each log(&lp’) bin were obtained by summing the individual log(&lp’) values weighted by their corresponding volume fractions. We then normalized these values by the total number of solutions. The individual distributions of both the largest and smallest domains constrained by ‘“Ar release prior to melting are indicated by different shades of gray. Note that a substantial portion of the larger domains in the distribution were not constrained by our data since their ‘“Ar content were released above melting (stripped pattern). The dispersion observed in the larger domains results from the lack of constraints in our data. Note also that the concentration of the smallest domains is quite small with their relative size well constrained. A weighted histogram of the log( D,,lp’ ) values calculated from the domain distributions obtained for selected samples distribution

I 12

10

1.4

I t 2

I.0

U.6

(I.2 rimental Results -0.2 0

20

40 Cumulative

60

80

100

% “Ar released

Fig. 6. Domain distribution parameters of typical low-temperature K-feldspars: ( a) Arrhenius plot for 93-NG- I7 K-feldspar. Measured log(Dlr’) values appears as open circles. Open squares represent the MDD model fit. The eight domains used to obtain this fit are shown as dotted lines. Note that the log (D/r’) values of the smallest domain plot 2.5 orders of magnitude above the reference line log (D/r i;) (bold line), while those of the largest domains occur 3 orders of magnitude beneath log(Dlrfj); (b) log( r/r,,) plot corresponding to the same sample. Measured log( r/r”) values appears as the bold, light gray curve. The sharp drop in apparent retentivity observed when degassing temperatures exceed I 100°C is due to the onset of melting. Bold curve represent the MDD model fit. Note that only gas obtained at temperatures below melting (-68%) was modeled.

Significance of activation energy determinations

3181

0.0

0.20 g ‘S $ 0.15 3 x 6 0.10

m

Unconshained‘9Ar

lntemxdiatedomains

$ g

0.05

Smallest domain

3 0.00

0

-2 N = 76

2

4

6

8

10

Bwwon Miner Database K-feldspars domain distributions

-2 0.08 P 8g 0.06

Above Melting m

V k 0.04 ”

m 0

unconstitied ‘p‘4r Below Melting Largest domains Intemxdii domains

1 4

6

8

10

Fig. 7. (a) 39Ar concentration vs. the corresponding size (log(L&,/p*)) for the domain distribution shown for 93NG-17 in Fig. 6. We have distinguished domains determined by 39Arreleased below melting from those corresponding to gas released at higher temperatures; (b) Weighted histogram from 100 equivalent domain distributions (all using eight domains). Distributions were calculated as in (a) above and both E (=41.6 kcal/mol) and log(D,,/ri) (= 3.7 s-‘) were held constant. These best-fit solutions yield similar x2 values between 0.06-0.09. Values of log(L&/p*) for each domain were weighted by their corresponding ‘9Ar fractions (4) and then summed into bins of 0.2 to calculate the histogram. Note that while larger domains are not uniquely constrained, smaller domains are sharply resolved; (c) Weighted histogram for database samples with E between 40-52 kcallmol calculated as in (b) above. The asymmetric distribution reflects the typical domain distribution properties of low-temperature potassium feldspars. Specifically the smallest domains constitute <5% volume fraction of the samples (black columns) and tend to plot -2 orders of magnitude above log(&/ri). Alternatively, fhe largest domains constrained by 39Ar released below melting (-20% by volume; gray columns) have log(D0/p2) values that are typically -3 orders of magnitude smaller than log(&/ri). The log(&/r*) value for Benson Mines orthoclase determined by reducing the Foland (1974) bulk loss experiments data using infinite slab diffusion geometry is shown for reference.

value determined by (Foland, thoclase (infinite slab diffusion ence in Fig. 7.

1974) for Benson Mines orgeometry) is shown for refer-

4. DISCUSSION Understanding the intersample variation in activation energies determined in 40Ar/39Ar K-feldspar step-heating experiments (Fig. 3a) is crucial to evaluating the accuracy of thermal histories calculated from the laboratory results. Whether or not the observed distribution represents real variation in the diffusion behavior of the materials examined or is an artifact that results from limitations in the model

systematics used to characterize sample properties or from hidden systematic errors in laboratory measurements is important for accurate determination of thermal histories. Below we explore possible causes that could produce large errors in estimating the diffusion parameters. We first examine the possible significance of the correlated relationship between E and log(&lri). In particular, we analyze the possible effects of hidden systematic laboratory errors in producing the observed correlation. We next test for the effects of random domain distributions and nonuniform 39Ar distribution upon E. Having investigated possible explanations for errors in the diffusion parameters, we explore the effects of these uncertainties upon thermal histories calcu-

0. M. Lovera et al

3182

lated with the MDD model. Finally we discuss other models proposed to explain Ar release from K-feldspar. We emphasize that while these alternative models might explain some of our experimental results. they fail to predict other important features. In any case, their generality precludes extrapolation of laboratory results to Ar loss in nature which in turn severely limits their applicability to determining thermal histories. 4.1. Correlated

Relationship

Between

E and log(&,lri)

The high degree of correlation between E and log(D,Jr g) shown in Fig. 3d is potentially a manifestation of the MeyerNeldel rule, (i.e., compensation between kinetic parameters in Arrhenius-type functions; Meyer and Neldel, 1937). Alternatively, the relationship could also result from underestimation of uncertainties that are highly correlated. In spite of these possibilities, we point out that the restricted range of experimental conditions encountered in our data (i.e., in 1 lT vs. log(Dlr*) space) could reduce or eliminate the apparent statistical significance of this correlation which limits our ability to confirm the existence of a Meyer-Neldel relationship. To test whether the observed relationship occurs as a result of a limited range of experimental conditions, we have performed the following numerical experiment. Using Monte Carlo simulations, we generated 100 values of E and log (D/r:) values at 450°C which define the Gaussian distributions shown in Fig. 8a and 8b. The median value of the Gaussian distributions were selected to resemble our K-feldspar distribution (Fig. 3, 46 5 6 kcal/mol and 8.7 ? I s-’ at 450°C). Each activation energy was then randomly paired with a 1og(D450sc/rg) value. From each pair of values, the corresponding log( D/r;) value at 1lT = 0 was calculated (i.e., log(D,Jri)). Because the resulting plot of E vs. log (Do/r:) (Fig. 8c) reproduces the same strong correlation as observed for our empirical results (Fig. 3d). it is possible that the empirical relationship simply reflects the narrow range of T - t conditions employed in our experiments. Thus while a compensation relationship cannot be ruled out, the statistical significance of the observed correlation is substantially muted. In spite of this conclusion, the fact that we were able to reproduce the slope of Fig. 3d using a Monte Carlo simulation has important implications concerning the possible existence of hidden systematic errors in our laboratory measurements. Note that the slope obtained in Fig. 3c, (i.e.. E (kcal/ mol) = 29.7 + 3.33 log(&lri)), is in agreement with the value obtained from the diffusion of a variety of species (e.g., Na, 0, K, Rb, Ar) in feldspar (Giletti, 1990) of E (kcal/mol) = 50.7 + 3.4 log& (Harrison et al., 1991). We interpret the offset (i.e., 50.7-29.7 = 2 1.O) between the two lines to reflect the average effective diffusion domain size of our samples. That is, an effective size (rO) of -6 pm is required to reconcile (Hart, 198 1) relationship with the line defined by our K-feldspar values in Fig. 4d. This is intermediate between our previous estimates of the size of the largest ( - 100 pm) and the smallest ( - 1 pm) domains of MH- 10 K-feldspar (Fitz Gerald and Harrison, 1993; Lovera et al., 1993). For an activation energy of 43.8 kcal/mol (i.e., Benson Mines orthoclase), this estimate of r0 yield a frequency

factor of Do = 0.0063 2 0.005 s -‘. somewhat lower but within uncertainty of the 0.00982 s -’ value reported by (Foland, 1974). Note that the Do value calculated from the E vs. log(D,,/r,,) relationship is only an effective frequency factor for these samples and may not reflect the intrinsic Do value of K-feldspar. 3.1. I. Hidden system&

experimentui erwrs

Although we consider it unlikely that our results are significantly influenced by hidden systematic experimental errors, we investigated the possible implications of unanticipated problems in experimental methods. The variable with the highest potential for producing spurious errors is temperature. The well-behaved nature of the isothermal duplicate measurements and the reproducibility found in our experiments (Lovera et al., 1993) rule out errors in relative temperature control greater than 55°C (Fig. 4). However, the accuracy of the absolute temperature calibration was not tested on a regular basis. If large systematic errors in temperature calibration (i.e., 2 250°C) that varied from sample to sample affected our experiments, then the resulting diffusion parameters would yield a distribution resembling that shown in Fig. 1. Specifically, runs performed with true temperatures systematically lower than that read from the thermocouple would yield anomalous low log(Dlr’) values while actual temperatures higher than those yielded by the thermocouple would produce the opposite effect. However, close inspection of our experimental results reveal important features that either cannot be explained by, or contradict, the hypothesis that large systematic errors in temperature varying from sample to sample occur in our laboratory. Specifically, systematic errors in temperature calibration (-50°C) would result in a E vs. log(DJrt) correlation characterized by a slope that is twice as large (-7 for E = 46 kcal/mol; log( Do/r i) = 5 s -‘) as that indicated by our data (3.3; Fig. 3d). The agreement between the slope value obtained from the experimental data (3.33, Fig. 3d) and that yield by our Monte Carlo simulation (3.13, Fig. 8c) indicates that any systematic errors affecting our data must be small. In addition, if the spread in E results solely from systematic errors in temperature, then a relatively narrow range of log (D45,,0cIr~) values (-8.8 t 0.4-l) should be associated with samples yielding similar activation energies (46 t 1 kcal/ mol ). However, samples from the database characterized by E = 46 t I kcal/mol yield the same spread of log(Dlri) values at a given temperature as is obtained from all samples (i.e., -2 log units, Fig. I ). 4.2. Intersample

Variation

in Activation

Energy

Consideration of the results summarized in Table 1 indicates that our approach to determining activation energies for individual samples normally results in an uncertainty on the order of l-2 kcal/mol. Errors of this magnitude are appreciably less than those required to explain the spread of values (-30 kcal/mol) for E indicated in Fig. 3a. Although certain effects, such as variation in diffusion geometry, will alter the calculated value of E, the magnitude of this effect is unlikely to exceed the quoted uncertainty in activation

Significance of activation energy determinations

- 30

35 40

45 so

55 60 l

-!O.O -9.5

-9.0

3183

-8.5

-8.0

-’

E (kcalhol)

intercept = 29.19 * 0.37 ’ slope = 3.13 f 0.06

20

0

1 2

I 4

I 6

I 8

N= 100 I 10

I 12

14

log(D/ro2) (s-l) Fig. 8. Monte Carlo simulation of the relationship between activation energy and log(&/r~); (a) Histogram of activation energies calculated from a Gaussian distribution simulating tire experimental results in Fig. 3c; (b) Histogram of log(D/ri) values at T = 723 K (logD,,,.c /r i)) calculated from a Gaussian distribution modeling the results shown in Fig. 1; (c) E vs. log(D,lr~) plot obtained from randomly pairing values of E and log(D45,&ri) from the distributions shown above. Note that a highly correlated array result. The slope and intercept of this array essentially reproduce the experimental results displayed in Fig. 3d.

energy (Lovera et al., 1991) . In this section, we investigate factors not taken into account in our calculations that can increase the uncertainties of the diffusion parameters. 4.2.1. Effect of nonuniform 39Ar concentrations One of the underlying assumptions of the MDD model is that K and, therefore, the initial 39Ar concentration inside each domain within the sample, is uniform (Lovera et al., 1989). However, the presence of perthite and other common microstructures in basement K-feldspars suggest that some departure from this ideal condition could be found in our samples. We analyzed the effect of nonuniform 3gArdistributions upon activation energies estimated following the method outlined in Appendix B and Fig. 2. In this approach (subsequently referred to as Model A) 39Ar concentrations within individual domains were described by c(x) = cos’(cu + mrrx) where x is distance from the boundary. Different “Ar distributions were obtained by randomly varying (Yand m using Monte Carlo methods (see inset in upper right comer of Fig. 9a). The activation energy was set to 46.5 kcal/mol in all the numerical simulations. and the same

domain distribution was used for all runs. Log( D/r’) values were then calculated using a heating schedule which began at 450°C and involved isothermal duplicate steps up to 650°C. Figure 9a shows a representative sampling of thirty Arrhenius plots calculated from Model A. Note that the second diffusion coefficient of successive isothermal steps plot either above or below the first value. A histogram of 100 activation energy results from Model A appears in the lower left comer of Fig. 9a. The activation energy recovered from 37% of the runs agree to within k2.5 kcal/mol with the activation energy used in the model. Note that in extreme instances, nonuniform 39Ar distribution will cause activation energies to be dramatically overestimated (>30 kcal/mol) when the approach of Appendix B is applied. In addition, significant underestimation of E (on the order of 3-4 kcal/ mol) can also be expected. 4.2.2. Effect of domain distribution parameters A second effect that could possibly interfere with our estimates of E involves our potential inability to separate the effects of domain distribution from those of the diffusion

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Model A above. Ten domains were used in all representations to allow greater variation in our distributions. The domain distribution parameters (p, $) were calculated using Monte Carlo methods. Only two restrictions were applied. The parameters were subject to the normalization condition ( Iif 4 = 1) and were permitted to differ by up to 20 orders of magnitude. Representative Arrhenius plots produced from Model B are shown in Fig. 9b. Note that in contrast to the behavior exhibited in Model A. log(l)/?) values corresponding to the second step of isothermal duplicate pairs are always lower than the first. A histogram of the calculated activation energies appears as an inset in Fig. 9b. The activation energies calculated for over half the runs agree within 22.5 kcal/mol with the input activation energy. Note that this model yields only calculated activation energies equal to, or lower than, the true value (i.e., 46.5 kcal/mol). In extreme instances, values of E more than 20 kcal/mol below the true value result from application of the method of Appendix B. 4.2.3. Implications

E~kca”nml, , r ’ a I 12 10 10000/T(K)

-11 8

’

*

’ 14

Fig. 9. Results of two models testing effect of nonuniform “‘Ar distribution (Model A) and variable domain distribution (Model B) upon calculated activation energy following method outlined in Appendix B. The input E of the models is 46.5 kcal/mol. (a) Representative Arrhenius plots produced by Model A. Inset appearing in upper right hand corner illustrates a representative “Ar distribution (see text for details). Note that the second diffusion coefficient of successive isothermal steps systematically plots either above or below the first value. Lower left inset is a histogram of activation energies calculated from Model A results. As indicated, values of E both larger and smaller than the input value are calculated. The activation energy recovered from 37% of the runs agree to within t2.5 kcallmol of the input value. (b) Representative Arrhenius plots produced from Model B. Inset shows histograms of the calculated E values. The activation energies recovered from ovei half the runs agree within 22.5 kcal/mol with the input value. Note that only values less or equal to 46.5 kcal/mol were obtained.

parameters include

(E and Do). extremely

small

Note that domain domains

distributions

are unlikely

that

to yield a lin-

ear Arrhenius array of log(Dlr*) values with slope proportional to the activation energy of the sample because the smallest diffusion domains will lose a substantial proportion of their K-derived Ar as the step-heating experiment progresses (see Appendix C of Lovera et al. ( 1989) Eqns. Cl 8). Because of the low ‘“Ar retentivity of extremely small domains, even minor temperature increase or protracted isothermal heating will exhaust these domains of 39Ar causing convex up curvature in the Arrhenius plot and resulting in underestimation of activation energy. Moreover, use of erroneously low activation energies results in miscalculation of the domain distribution parameters. Specifically, the smallest domains are overlooked in the fitting process because of the compensating effect of low E. In order to evaluate these effects, we have produced random domain distributions (herein referred to as Model B) and subjected them to the same heating schedule as used in

jbr experimental

results

Neither Model A nor Model B alone are capable of describing the observed range of activation energies calculated from the initial low-temperature steps in 40Ar/‘9Ar step-heating experiments. Of the two, Model A comes closest. However. combination of features in Model A and Model B could produce 225 kcal/mol apparent variation in E which is comparable to the spread of our experimental results (Fig. 3a). To further evaluate similarities of Model A and Model B behavior to the experimental results, we have focused upon the relationship between activation energy and Alog( Dl r’) values calculated for duplicate isothermal steps. These relationships are illustrated in Fig. 10. Calculated values of E vs. the weighted mean of the 450 and 500°C Alog(Dlv’) values for Models A and B are shown in Fig. 10. Note the overall strong correlation between E and Alog( D/r’). Specifically, underestimation of E corresponds with positive values of Alog(Dlr*) and vice versa. We emphasize that this correlation does not exist for the experimental data (Fig. 3d). For example, the total variation in E for Alog(Dlr’) = 0 in Fig. 4b is over 30 kcal/mol whereas the spread in Fig. IO is on the order of 5 kcal/mol. In addition, note that the span in activation energy observed in our experimental data for 1Alog( D/r’) 1 values in excess of 0. I is also large (Fig. 4). Such dispersion around the mean value of E (46.5 kcal/mol) at almost any given value of Alog(Dlr’) is not represented by the numerical simulations (Fig. 10). It is clear that in both models, over or underestimation of E of more than 5 kcal/mol is directly correlated to the Alog(Dlr*) values of the first two duplicates steps and can, therefore, be linked to the method (i.e., Appendix B) used to calculate the activation energy. This does not appear to be the case in our K-feldspar samples where the same dispersion in E was obtained for almost any value of Alog( D/r’). 4.3. Step-Heating vs. Bulk Loss Experiments Differences in the experimental step-heating experiments presented

approach between the here with isothermal,

Significance of activation energy determinations

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mean Alog(Dlr*) values calculated from 450°C and 500°C Model A and Model B results. A constructed from Model A and B results (see Fig. 9) is shown in the inset. Although the E that shown in Fig. 3a, the systematic relationship between E and Alog(D/r’) contrasts with the seen in the experimental data (Fig. 4d).

bulk loss experiments (i.e., Foland, 1974) need to be considered when comparing results from the two approaches. In a bulk loss experiment, a diffusion coefficient is calculated from the measured fractional loss and the duration of heating. A number of such measurements performed at different temperatures and heating durations are required to characterize the Arrhenius plot of a sample. Although it is often asserted that a linear Arrhenius array can only be produced from such an experiment if the sample contains a single diffusion length scale (Foland, 1974; Harrison et al., 1985; Foland, 1994), this is not necessarily the case. Because the argon is extracted in a single step, the result tends to emphasize the average diffusion properties of the sample and can yield apparently linear arrays for samples containing multiple diffusion domains. This is particularly true when the sample contains a dominant intermediate-sized domain (i.e., one that comprises >70% of the 39Ar). In such cases, the measured loss from the aggregate is insensitive to the two lower volume fraction components (i.e., the larger and smaller domains) which tend to self-compensate yielding an average result close to that of the middle domain value. For example, bulk loss experiments on Benson Mines orthoclase appears com-

patible with a sample containing only a single diffusion length-scale (Foland, 1974). This conclusion was based upon the following observations: (1) diffusion coefficients calculated between 400 and 800°C yield a linear Arrhenius array; (2) the measured fractional loss from a single grain size at 700°C for varying lengths of time plot close to the curve predicted for a single domain size; and (3) results from different grain sizes yield the similar diffusion coefficients at a given temperature when normalized for size. Although these arguments are consistent with the Benson Mines orthoclase containing a single length scale, they are not diagnostic. Specifically, step-heating experiments carried out on a 127 pm aliquot of Benson Mines orthoclase that had been previously heated at 700°C produced both an age spectrum and an Arrhenius plot that exhibit a diffusion behavior compatible with multiple diffusion domains (see Figs. 13 and 14 of Foland, 1994). To illustrate why the above criteria are in fact not diagnostic of a single diffusion length scale, we have numerically simulated bulk loss and step-heating experiments (Fig. 11) for a sample with a domain distribution that produces bulk loss results similar to that found for Benson Mines orthoclase

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10000/T(K) T=700"C

(b)

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500

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1500

)

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t'h(sy Fig. I 1, Comparison of step-heating (open circles) and bulk loss (open squares) numerical experiments with hydrothermal experiments (filled triangles) performed by Foland ( 1974) (a) The indicated domain distribution produced the step heating Arrhenius results shown as open circles. The same distribution also yielded essentially the hydrothermal bulk loss results (filled squares) obtained by Foland ( 1974) for Benson Mines orthoclase (represented by line segment and filled triangles). Bulk loss results were calculated using the same temperature and heating duration used by Foland ( 1974). (b) Isothermal bulk loss numerical experiments at 700°C (open squares) also reproduce the f vs. t”’ array defined by Benson Mines orthoclase (bold line). Filled triangles represent the values obtained by Foland (1974) from Benson Mines.

(Foland, 1994). At the same time, the Arrhenius data for this domain distribution (Fig. 1 la) produces the same characteristic curvilinear array of diffusion coefficients yielded by multi-domain samples in step-heating experiments (Fig. 1). Note that rigorous modeling of the Benson Mines stepheating results (Foland, 1994) could not be attempted because laboratory temperature-time measurements were not available. However, a simulated bulk loss experiment for this sample using the same temperatures and heating durations as those employed by (Foland, 1974) closely reproduces (Fig. 1 la) the apparent linear array obtained for Benson Mines orthoclase (point 1 above). Similarly, our synthetic bulk loss results from this sample also display a fractional loss vs. t”* curve at 700°C (Fig. 1 lb) similar to the result of Foland ( 1974) indicating that point 2 is not indisputable evidence for the presence of a single diffusion length scale. Regarding point 3, we note that concordant measurements of different size fractions are not diagnostic because multidomain samples are similarly affected. For example, a sizing experiment on MH-10 K-feldspar show a clear correlation between grain sizes and retentivity when crushing experiments were performed below I50 km (Lovera et al., 1993 )

Because of the averaging process intrinsic to bulk loss experiments, the activation energy calculated for a sample possessing multiple diffusion domains following this approach will likely be lower than the value obtained from step-heating results. For example, the value of E used in our synthetic K-feldspar was 47 kcal/mol while the Arrheniua data from the bulk loss simulation yielded only 44 kcal/mol, similar to the value obtained by Foland ( 1994). We point out that depending upon the heating schedule, step-heating results from the same synthetic sample could also result in an erroneously low activation energy. Finally we note that an 4”Ar diffusivity calculated directly from a measured depth profile of the homogeneous Madagascar orthoclase yields D,ooOc = 2.4 X IO-” cm*/s (Kelley et al., 1994) which is a factor of 6 lower than the value predicted by Foland ( 1974). Assuming similar argon diffusion behavior in the two gem quality samples, this suggests that the Foland ( 1974) use of the measured grain size resulted in an overestimation of the argon diffusivity because the effective domain size was significantly smaller than the measured value. To summarize, the Benson Mines orthoclase bulk loss results

(both

the Arrhenius

and

f vs. t”’

plots)

and subse-

‘“Ar/‘“Ar age spectrum and Arrhenius data (Foland, 1974, 1994) all appear consistent with the sized aggregates containing a modest but significant distribution of diffusion domain sizes, similar to the one given in the caption of Fig. 1I It seems apparent that the Benson Mines orthoclase contains a mosaic of internal diffusion boundaries (e.g., cleavage) that endow this sample with minor multi-diffusion domain properties. We conclude that the multi-domain diffusion theory is better able to explain all the Benson Mines orthoclase data than the mechanisms invoked by Foland ( 1994) to reconcile the bulk loss and step-heating results (e.g., synheating grain fracturing, defect trapping of argon, annealing at high temperatures). In addition, these mechanisms appear inconsistent with cycling and double irradiation experiments designed to test for, but that did not reveal, these phenomena in MH-10 K-feldspar (Harrison et al., 1991; Lovera et al., 1991, 1993). quent

4.4. Effect of Uncertainties in Kinetic Parameters Calculation of Thermal Histories

upon

Having described the range of variation of the diffusion parameters calculated from K-feldspar 40Ar/3sAr step-heating experiments, it is now important to investigate the effect of uncertainties in these parameters upon our primary goal, the reconstruction of a thermal history. Until recently, a lack of reasonable estimates for the uncertainties of the kinetic parameters had prevented us from performing calculations of confidence intervals required to address the statistical significance of calculated K-feldspar thermal histories. Instead, we based the uncertainty estimates for our thermal history almost exclusively upon the measured uncertainty in fitting the age spectrum. With the insights into uncertainties of kinetic parameters we have drawn from the K-feldspar database and the development of automated routines to mode1 the kinetic and age properties of K-feldspar 4”Ar/3yAr step-heating results (Lovera

Significance of activation energy determinations

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Fig. 12. Weighted histograms calculated as in Fig. 7 from eight domain distributions obtained from repeated attempts to model measured Arrhenius data from 93-NG-17 K-feldspar using a range of values for E and log(D,,/ri). We used Monte Carlo simulation to calculate Gaussian distributions of E and log( L&/r i) centered on the measured values (E = 41.6 and log(&lr~) = 3.7) with half widths in agreement with the indicated 1~ uncertainty. Values of E and log(&/ri) were constrained to have the same correlation seen in Fig. 3d (R = 0.97). (a) Histogram of the domain distributions obtained using the individual lo uncertainty calculated for 93-NC-17. Note that some individual domain sizes appear clearly resolved (compare with Fig. 7b). (b) Histogram calculated using lo uncertainties similar to the dispersion observed for all samples (Fig. 3). Note that the dispersion in the domain distribution increases such that individual domains are no longer resolved.

et al., 1995; Quidelleur

et al., 1997), our ability to address the statistical significance of K-feldspar thermal history results has been significantly improved. Below we describe this new approach using results from 93-NG- 17 K-feldspar as an example (see Grove and Lovera ( 1996) for additional information regarding this sample). As previously described, regression of the initial low-temperature heating steps following the method described in Appendix B yielded and activation energy and log(DOlrt) of 42 2 1 kcal/mol and 3.7 + 0.3 s-l, respectively (Fig. 2). Using Monte Carlo simulations, we produced Gaussian distributions for E and log(&,lr i) in agreement with their respective uncertainties. Values of E and log(D,Jri) possessed the same correlation as seen in Fig. 3d (R = 0.97). Size distributions consisting of eight domains were then calculated for each pair of E and log(D,,/r i) values by fitting the 39Ar data in the usual manner. A weighted histogram of the log(&lr*) values (calculated in the same manner as Fig. 7) is shown in Fig. 12a. Comparison of the two plots illustrates the effect of introducing uncertainty in the diffusion parameters upon the domain distribution. Variation in the diffusion parameters and uncertainty in the 39Ar data combined with the fact that about 30% of the ‘9Ar in the sample was released at temperatures beyond 1100°C (i.e., above incongruent melting) results in multiple equivalent solutions for the distribution parameters. Despite this, maximums corresponding to the smallest domains are clearly resolved. Thermal histories were then calculated for each set of diffusion parameters (E, D,,) and their corresponding set of domain distribution parameters (p, 4) using a modified Levenberg-Marquardt method described in Quidelleur et al.

(1997). The form of the cooling history was assumed to be monotonic. Five runs were performed for each set of parameters with different random input values used to initiate each run. The solution that yielded the lowest x2 value was accepted as the best-fit result. This process was repeated for all 100 sets of parameters. Representative calculated age spectra and their corresponding thermal history solutions are shown in Fig. 13a. In addition, we have also calculated the 90% confidence intervals for the median and overall distribution of T - t solutions as a function of age. These results are represented graphically in Fig. 14. In order to demonstrate the effect of larger uncertainties in the diffusion parameters we calculated an additional set of thermal history solutions following the same process described above but employing uncertainties in E and log (DOlr~) of +6 kcal/mol and ?3 s-‘, respectively. The resulting domain distributions are shown in Fig. 12b. Our use of larger uncertainties in the diffusion parameters was motivated by the dispersion in these parameters exhibited by our database samples (Fig. 3). Propagation of the larger uncertainties in the diffusion parameters results in significantly greater dispersion in calculated T - t histories Fig. 13c than that illustrated in Fig. 13b. Finally, we demonstrate an alternative approach that exploits the fact that by simply taking experimental uncertainty into account, a single set of diffusion and distribution pammeters is capable of producing a whole family of equivalent solutions. We generated the solutions shown in Fig. 13d from the domain distribution parameters of Fig. 6 by considering all solutions for which 1Ax2 1/xii” < 0.3 to be equivalent. The parameter xii, is the minimum value of x obtained for all runs (Fig, 13d). Note that the resulting confidence

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is

Ages&a)

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6-O

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Age’tMa)

Fig. 13.Thermal history results for 93.NG- I7 K-feldspar. (a) Measured (filled gray boxes) and calculated (bold line) age spectra. (b) Thermal histories calculated for 93.NG- I7 K-feldspar using domain distributions shown in Fig. 12a (E = 41.6 2 1.3 kcal/mol and log(&lr~) = 3.7 t 0.3 s ‘). (c) Calculated thermal histories using domain distributions shown in Fig. 12b (E = 42 f 6 kcal/mol and log(DJri) = 4 2 3 s-l). (d) Thermal histories calculated

from a single domain distribution by allowing the degree of approximation to vary according to 1Ax’ I/x& where ~2,. was the minimum criteria.

value obtained

for all the runs. Ninety-seven

intervals (Fig. 14~) are similar to those obtained propagating the laboratory determined uncertainties into the thermal histories (Fig. 14a). Calculation of a set of cooling history following the later approach (Fig. 14~) is appealing in that it involves significantly less computational time. Note that both methods (Fig. 14a,c) will yield equivalent results only when the uncertainties in E are -1 kcal/mol. In addition,

solutions

< 0.3,

out of 200 runs satisfied

this

appropriate x L,, values will need to be determined empiritally for each sample. In considering all results in Fig. 14, it is clear that under the assumption of monotonic cooling, the 40Ar/“Ar results from 93-NG-17 K-feldspar tightly constrain the thermal history of the sample between 55 and 40 Ma but provide signilicantly less information at earlier or later times. In addi-

Significance of activation energy determinations

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600 -5 500 & $ f

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Fig. 14. Maximum probability estimates of cooling histories calculated from 93-NG-17 kinetic parameters and domain distributions of Fig. 12. Shown are the 90% confidence interval for the median (black) and overall distribution (gray) : Results shown in (a), (b), and (c) were calculated from thermal history solutions in Fig. 12b, c, and d, respectively. Note that the less intensive computational calculations used to obtain (c) yield results similar to (a).

tion, even though assuming large uncertainties in the diffusion parameters produces significant dispersion in temperature, the overall form of the cooling history is preserved (Fig. 14b). 4.5. Role of Nondiffusive

Argon Transport

The multi-path (MP) hypothesis features nondiffusive transport between lattice and regions of enhanced Ar diffusion (short-circuit pathways). The most highly developed form of the MP model is that presented by Ainfantis ( 1979) and refined by Lee and Aldama ( 1992) and Lee ( 1995). From a mathematical point of view, their approach actually represents a generalization of the MDD model. The principle difference is that interaction is permitted between the lattice and short-circuit pathways in the MP model whereas domains are noninteracting in the MDD model. Eliminating interaction reduces the MP model to a mathematical formulation equivalent to the MDD model with two domains (Lee, 1995 ) . The chief difficulty in applying the MP model is that while argon transport in both the lattice and within shortcircuit pathways is characterized by volume diffusion parameters that differ in magnitude, the process that permits interaction (i.e., mass transfer between the regions) is not specified. This generality prevents the exchange coefficients from being constrained by laboratory experiments.

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Lee ( 1995) demonstrated that inflections can be expected from Arrhenius plots calculated by the MP model using heating steps involving monotonic increase in temperatures. This behavior reflects transition from conditions under which diffusion in high-diffusivity pathways predominates due to a large Ar transfer towards SC-paths to conditions for which diffusion through the lattice is most important. Beginning at low temperature, Arrhenius plots exhibit either convex upwards or convex downward curvature depending upon whether exchange coefficients are held constant or are allowed to increase with temperature (see Fig. 6 of Lee, 1995). This behavior renders the MP model a potential candidate to explain some of the effects observed in our samples. Specifically, the observed dispersion in calculated activation energies (Fig. 3) could result from samples having the same activation energy for lattice diffusion if they were characterized by significantly different exchange coefficients. A crucial property of the MP model as defined by Lee and Aldama (1992) and Lee (1995) is that the value of log(Dlr’) at a given temperature is a constant determined by the exchange coefficients. This implies that duplicate isothermal steps will not show separation since the exchange between the lattice and SC-paths will replace the argon loss from the SC-paths during the heating steps. Such behavior potentially accounts for the behavior shown in Fig. 5 where a spread in activation energy exists despite no separation in log(Dlr*) values obtained from successive isothermal steps. In spite of these properties, the MP model fails to describe the fundamental behavior observed for K-feldspars subjected to temperature cycling in step-heating experiments. The interaction requirement in the MP model causes depleted SCpaths to be replenished during backward cycling of temperature such that the resulting log( D/r’) values reproduce those previously measured as temperature was increased during the forward cycle. Although over a third of the K-feldspars in Table 1 were analyzed with heating schedules involving temperature cycling backwards from elevated temperature ( -750°C or higher), the log(Dlr’) values obtained during initial, low-temperature 39Ar release were never reproduced (Harrison et al., 1991; Lovera et al., 1993). Instead, log(Dlr*) values measured during backward temperature cycling are l-2 orders of magnitude lower than those previously determined at a given temperature. The actual difference observed is a function of the heating schedule and the properties of the sample. Amaud and Kelley (1997) proposed that the failure of the Lee and Aldama (1992) MP model to reproduce the effects observed in the temperature cycling experiments can be explained if annealing of SC-paths prior to backward cycling of temperature occurs. The magnitude of the effect of annealing under these conditions appears to be relatively small for at least some samples, however. The effect of annealing in low temperature 39Ar release from MH-10 Kfeldspar, for example, was addressed in a double-irradiation experiment described in Lovera et al. ( 1993). In this experiment, a split of MH- 10 K-feldspar previously at temperatures reaching 850°C for several hours was re-irradiated to replenish out-gassed regions with 39Ar. Near duplication of the previously measured Arrhenius plot in the subsequent stepheating experiment clearly indicated a minimal role for an-

3190 nealing during low-temperature (Lovera et al., 1993).

0. M. Lovera et al “Ar release for this sample

4.6. Final Considerations The lack of viability of other potential descriptions of the transport of argon in K-feldspars such as multi-path behavior, force us to conclude that the observed intersample variation in diffusion parameters reflects real differences in the diffusion properties of these materials and that uncertainties calculated via the method of Appendix B are valid estimates. We recognize that a minority of our samples (represented by the examples of Fig. 5b and c) exhibit behavior that is quite consistent with the predictions of Models A and B. Although we cannot rule out minor tendencies toward Model A or Model B behavior for the majority of our samples (i.e.. those for which mean square root nlog(Dlr’) for successive isothermal duplicate pairs is less than 0.1)) we point out that the likely error in activation energy will be less than 5 kcal/ mol (Fig. 10). Because the observed dispersion in E and & ( Fig. 3a) is well in excess of what could be explained by experimental error, and because the processes considered in Model A or Model B (Fig. IO) do not reproduce our empirical observations (Fig. 4)) we conclude that there is a real variation in calculated activation energy of argon in Kfeldspar of at least IO kcal/mol. The observed spread in the values of E and Do may reflect intrinsic differences in diffusion mechanism at the atomic scale (Harrison et al., 1991), but could also simply result from the macroscopic nature of the diffusion model aimed at describing the complex microscopic behavior. The macroscopic parameters only reflect the average properties of the sample, and it is thus conceivable that this averaging changes from sample to sample. Furthermore, the separation between the macroscopic parameter governing the exponential temperature dependence (E) and the parameter which should be related to the geometry and scaling of the sample (Do) is probably not fully accomplished due to the limited range of our experimental conditions (i.e., in l/T and log(Dlr*)), as the clear compensation correlation observed in Fig. 3 demonstrates. For example, if E and D, each have a weak temperature dependence, our inability to accurately partition these effects into the two macroscopic parameters would contribute to a measurable spread in observed activation energies. However, the impact of this broad range in calculated activation energies when extrapolating diffusivity to geological temperatures will be significantly diminished because of the correlated compensation relationship between E and Do. For example, a difference in E of 1 kcal/mol corresponds to a temperature difference of only ~10°C rather than >3o”C if E were changed independent of D,,. Furthermore, the form of the calculated thermal history remains unaffected by the manner in which E is varied. Finally, it is important to ask whether the observed dispersion in the diffusion parameters (Fig. 3) is a consequence of the method used to calculate them (Appendix B). Although it is possible that more elastic criteria could produce better results with less dispersion, we point out that while

we have tried other methods, they also produced essentially the same result. In fact, the appearance of Fig. 3 does not change materially if it is compiled from diffusion parameters obtained by nonuniform and subjective criteria used by different individual analysts. 5. CONCLUSIONS Determining the diffusion parameters in K-feldspar ‘“Ar/ “Ar step-heating experiments is one of the most crucial, yet problematic tasks involved in recovering thermal history information. According to the MDD model, it should be possible to estimate the diffusion parameters from the initial stages of a step-heating experiment. Despite this, use of an inappropriate heating schedule or selection of an unsuitable number of heating steps to calculate the diffusion parameters can yield misleading results. In particular, a heating schedule that degasses the sample too quickly (starting at too high a temperature for example) will likely cause the smallest domains to be exhausted in the first or second heating steps and result in an erroneously low value for E. Regardless of the heating schedule employed, it appears that a minority of samples will challenge our ability to estimate the diffusion parameters if a single set of criteria to select values for E and log(&lr~) is applied. Our principal conclusions are: ( I ) Calculated activation energies among basement K-feldspars vary by more than IO kcal/mol. (2) Results from all ““Ar/“‘Ar step-heating measurements of K-feldspar exhibit a high degree of correlation between E and log(DOlrs) with a slope that reproduces the previously documented feldspar compensation relationship. The statistical significance of the correlation is not sufficient to confirm the existence of a Meyer-Neldel law for argon diffusion in basement K-feldspars. Because systematic errors in temperature produce a correlated relationship between E and log(&lri) that is significantly different from what we observe, this type of laboratory error can be ruled out. ( 3 ) The MDD model is able to account for the majority of data we have obtained from basement K-feldspars. (4) We identify general tendencies in the domain distribution of basement K-feldspars. For the majority of samples, most ‘“Ar resides in the larger domains. The smallest domains generally constitute
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