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Extrapolation of zircon fission-track annealing models
Radiation Measurements 50 (2013) 192e196
Contents lists available at SciVerse ScienceDirect
Radiation Measurements journal homepage: www.elsevier.com/locate/radmeas
Extrapolation of zircon fission-track annealing models R. Palissari a, *, S. Guedes a, E.A.C. Curvo b, P.A.F.P. Moreira a, C.A. Tello b, J.C. Hadler a a b
Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, UNICAMP, 13083-970 Campinas, SP, Brazil Departamento de Física Química e Biologia, Universidade Estadual Paulista, UNESP, 19060-900 Presidente Prudente, SP, Brazil
h i g h l i g h t s < Geological data were used along with lab data for improving model extrapolation. < Index temperatures were simulated for testing model extrapolation. < Curvilinear Arrhenius models produced better geological temperature predictions.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 December 2011 Received in revised form 4 June 2012 Accepted 6 June 2012
One of the purposes of this study is to give further constraints on the temperature range of the zircon partial annealing zone over a geological time scale using data from borehole zircon samples, which have experienced stable temperatures for w1 Ma. In this way, the extrapolation problem is explicitly addressed by fitting the zircon annealing models with geological timescale data. Several empirical model formulations have been proposed to perform these calibrations and have been compared in this work. The basic form proposed for annealing models is the Arrhenius-type model. There are other annealing models, that are based on the same general formulation. These empirical model equations have been preferred due to the great number of phenomena from track formation to chemical etching that are not well understood. However, there are two other models, which try to establish a direct correlation between their parameters and the related phenomena. To compare the response of the different annealing models, thermal indexes, such as closure temperature, total annealing temperature and the partial annealing zone, have been calculated and compared with field evidence. After comparing the different models, it was concluded that the fanning curvilinear models yield the best agreement between predicted index temperatures and field evidence. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Fission tracks Annealing model Zircon Field evidence Geological extrapolation
1. Introduction Fission tracks are the damaged structure caused by passage of fission fragments. These regions in some minerals as apatite and zircon can last for geological times, depending on the host rock thermal history. If a polished surface is properly etched, the fission tracks can be observed under an optical microscope. The number of etched fission tracks is a good measure of the time the mineral is retaining them. It is well known that the fission-track lengths are shortened by the combined action of time and temperature. To this phenomenon is given the name annealing. Thus, the cooling history of the host rock can be inferred by measuring the length distribution in a given mineral sample, provided the annealing rates are
* Corresponding author. E-mail address: paliss@ifi.unicamp.br (R. Palissari). 1350-4487/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radmeas.2012.06.004
known. Although apatite is by far the most used mineral, the use of zircon is also desirable, since this mineral presents higher closure temperature (Rahn et al., 2004). Unfortunately, the processes involved in track formation, annealing and etching are not well understood. To overcome this difficult, empirical annealing models have been proposed (Laslett et al., 1987; Laslett and Galbraith, 1996) and their parameters found by fitting annealing datasets. Models, in which tentative annealing mechanisms are presented, have also been proposed (Carlson, 1990; Guedes et al., 2005). However, they are essentially empirical, since their parameters must be found by fitting procedures. For zircon, laboratory datasets range from 3.5 s to 10,000 h (Yamada et al., 1995; Tagami et al., 1998; Murakami et al., 2006). Strictly, the model equations calibrated by these datasets hold only in the range covered by data. Not the less, geological applications require extrapolation to times of millions of years. Thus, the
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
geological predictions of the model equations must be compared with field evidence. Rahn et al. (2004) carried out a comprehensive survey, in which they summarized field evidence for annealing of zircon fission-tracks and present constraints for the zircon partial annealing zone (ZPAZ) and closure temperatures for several cooling rates. Comparing the predictions of the model equations with field data, they noted that models preview higher closure temperatures. They attributed this discrepancy to radiation damage in zircon, which has been shown to modify the crystalline structure of this mineral (Palenik et al., 2003). Although radiation damage cannot be discarded, the error in extrapolating model equations has to be considered. Model equations present different trends producing therefore different geological extrapolations, even when calibrated with the same laboratory dataset. The present contribution is inserted in this context. It is proposed to incorporate geological data in the fitting procedure in order to improve extrapolation. The borehole data presented by Hasebe et al. (2003) are used. This dataset is based on measurements of fission tracks in samples that experienced stable paleotemperatures during the last one million years. In this way, it will be treated (as an approximation) as a “1 My dataset”. In the following, parameters for model equations presented in literature are obtained using jointly laboratory and borehole data and their geological predictions are compared with the field data survey from Rahn et al. (2004). Then, the implications of the applied procedure are discussed.
gðrÞ ¼
193
C4 1 r C5 C5 1 C4
(5)
lnðtÞ C2 f t; T ¼ C0 þ C1 lnð1=TÞ C3
(6)
In Eqs. (3)e(6), C0, C1, C2, C3, C4, and C5 are fitting parameters. [c] Improved Fanning Arrhenius (Laslett and Galbraith, 1996), LGFL:
i h gðlÞ ¼ ln 1 ðl=C0 Þ1=C5
(7)
lnðtÞ C3 f t; T ¼ C1 þ C2 ð1=TÞ C4
(8)
[d] Improved Parallel Arrhenius (Laslett and Galbraith, 1996), LGPL:
i h gðlÞ ¼ ln 1 ðl=C0 Þ1=C5
(9)
C f t; T ¼ C1 þ C2 ln t 3 T
2. Calibration of the model equations 2.1. Model equations
(10)
In Eqs. (7)e(10), C0, C1, C2, C3, C4, and C5 are fitting parameters.
The empirical models used in previous works for fitting annealing datasets present the general format proposed by Laslett et al. (1987):
[e] Fanning Linear (Rahn et al., 2004), RHFL:
gðrÞ ¼ f ðt; TÞ
gðrÞ ¼ ln½1 r
(11)
f ðt; TÞ ¼ fC0 þ C1 TlnðtÞ þ C2 Tg
(12)
(1)
or alternatively the format proposed by Laslett and Galbraith (1996):
gðlÞ ¼ f ðt; TÞ
(2)
In the above equations r is the reduced mean track length, l/l0, with l being the track length after the heating experiment and l0 the initial mean track length; t is the duration (in seconds), T the temperature (in Kelvin) of the heating experiment and g is a function that transforms l or r. The function f(t, T) carries the specific geometric properties and assumes different forms. In this work, we kept the form, in which each model is better known, g(r) or g(l), in order to make the analyses more understandable for the researchers that often apply these models. The following empirical model equations will be considered:
f t; T
lnðtÞ C2 ¼ C0 þ C1 ð1=TÞ C3
[b] Fanning curvilinear (Crowley et al., 1991), FC:
gðrÞ ¼ ln½1 r
(13)
f ðt; TÞ ¼ fC0 þ C1 ð1=TÞ þ C2 lnðtÞg
(14)
In Eqs. (12) and (14), C0, C1 and C2 are fitting parameters. [g] Fanning Linear (Yamada et al., 2007), YFL:
1=C5
gðlÞ ¼ ln 1 ufl =C0
[a] Fanning Arrhenius (Laslett et al., 1987), FA:
C4 1 r C5 C5 1 gðrÞ ¼ C4
[f] Parallel Linear (Rahn et al., 2004), RHPL:
(3)
f ðt; TÞ ¼
(4)
C1 þ C2
lnðtÞ C3 lnð1=TÞ C4
(15)
(16)
In Eqs. (15) and (16), C0, C1, C2, C3, C4,C5 are fitting parameters and mfl is the mean length at the annealing condition of (t, T) for the fanning equation.
194
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
[h] Parallel Linear (Yamada et al., 2007), YPL:
h 1=C5 i gðlÞ ¼ ln 1 upl =C0 C f ðt; TÞ ¼ C1 þ C2 lnðtÞ 3 T
parameters. The units used for the constants are the same presented in the original presentation Carlson (1990).
(17)
(18)
In Eqs. (17) and (18), C0, C1, C2, C3, C4, C5 are fitting parameters, and mpl is the mean length at the annealing condition of (t, T) for the parallel equation.
This model is expressed by the weighted sum of mfl and mpl.
(19)
Where
1=2 lnðt=t0 Þ kB T 10:24w U0 A1 lnðT0 =TÞ
(23)
(24)
where t0 ¼ (Z/2U0), Z¼ 6,5822 1016 eV. In Eqs. (23) and (24), n, w, U0, A1, T0, are fitting parameters.
2.2. Calibration datasets
wðTÞ ¼ 1=ð1 þ expð 11:08ð1000=TÞ 1:169ÞÞ
(20)
Two other models will be considered, which are based on hypotheses about mechanisms of the annealing process. They are: [a] The model proposed by Carlson (1990), CS:
gðrÞ ¼ ln½1 r
f ðt; TÞ ¼ ln
lnð1=rÞ gðrÞ ¼ ln n
f ðt; TÞ ¼
[i] Hybrid model (Yamada et al., 2007), YHY:
ul ðt; TÞ ¼ wðTÞufl ðt; TÞ þ ð1 wðTÞÞupl ðt; TÞ
[b] The model proposed by Guedes et al. (2005, 2006) with the modifications proposed by Guedes et al. (submitted), CR:
A k T Q þ n lnt þ nln B l0 h RT
(21)
(22)
where kB is Boltzmann’s constant (3.2997 1027 kcal K1), h is Planck’s constant (1.5836 1037 kcal s), R is the Universal gas constant (1.987 103 kcal mol1 K1) and l0, A, n, Q are fitting
For this study, we chose the dataset composed of a series of isochronal (the samples were annealed at different temperatures for a fixed time) annealing experiments for 4.5 min, 1 h, 11 h, 100 h and 1,000 h on spontaneous fission tracks in Nisatai Dacite (NST) zircon, presented by Yamada et al. (1995), experiments carried out for 10,000 h (Tagami et al., 1998) and those by Murakami et al. (2006), an experimental data set obtained for short term (<4 min) heating condition of spontaneous fission tracks in NST zircon at 550e910 C for w4, 10 and 100 s. It is proposed to incorporate geological data in the fitting procedure in order to improve extrapolation. For this purpose, it is used the bore hole data presented by Hasebe et al. (2003). This dataset is based on measurements of fission tracks in samples that experienced stable paleotemperatures (up to 230 C) for the last one million years. In this way, it will be treated (as an approximation) as a “1 Ma dataset”.
Fig. 1. Results of the fit for the zircon annealing models in an Arrhenius diagram. Different degrees of length reduction of experimental data are shown with different symbols. [a] FA and FC [b] LGFL and LGPL [c] RHFL and RHPL [d] YHY [e] CS [f] CR. The initials on the graphs mean: FA: Fanning Arrhenius (Laslett et al., 1987); FC: Fanning curvilinear (Crowley et al., 1991); LGFL: Improved Fanning Arrhenius (Laslett and Galbraith, 1996); LGPL: Improved Parallel Arrhenius (Laslett and Galbraith, 1996); RHFL: Fanning Linear (Rahn et al., 2004); RHPL: Parallel Linear (Rahn et al., 2004); YFL: Fanning Linear (Yamada et al., 2007); YPL: Parallel Linear (Yamada et al., 2007); YHY: Hybrid model (Yamada et al., 2007); CS: model by Carlson (1990); CR: Guedes et al. (submitted).
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
195
Fig. 2. Comparison between the zircon FT annealing model predictions and geological constraints (compiled by Rahn et al., 2004) on the zircon PAZ from the literature. [a] FA and FC [b] LGFL and LGPL [c] RHFL and RHPL [d] YHY [e] CS [f] CR. Light gray boxes indicate conditions of total track retention, dark gray boxes indicate partial annealing conditions and one black box indicates total annealing.
3. Fitting procedures and results Fitting of the model equation to experimental datasets was carried out using a c2 minimization scheme. The standard deviation has been used to weight the data. Numerical minimization was performed with combined Simplex and LangevineMarquardt interactions using the software OriginÒ. The Arrhenius plot of the annealing models equations is shown in Fig. 1 where the isoannealing lines represent two different degrees of length reduction, l/l0 ¼ 0.4 and 0.9. As stated before, empirical models hold only in the range of the experimental data used for calibration. The use of such long term data expands the range in which the empirical model is valid to times closer to the range of interest for practical applications. This is the reason to use the borehole data together with the laboratory dataset even knowing this is an approximation. Note in Fig. 1, for most models, the pair of contour lines is very similar over the range of time of laboratory experiments, but for geological timescale they make different predictions. Including the Hasebe et al. (2003) data, the fits were obtained with the curvilinear equations. However, the ultimate criterion to choose a model is its ability to reproduce geological data.
shown (Fig. 2) along with the experimental evidence presented by Rahn et al. (2004). Satisfactory temperature indexes are obtained for the fanning curvilinear (Crowley et al., 1991) and the hybrid (Yamada et al., 2007) models. Among the physical models, the best results are obtained by the Guedes et al. (submitted) model. Besides constraints for the ZPAZ, Rahn et al. (2004) presented field data for closure temperatures for several cooling rates. To estimate closure temperatures from the models, the definition given by Dodson (1973) was applied, namely, that the closure temperature is the one corresponding to the apparent age of the mineral in a constant cooling thermal history. To implement this definition, the principle of equivalent time, as defined by Duddy et al. (1988) was used in order to apply the model equations to
4. Extrapolation to geological timescale In order to compare model geological predictions with field evidence, the boundaries of the Zircon Partial Annealing Zone (ZPAZ) and the cooling rate dependent closure temperature were calculated for each model equation. The ZPAZ boundaries have been defined as 10 and 90% track-density reduction (Rahn et al., 2004). The transformation from length reduction (l/l0) to density reduction (r/r0) was done using the equation presented by Guedes et al. (2004) fitted to the dataset of Tagami et al. (1990). To take into account the tracks generated during heating, the heating duration, t, was divided in various intervals and the density reductions were computed individually from the time tracks were generated to present. Then, the mean reduction was calculated. In this way, temperatures leading to 10 and 90% track-density reduction were calculated for several values of t. The results are
Fig. 3. Comparison between the zircon FT annealing model predictions and geological constraints (compiled by Rahn et al., 2004) on the closure temperatures from the literature.
196
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
variable temperatures. The results of these calculations are presented in Fig. 3. Field evidences for closure temperatures stand below the values simulated using the model equations. The models that give closer results are fanning curvilinear (Crowley et al., 1991), fanning and parallel linear models of Rahn et al. (2004), fanning linear of Yamada et al. (2007) and the Guedes et al. (submitted) model. 5. Conclusion Fission track annealing models were calibrated using zircon laboratory and geological scale data. Data sets were used with temperatures and times ranging from 350 to 910 C and w4 s to 10,000 h, respectively. As it can be seen Fig. 1, the annealing models diverge out of the laboratory range. Particularly, in the geological timescale the length shortening predictions are very different. In this way, the modeled thermal histories are dependent on the particular model to be used, which may lead to very different temperature predictions. It was proposed in this work to use the dataset presented by Hasebe et al. (2003) that is composed of geological scale data (1 Ma) of zircon samples from MITI-Nishikubiki and MITIMishima deep drilling, Japan. Thus, the calibration range is extended to more convenient time scale. Although the zircon ages and therefore modeled thermal histories may be in the order of several hundreds of millions of years, which are still out the calibration data range, the extrapolation is less uncertain. In addition, comparison of model predictions with field data for ZPAZ and closure temperatures made possible the evaluation of model extrapolation. ZPAZ temperatures are better predicted by Crowley et al. (1991), Yamada et al. (2007) and Guedes et al. (submitted) models. Closure temperature predictions, shown in Fig. 3, are similar for the following models: Crowley et al. (1991), Yamada et al. (2007), Guedes et al. (submitted), Rahn et al. (2004), which predict temperature slightly higher than field evidence. In this way, based on agreement with field evidence, we recommend that the curvilinear models by Crowley et al. (1991), Yamada et al. (2007) and Guedes et al. (submitted) should be used for modeling thermal histories in zircon fission-track systems. The other models, although also fitted with Hasebe et al. data, present divergent temperature predictions when extrapolated to times out of calibration data range.
References Carlson, W.D., 1990. Mechanisms and kinetics of apatite fission-track annealing. Am. Mineral 75, 1120e1139. Crowley, K.D., Cameron, M., Shaefer, R.L., 1991. Experimental studies of annealing of etched fission tracks in fluor apatite. Geochim. Cosmochim. Acta 55, 1449e1465. Dodson, M.H., 1973. Closure temperature in cool geochronological and petrological systems. Contrib. Mineral Petrol. 40, 259e274. Duddy, I.R., Green, P.F., Laslett, G.M., 1988. Thermal annealing of fission tracks in apatite. 3. Variable temperature behaviour. Chem. Geol. 73, 25e38. Guedes, S., Hadler, J.C., Iunes, P.J., Tello, C.A., 2004. Kinetic model for the relationship between confined fission-track length shortening and fission track age reduction in minerals. Nucl. Instrum. Methods B 217, 627e636. Guedes, S., Hadler, J.C., Iunes, P.J., Oliveira, K.M.G., Moreira, P.A.F.P., Tello, C.A., 2005. Kinetic model for the annealing of fission tracks in zircon. Radiat. Meas. 40, 517e521. Guedes, S., Hadler, J.C., Oliveira, K.M.G., Moreira, P.A.F.P., Iunes, P.J., Tello, C.A., 2006. Kinetic model for the annealing of fission tracks in minerals and its application to apatite. Radiat. Meas. 41, 392e398. Guedes S., Moreira P.A.F.P., Devanathan R., Weber W.J., Hadler J.C. , submitted. Improved fission-track annealing model based on reevaluation of annealing data. Hasebe, N., Mori, S., Tagami, T., Matsui, R., 2003. Geological partial annealing zone of zircon fission-track system: additional constraint from the deep drilling MITINischikubi and MITIMishima. Chem. Geol. 199, 45e52. Laslett, G.M., Green, P.F., Duddy, I.R., Gleadow, J.W.A., 1987. Thermal annealing of fission tracks in apatite 2. A quantitative analysis. Chem. Geol. Isot. Geosci. Sect. 65, 1e13. Laslett, G.M., Galbraith, R.F., 1996. Statistical modeling of thermal annealing of fission tracks in apatite. Geochim. Cosmochim. Acta 60, 5117e5131. Murakami, M., Yamada, R., Tagami, T., 2006. Short-term annealing characteristics of spontaneous fission tracks in zircon: a qualitative description. Chem. Geol. Isot. Geosc. Sect. 227, 214e222. Palenik, C.S., Nasdala, L., Ewing, R.C., 2003. Radiation damage in zircon. Am. Mineral 88, 770e781. Rahn, M.K., Brandon, M.T., Batt, G.E., Garver, J.I., 2004. A zero-damage model for fission-track annealing in zircon. Am. Mineral 89, 473e484. Tagami, T., Ito, H., Nishimura, S., 1990. Thermal annealing characteristics of fission tracks in zircon. Chem. Geol. Isot. Geosci. Sect. 80, 159e169. Tagami, T., Galbraith, R.F., Yamada, R., Laslett, G.M., 1998. Revised Annealing Kinetics of Fission Tracks in Zircon and Geological Implications. Advances in FissionTrack Geochronology. Kluwer Academic Publishing, The Netherlands, pp. 99e112. Yamada, R., Tagami, T., Nishimura, S., Ito, H., 1995. Annealing kinetics of fission tracks in zircon: an experimental study. Chem. Geol. Isot. Geosci. Sect. 122, 249e258. Yamada, R., Murakami, M., Tagami, T., 2007. Statistical modeling of annealing kinetics of fission tracks in zircon; Reassessment of laboratory experiments. Chem. Geol. 236, 75e91.
Contents lists available at SciVerse ScienceDirect
Radiation Measurements journal homepage: www.elsevier.com/locate/radmeas
Extrapolation of zircon fission-track annealing models R. Palissari a, *, S. Guedes a, E.A.C. Curvo b, P.A.F.P. Moreira a, C.A. Tello b, J.C. Hadler a a b
Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, UNICAMP, 13083-970 Campinas, SP, Brazil Departamento de Física Química e Biologia, Universidade Estadual Paulista, UNESP, 19060-900 Presidente Prudente, SP, Brazil
h i g h l i g h t s < Geological data were used along with lab data for improving model extrapolation. < Index temperatures were simulated for testing model extrapolation. < Curvilinear Arrhenius models produced better geological temperature predictions.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 December 2011 Received in revised form 4 June 2012 Accepted 6 June 2012
One of the purposes of this study is to give further constraints on the temperature range of the zircon partial annealing zone over a geological time scale using data from borehole zircon samples, which have experienced stable temperatures for w1 Ma. In this way, the extrapolation problem is explicitly addressed by fitting the zircon annealing models with geological timescale data. Several empirical model formulations have been proposed to perform these calibrations and have been compared in this work. The basic form proposed for annealing models is the Arrhenius-type model. There are other annealing models, that are based on the same general formulation. These empirical model equations have been preferred due to the great number of phenomena from track formation to chemical etching that are not well understood. However, there are two other models, which try to establish a direct correlation between their parameters and the related phenomena. To compare the response of the different annealing models, thermal indexes, such as closure temperature, total annealing temperature and the partial annealing zone, have been calculated and compared with field evidence. After comparing the different models, it was concluded that the fanning curvilinear models yield the best agreement between predicted index temperatures and field evidence. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Fission tracks Annealing model Zircon Field evidence Geological extrapolation
1. Introduction Fission tracks are the damaged structure caused by passage of fission fragments. These regions in some minerals as apatite and zircon can last for geological times, depending on the host rock thermal history. If a polished surface is properly etched, the fission tracks can be observed under an optical microscope. The number of etched fission tracks is a good measure of the time the mineral is retaining them. It is well known that the fission-track lengths are shortened by the combined action of time and temperature. To this phenomenon is given the name annealing. Thus, the cooling history of the host rock can be inferred by measuring the length distribution in a given mineral sample, provided the annealing rates are
* Corresponding author. E-mail address: paliss@ifi.unicamp.br (R. Palissari). 1350-4487/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radmeas.2012.06.004
known. Although apatite is by far the most used mineral, the use of zircon is also desirable, since this mineral presents higher closure temperature (Rahn et al., 2004). Unfortunately, the processes involved in track formation, annealing and etching are not well understood. To overcome this difficult, empirical annealing models have been proposed (Laslett et al., 1987; Laslett and Galbraith, 1996) and their parameters found by fitting annealing datasets. Models, in which tentative annealing mechanisms are presented, have also been proposed (Carlson, 1990; Guedes et al., 2005). However, they are essentially empirical, since their parameters must be found by fitting procedures. For zircon, laboratory datasets range from 3.5 s to 10,000 h (Yamada et al., 1995; Tagami et al., 1998; Murakami et al., 2006). Strictly, the model equations calibrated by these datasets hold only in the range covered by data. Not the less, geological applications require extrapolation to times of millions of years. Thus, the
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
geological predictions of the model equations must be compared with field evidence. Rahn et al. (2004) carried out a comprehensive survey, in which they summarized field evidence for annealing of zircon fission-tracks and present constraints for the zircon partial annealing zone (ZPAZ) and closure temperatures for several cooling rates. Comparing the predictions of the model equations with field data, they noted that models preview higher closure temperatures. They attributed this discrepancy to radiation damage in zircon, which has been shown to modify the crystalline structure of this mineral (Palenik et al., 2003). Although radiation damage cannot be discarded, the error in extrapolating model equations has to be considered. Model equations present different trends producing therefore different geological extrapolations, even when calibrated with the same laboratory dataset. The present contribution is inserted in this context. It is proposed to incorporate geological data in the fitting procedure in order to improve extrapolation. The borehole data presented by Hasebe et al. (2003) are used. This dataset is based on measurements of fission tracks in samples that experienced stable paleotemperatures during the last one million years. In this way, it will be treated (as an approximation) as a “1 My dataset”. In the following, parameters for model equations presented in literature are obtained using jointly laboratory and borehole data and their geological predictions are compared with the field data survey from Rahn et al. (2004). Then, the implications of the applied procedure are discussed.
gðrÞ ¼
193
C4 1 r C5 C5 1 C4
(5)
lnðtÞ C2 f t; T ¼ C0 þ C1 lnð1=TÞ C3
(6)
In Eqs. (3)e(6), C0, C1, C2, C3, C4, and C5 are fitting parameters. [c] Improved Fanning Arrhenius (Laslett and Galbraith, 1996), LGFL:
i h gðlÞ ¼ ln 1 ðl=C0 Þ1=C5
(7)
lnðtÞ C3 f t; T ¼ C1 þ C2 ð1=TÞ C4
(8)
[d] Improved Parallel Arrhenius (Laslett and Galbraith, 1996), LGPL:
i h gðlÞ ¼ ln 1 ðl=C0 Þ1=C5
(9)
C f t; T ¼ C1 þ C2 ln t 3 T
2. Calibration of the model equations 2.1. Model equations
(10)
In Eqs. (7)e(10), C0, C1, C2, C3, C4, and C5 are fitting parameters.
The empirical models used in previous works for fitting annealing datasets present the general format proposed by Laslett et al. (1987):
[e] Fanning Linear (Rahn et al., 2004), RHFL:
gðrÞ ¼ f ðt; TÞ
gðrÞ ¼ ln½1 r
(11)
f ðt; TÞ ¼ fC0 þ C1 TlnðtÞ þ C2 Tg
(12)
(1)
or alternatively the format proposed by Laslett and Galbraith (1996):
gðlÞ ¼ f ðt; TÞ
(2)
In the above equations r is the reduced mean track length, l/l0, with l being the track length after the heating experiment and l0 the initial mean track length; t is the duration (in seconds), T the temperature (in Kelvin) of the heating experiment and g is a function that transforms l or r. The function f(t, T) carries the specific geometric properties and assumes different forms. In this work, we kept the form, in which each model is better known, g(r) or g(l), in order to make the analyses more understandable for the researchers that often apply these models. The following empirical model equations will be considered:
f t; T
lnðtÞ C2 ¼ C0 þ C1 ð1=TÞ C3
[b] Fanning curvilinear (Crowley et al., 1991), FC:
gðrÞ ¼ ln½1 r
(13)
f ðt; TÞ ¼ fC0 þ C1 ð1=TÞ þ C2 lnðtÞg
(14)
In Eqs. (12) and (14), C0, C1 and C2 are fitting parameters. [g] Fanning Linear (Yamada et al., 2007), YFL:
1=C5
gðlÞ ¼ ln 1 ufl =C0
[a] Fanning Arrhenius (Laslett et al., 1987), FA:
C4 1 r C5 C5 1 gðrÞ ¼ C4
[f] Parallel Linear (Rahn et al., 2004), RHPL:
(3)
f ðt; TÞ ¼
(4)
C1 þ C2
lnðtÞ C3 lnð1=TÞ C4
(15)
(16)
In Eqs. (15) and (16), C0, C1, C2, C3, C4,C5 are fitting parameters and mfl is the mean length at the annealing condition of (t, T) for the fanning equation.
194
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
[h] Parallel Linear (Yamada et al., 2007), YPL:
h 1=C5 i gðlÞ ¼ ln 1 upl =C0 C f ðt; TÞ ¼ C1 þ C2 lnðtÞ 3 T
parameters. The units used for the constants are the same presented in the original presentation Carlson (1990).
(17)
(18)
In Eqs. (17) and (18), C0, C1, C2, C3, C4, C5 are fitting parameters, and mpl is the mean length at the annealing condition of (t, T) for the parallel equation.
This model is expressed by the weighted sum of mfl and mpl.
(19)
Where
1=2 lnðt=t0 Þ kB T 10:24w U0 A1 lnðT0 =TÞ
(23)
(24)
where t0 ¼ (Z/2U0), Z¼ 6,5822 1016 eV. In Eqs. (23) and (24), n, w, U0, A1, T0, are fitting parameters.
2.2. Calibration datasets
wðTÞ ¼ 1=ð1 þ expð 11:08ð1000=TÞ 1:169ÞÞ
(20)
Two other models will be considered, which are based on hypotheses about mechanisms of the annealing process. They are: [a] The model proposed by Carlson (1990), CS:
gðrÞ ¼ ln½1 r
f ðt; TÞ ¼ ln
lnð1=rÞ gðrÞ ¼ ln n
f ðt; TÞ ¼
[i] Hybrid model (Yamada et al., 2007), YHY:
ul ðt; TÞ ¼ wðTÞufl ðt; TÞ þ ð1 wðTÞÞupl ðt; TÞ
[b] The model proposed by Guedes et al. (2005, 2006) with the modifications proposed by Guedes et al. (submitted), CR:
A k T Q þ n lnt þ nln B l0 h RT
(21)
(22)
where kB is Boltzmann’s constant (3.2997 1027 kcal K1), h is Planck’s constant (1.5836 1037 kcal s), R is the Universal gas constant (1.987 103 kcal mol1 K1) and l0, A, n, Q are fitting
For this study, we chose the dataset composed of a series of isochronal (the samples were annealed at different temperatures for a fixed time) annealing experiments for 4.5 min, 1 h, 11 h, 100 h and 1,000 h on spontaneous fission tracks in Nisatai Dacite (NST) zircon, presented by Yamada et al. (1995), experiments carried out for 10,000 h (Tagami et al., 1998) and those by Murakami et al. (2006), an experimental data set obtained for short term (<4 min) heating condition of spontaneous fission tracks in NST zircon at 550e910 C for w4, 10 and 100 s. It is proposed to incorporate geological data in the fitting procedure in order to improve extrapolation. For this purpose, it is used the bore hole data presented by Hasebe et al. (2003). This dataset is based on measurements of fission tracks in samples that experienced stable paleotemperatures (up to 230 C) for the last one million years. In this way, it will be treated (as an approximation) as a “1 Ma dataset”.
Fig. 1. Results of the fit for the zircon annealing models in an Arrhenius diagram. Different degrees of length reduction of experimental data are shown with different symbols. [a] FA and FC [b] LGFL and LGPL [c] RHFL and RHPL [d] YHY [e] CS [f] CR. The initials on the graphs mean: FA: Fanning Arrhenius (Laslett et al., 1987); FC: Fanning curvilinear (Crowley et al., 1991); LGFL: Improved Fanning Arrhenius (Laslett and Galbraith, 1996); LGPL: Improved Parallel Arrhenius (Laslett and Galbraith, 1996); RHFL: Fanning Linear (Rahn et al., 2004); RHPL: Parallel Linear (Rahn et al., 2004); YFL: Fanning Linear (Yamada et al., 2007); YPL: Parallel Linear (Yamada et al., 2007); YHY: Hybrid model (Yamada et al., 2007); CS: model by Carlson (1990); CR: Guedes et al. (submitted).
R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
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Fig. 2. Comparison between the zircon FT annealing model predictions and geological constraints (compiled by Rahn et al., 2004) on the zircon PAZ from the literature. [a] FA and FC [b] LGFL and LGPL [c] RHFL and RHPL [d] YHY [e] CS [f] CR. Light gray boxes indicate conditions of total track retention, dark gray boxes indicate partial annealing conditions and one black box indicates total annealing.
3. Fitting procedures and results Fitting of the model equation to experimental datasets was carried out using a c2 minimization scheme. The standard deviation has been used to weight the data. Numerical minimization was performed with combined Simplex and LangevineMarquardt interactions using the software OriginÒ. The Arrhenius plot of the annealing models equations is shown in Fig. 1 where the isoannealing lines represent two different degrees of length reduction, l/l0 ¼ 0.4 and 0.9. As stated before, empirical models hold only in the range of the experimental data used for calibration. The use of such long term data expands the range in which the empirical model is valid to times closer to the range of interest for practical applications. This is the reason to use the borehole data together with the laboratory dataset even knowing this is an approximation. Note in Fig. 1, for most models, the pair of contour lines is very similar over the range of time of laboratory experiments, but for geological timescale they make different predictions. Including the Hasebe et al. (2003) data, the fits were obtained with the curvilinear equations. However, the ultimate criterion to choose a model is its ability to reproduce geological data.
shown (Fig. 2) along with the experimental evidence presented by Rahn et al. (2004). Satisfactory temperature indexes are obtained for the fanning curvilinear (Crowley et al., 1991) and the hybrid (Yamada et al., 2007) models. Among the physical models, the best results are obtained by the Guedes et al. (submitted) model. Besides constraints for the ZPAZ, Rahn et al. (2004) presented field data for closure temperatures for several cooling rates. To estimate closure temperatures from the models, the definition given by Dodson (1973) was applied, namely, that the closure temperature is the one corresponding to the apparent age of the mineral in a constant cooling thermal history. To implement this definition, the principle of equivalent time, as defined by Duddy et al. (1988) was used in order to apply the model equations to
4. Extrapolation to geological timescale In order to compare model geological predictions with field evidence, the boundaries of the Zircon Partial Annealing Zone (ZPAZ) and the cooling rate dependent closure temperature were calculated for each model equation. The ZPAZ boundaries have been defined as 10 and 90% track-density reduction (Rahn et al., 2004). The transformation from length reduction (l/l0) to density reduction (r/r0) was done using the equation presented by Guedes et al. (2004) fitted to the dataset of Tagami et al. (1990). To take into account the tracks generated during heating, the heating duration, t, was divided in various intervals and the density reductions were computed individually from the time tracks were generated to present. Then, the mean reduction was calculated. In this way, temperatures leading to 10 and 90% track-density reduction were calculated for several values of t. The results are
Fig. 3. Comparison between the zircon FT annealing model predictions and geological constraints (compiled by Rahn et al., 2004) on the closure temperatures from the literature.
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R. Palissari et al. / Radiation Measurements 50 (2013) 192e196
variable temperatures. The results of these calculations are presented in Fig. 3. Field evidences for closure temperatures stand below the values simulated using the model equations. The models that give closer results are fanning curvilinear (Crowley et al., 1991), fanning and parallel linear models of Rahn et al. (2004), fanning linear of Yamada et al. (2007) and the Guedes et al. (submitted) model. 5. Conclusion Fission track annealing models were calibrated using zircon laboratory and geological scale data. Data sets were used with temperatures and times ranging from 350 to 910 C and w4 s to 10,000 h, respectively. As it can be seen Fig. 1, the annealing models diverge out of the laboratory range. Particularly, in the geological timescale the length shortening predictions are very different. In this way, the modeled thermal histories are dependent on the particular model to be used, which may lead to very different temperature predictions. It was proposed in this work to use the dataset presented by Hasebe et al. (2003) that is composed of geological scale data (1 Ma) of zircon samples from MITI-Nishikubiki and MITIMishima deep drilling, Japan. Thus, the calibration range is extended to more convenient time scale. Although the zircon ages and therefore modeled thermal histories may be in the order of several hundreds of millions of years, which are still out the calibration data range, the extrapolation is less uncertain. In addition, comparison of model predictions with field data for ZPAZ and closure temperatures made possible the evaluation of model extrapolation. ZPAZ temperatures are better predicted by Crowley et al. (1991), Yamada et al. (2007) and Guedes et al. (submitted) models. Closure temperature predictions, shown in Fig. 3, are similar for the following models: Crowley et al. (1991), Yamada et al. (2007), Guedes et al. (submitted), Rahn et al. (2004), which predict temperature slightly higher than field evidence. In this way, based on agreement with field evidence, we recommend that the curvilinear models by Crowley et al. (1991), Yamada et al. (2007) and Guedes et al. (submitted) should be used for modeling thermal histories in zircon fission-track systems. The other models, although also fitted with Hasebe et al. data, present divergent temperature predictions when extrapolated to times out of calibration data range.
References Carlson, W.D., 1990. Mechanisms and kinetics of apatite fission-track annealing. Am. Mineral 75, 1120e1139. Crowley, K.D., Cameron, M., Shaefer, R.L., 1991. Experimental studies of annealing of etched fission tracks in fluor apatite. Geochim. Cosmochim. Acta 55, 1449e1465. Dodson, M.H., 1973. Closure temperature in cool geochronological and petrological systems. Contrib. Mineral Petrol. 40, 259e274. Duddy, I.R., Green, P.F., Laslett, G.M., 1988. Thermal annealing of fission tracks in apatite. 3. Variable temperature behaviour. Chem. Geol. 73, 25e38. Guedes, S., Hadler, J.C., Iunes, P.J., Tello, C.A., 2004. Kinetic model for the relationship between confined fission-track length shortening and fission track age reduction in minerals. Nucl. Instrum. Methods B 217, 627e636. Guedes, S., Hadler, J.C., Iunes, P.J., Oliveira, K.M.G., Moreira, P.A.F.P., Tello, C.A., 2005. Kinetic model for the annealing of fission tracks in zircon. Radiat. Meas. 40, 517e521. Guedes, S., Hadler, J.C., Oliveira, K.M.G., Moreira, P.A.F.P., Iunes, P.J., Tello, C.A., 2006. Kinetic model for the annealing of fission tracks in minerals and its application to apatite. Radiat. Meas. 41, 392e398. Guedes S., Moreira P.A.F.P., Devanathan R., Weber W.J., Hadler J.C. , submitted. Improved fission-track annealing model based on reevaluation of annealing data. Hasebe, N., Mori, S., Tagami, T., Matsui, R., 2003. Geological partial annealing zone of zircon fission-track system: additional constraint from the deep drilling MITINischikubi and MITIMishima. Chem. Geol. 199, 45e52. Laslett, G.M., Green, P.F., Duddy, I.R., Gleadow, J.W.A., 1987. Thermal annealing of fission tracks in apatite 2. A quantitative analysis. Chem. Geol. Isot. Geosci. Sect. 65, 1e13. Laslett, G.M., Galbraith, R.F., 1996. Statistical modeling of thermal annealing of fission tracks in apatite. Geochim. Cosmochim. Acta 60, 5117e5131. Murakami, M., Yamada, R., Tagami, T., 2006. Short-term annealing characteristics of spontaneous fission tracks in zircon: a qualitative description. Chem. Geol. Isot. Geosc. Sect. 227, 214e222. Palenik, C.S., Nasdala, L., Ewing, R.C., 2003. Radiation damage in zircon. Am. Mineral 88, 770e781. Rahn, M.K., Brandon, M.T., Batt, G.E., Garver, J.I., 2004. A zero-damage model for fission-track annealing in zircon. Am. Mineral 89, 473e484. Tagami, T., Ito, H., Nishimura, S., 1990. Thermal annealing characteristics of fission tracks in zircon. Chem. Geol. Isot. Geosci. Sect. 80, 159e169. Tagami, T., Galbraith, R.F., Yamada, R., Laslett, G.M., 1998. Revised Annealing Kinetics of Fission Tracks in Zircon and Geological Implications. Advances in FissionTrack Geochronology. Kluwer Academic Publishing, The Netherlands, pp. 99e112. Yamada, R., Tagami, T., Nishimura, S., Ito, H., 1995. Annealing kinetics of fission tracks in zircon: an experimental study. Chem. Geol. Isot. Geosci. Sect. 122, 249e258. Yamada, R., Murakami, M., Tagami, T., 2007. Statistical modeling of annealing kinetics of fission tracks in zircon; Reassessment of laboratory experiments. Chem. Geol. 236, 75e91.