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Kinetic model for the annealing of fission tracks in minerals and its application to apatite
Radiation Measurements 41 (2006) 392 – 398 www.elsevier.com/locate/radmeas
Kinetic model for the annealing of fission tracks in minerals and its application to apatite S. Guedes a, ∗ , J.C. Hadler N a , K.M.G. Oliveira b , P.A.F.P. Moreira a , P.J. Iunes a , C.A. Tello S a a Departamento de Raios Cósmicos e Cronologia, Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, CP 6165, Campinas SP
13083-970, Brazil b Departamento de Físico-Química, Instituto de Química, Universidade Estadual de Campinas, CP 6154, Campinas SP 13083-970, Brazil
Received 11 October 2004; received in revised form 28 April 2005; accepted 10 June 2005
Abstract Fission tracks are formed in apatite and other minerals after the passage of fission fragments, which deliver locally intense amounts of energy to the crystal lattice. It is well known that the observable mean track lengths are reduced due to thermal treatment. If the annealing kinetics are known, it is sometimes possible to infer the thermal history a given sample experienced. Given the present lack of appropriate information on track formation, annealing and etching, researchers have used empirical models fitted to laboratory data on annealing to describe the annealing kinetics. In this work, a kinetic model is presented to describe the annealing process. It is based upon some experimental evidence. Instead of furnishing a complete and detailed description, it is intended to relate the observable quantities, namely, etched confined fission tracks, time and temperature based on simple hypotheses using a simplified view of the track. A kinetic model equation for the reduced mean track length, L/L0 , as a function of temperature, T, and heating duration, t, which fits quite well the available literature, has been derived and is given by (L/L0 ) = exp{−n exp[−w (U0 − A1 ln(t) + A2 ln2 (t) − kB T )1/2 ]} in which n is a parameter related to etching and track geometry, w and U0 are the width and the energy of a newly hypothesized potential barrier, respectively. A1 and A2 account for the dependence of the energy barrier on the duration of heating. Correlations with cell parameters of compositionally different apatites show that the barrier energy is the principal model descriptor for annealing. © 2005 Elsevier Ltd. All rights reserved. PACS: 29.40.Gx; 61.82.Ms; 61.85.+p; 93.85.+q Keywords: Apatite; Fission track; Annealing; Potential barrier; Model
1. Introduction Fission fragments released during fission (spontaneous or neutron-induced) of uranium atoms deliver high specific damage to the lattice of host minerals. The net result is the creation of a region, with cylindrical symmetry around the fragment trajectory, permeated with vacant atomic sites and interstitial atoms. The combined range of the two fragments is the length of the so-called latent fission track. Transmission ∗ Corresponding author. Tel.: +55 19 3788 5362; fax: +55 19 3788 5512.
E-mail address: guedes@ifi.unicamp.br (S. Guedes). 1350-4487/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2005.06.040
electron microscope (TEM) studies (Paul and Fitzgerald, 1992) show that the track diameters are in the magnitude of several nanometers while they are about 20 m in length in apatite. It has been early realized that fission tracks that reach a polished mineral surface can be enlarged by chemical etching to sizes where they are observable in optical microscopes and that latent tracks are often stable for geological times. Thus, as spontaneous fission obeys the decay law with a decay constant of ∼ 8.5 × 10−17 a−1 (Holden and Hoffman, 2000; Guedes et al., 2003a, b), the number of tracks is a suitable measure of the time minerals have
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
been recording tracks in the geological timescale. This is the principle of the fission-track dating method (Fleischer et al., 1975). Another important feature is that fission tracks have their etchable lengths reduced by thermal treatment. This process is called annealing. The annealing of fission tracks can be used to infer the thermal history of the host mineral provided the annealing rates are known, i.e. it is necessary to know quantitatively how tracks shorten due to temperature and time. The best measurable length of fission tracks is the horizontal confined fission-track length, which is measured in tracks fully contained in the crystal, parallel to the observation surface. These tracks are etched through surface tracks or crystal cleavages. Typical values of fresh neutron-induced fission-track lengths in apatite are around 16 m, considerably shorter than the latent track length measured using TEM (Paul and Fitzgerald, 1992). For the purpose of studying annealing rates, laboratory data sets relating confined track lengths in apatite with time and temperature of thermal treatment have been obtained (e.g., Green et al., 1986; Crowley et al., 1991; Carlson et al., 1999; Barbarand et al., 2003) and fitted with empirical equations (e.g., Laslett et al., 1987; Crowley et al., 1991; Laslett and Galbraith, 1996; Ketcham et al., 1999). Strictly, any purely empirical equations hold only in the range of time of the laboratory experiments, which are no longer than 10,000 h. However, to obtain thermal histories, these equations are extrapolated to the geological timescale, which can reach ∼ 500 Ma in the case of apatite. This approach has been validated by borehole data from the Otway Basin (Green et al., 1989). Nevertheless, it has been proposed that results from another borehole, the KTB (Kontinetales Tiefbohrprogramm der Bundesrepublik Deutschland), do not corroborate the Otway predictions (Jonckheere, 2003). On the other hand, few attempts to propose mechanisms of annealing have been done. Carlson (1990) proposed a model for annealing of fission tracks based on the migration of lattice defects. Although it has succeeded in fit the laboratory data, it has not been well accepted mainly because of the ad hoc introduction of unetchable gaps for track length reduction (L/L0 ) < 0.65 (Green et al., 1993; Carlson, 1993), where L is the track length after annealing and L0 is the track length measured in fresh induced fission tracks, in order to make the fit possible. Sood et al. (2000) also proposed a model for migration of defects and were able to reproduce the empirical fanning Arrhenius equation proposed by Laslett et al. (1987). It is in this context that the present work is inserted. It is aimed to contribute to the efforts of understanding and interpreting the experimental data on mean etched fission-track length reduction due to time and temperature. In the following, a model is proposed, based on general experimental observations. It is not the objective of this work to describe the annealing process in detail, but propose a general mechanism. Because of this, the track is presented in a simplified fashion. The final goal is to relate the observable quantities, namely, time, temperature and the length of etched confined fission tracks. The resulting
393
equation is compared with the experimental data and the model implications are discussed.
2. The model 2.1. Conceptual picture Studies with small angle X-ray scattering methods (Dartyge et al., 1981) revealed the existence of extended defect zones, separated by gap zones loaded with point defects in the track. Yada et al. (1987) showed that fission tracks in zircon are composed of an almost amorphous inner region with a clear separation from the undamaged crystal lattice. Paul (1993) observed the same structure for fission tracks in apatite. Jaskierowics et al. (2004) found similar behavior for C60 cluster tracks in apatite. Based upon these observations, it is assumed that the track is composed of a region permeated with vacant atomic sites (Yada et al., 1987) and interstitial atoms, which cause the lattice distortion. The lattice regeneration occurs when displaced atoms occupy vacant sites. It is proposed, in this work, that the track stability is assured by the existence of a potential energy barrier, which prevents immediate track restoration. It is additionally supposed that the barrier characteristics (width and energy) depend on the amount of vacant sites and interstitial atoms. The justification for the hypothesized energy barrier is that this mechanism can account for an unexplained experimental result: when temperature is increased in fission-track annealing experiments, the tracks are gradually reduced until ∼ 40% (in case of apatite) of their initial lengths (Ketcham et al., 1999), when they suddenly cease to exist. According to the proposed model, displaced atoms will just recombine with vacant sites if they succeed in transiting the energy barrier. The barrier could be the energy to move an interstitial to a vacant site or that to move a vacancy to an interstitial—whichever is smaller. If every transmission event results in recombination, the number of vacant sites is diminished, leading to a new state of equilibrium with consistent decreasing in the energy of the potential barrier. In addition, if the apatite experiences higher temperatures, the displaced atom thermal energy is increased and consequently the transmission probability is also increased. Note that if the energy of the displaced atoms is higher than the energy barrier, they are free to recombine with the vacant sites. When this threshold energy is reached, the number of vacant sites in the track experiences an abrupt fall to near zero and the observable etched track ceases to exist. As mentioned above, the observable quantities are lengths of etched confined fission tracks. Thus, the amount of residual vacant sites in the track must be transformed into an etched confined track. Guedes and coworkers (2004, 2005) proposed that chemical etching obeys rate laws as most chemical reactions and were able to explain the experimental results for the relationship between surface track density reduction and length shortening of fission tracks in minerals, which shows a deviation from the behavior predicted by geometrical considerations.
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The principal implication of their model is that the etching rates depend on the residual amount of defects compounding the track. As tracks have cylindrical symmetry, if an axial track length shortening mechanism is accepted (Carlson, 1990), in which length shortening and diameter reduction occur at the same rate, the track length can be chosen as the main characteristic of the track volume. Taking the number of vacancy-interstitial pairs, N, to be proportional to the track volume and assuming that the defect etching velocity is much higher than that in the undamaged structure, it is proposed that L = kN
n
Applying the transformation given by Eq. (2) m−1 L = exp −n (Ni , T ) . L0
Note however that the only measurable quantities are L and L0 . There is no way to access the intermediary Ni ’s. For simplicity, we can define a new quantity given by (Ni s; T ) = (N0 , N1 , N2 , . . . , Nm−1 ; T ) =
(1)
with k and n being parameters related to the track volume. If the view of a chemical etching obeying rate laws is also adopted, the parameters k and n must also depend on etching conditions. Taking L0 and N0 to be the initial etched fission-track length and the initial number of vacancies, i.e. soon after the track attain its first equilibrium, the reduced confined fission-track length is n L N = . (2) L0 N0
N0 = (N0 , T )N0 .
(3)
In Eq. (3) it has been assumed that all displaced atoms have the same probability to transmit and that the transmission coefficient, , varies consistently with the number of vacant sites and is a function of temperature, mainly due to the displaced atom energy, which must be of the magnitude of kB T . Then, the amount of vacant sites is N1 = N0 − N0 = [1 − (N0 , T )]N0 .
(4)
Further variation in the amount of vacant sites is expressed by N1 = (N1 , T )N1
(5)
and N2 = N1 − N1 = [1 − (N1 , T )][1 − (N0 , T )]N0 .
i=0
(Ni , T ). (9)
i=0
1/2 ,
(10)
in which kB T is the energy of a displaced atom at temperature T and w is the width of the barrier. The constants mef and h¯ are the displaced atom effective mass and the Planck’s constant divided by 2. With energies in eV and w in nm, Eqs. (8)–(10) can be combined to obtain L = exp{−n exp[−w (U − kB T )1/2 ]}, L0
(11)
in which w is a constant embracing mef , h¯ and w. It is of interest to find how the barrier mean energy varies for the different times employed in annealing experiments. Let U0 be the barrier energy at the time the track attained the equilibrium. It is expected that insofar as the number of vacancies in the track diminishes, the barrier energy diminish consistently. Close inspection of the curves in Fig. 1 shows that annealing has logarithm dependence on the duration of the experiments, i.e. the distance between curves is approximately constant although the duration of the experiments varies in a logarithm scale. Thus, a suitable expression for the dependence of U on the duration of heating experiments is
(6)
Repeating this procedure for m steps during the heating experiment, it can be shown that m−1 m−1 N = [1 − (Ni , T )] exp − (Ni , T ) . (7) N0 i=0
2mef (U − kB T ) (U, T ) ≈ exp −2w h¯ 2
Consider now a fission track and its surroundings, initially in equilibrium at temperature T. The variation in N is given by the number of displaced atoms that succeeded in transmit through the potential barrier.
m−1
Suppose that each Ni is associated with an intermediary value of the barrier energy, Ui . If a new variable U can be defined to represent some association of Ui ’s, this new parameter can be associated with the Ni ’s. Thus, a transmission coefficient given in terms of a barrier energy and temperature can be used to represent the probability of displace atoms transmit through the potential barrier. In order to obtain an analytical expression, let the potential barrier be approximated by a square barrier and the approximate JWKB solution for the transmission coefficient (Sakurai, 1994, p. 449) be used.
2.2. Kinetic picture
(8)
i=0
U = U0 −
Aj [ln(t)]j .
(12)
j
In Eq. (12), t the duration of the annealing experiment (in seconds).The time t is a normalized time and the normalizing factor is set to 1 s. A better justification for Eq. (12) is provided
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
time, Eq. (14) would reduce to t N = exp − (t , T ) dt , N0 0
reduced mean track length, L/L0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
395
heating duration 1 hour 10 hours 100 hours 1,000 hours
0.0 250 300 350 400 450 500 550 600 650 temperature, T (K) Fig. 1. Results for the annealing of fission tracks in Durango apatite. The scatter plot represents the experimental data found in literature (Carlson et al., 1999) and the continuous line represents the model fit.
in the next topic. Then, it follows from Eqs. (11) and (12) that ⎧ ⎡ ⎛ ⎞1/2 ⎤⎫ ⎪ ⎪ ⎬ ⎨ L ⎥ ⎢ ⎝ j ⎠ = exp −n exp ⎣−w U0 − Aj [ln(t)] −kB T ⎦ . ⎪ ⎪ L0 ⎭ ⎩ j (13) As it will be shown in Section 3, Eq. (13) provides good fit to experimental results. 2.3. Further justification for the time variation of the barrier energy As mentioned before, it was preferred to construct a model based on the observations relating etched mean confined tracks and a few general observations on track structure. In this sense, Eq. (13) have been built up rather than demonstrated. The time dependence has been tacitly suppressed in Section 2.2, and then reintroduced as a relation for the dependence of the barrier energy on heating duration. Since the potential barrier is not constant in time, a more rigorous treatment would demand the self-consistent solution of the time dependent Schröndinger equation along with a semi-classical equation as dN = −(N, T )N, dt
(14)
in which is the frequency of collision of the displaced atoms on the barrier. Eq. (14) accounts for the dependence of the barrier energy on the amount of remaining vacant sites. This equation describes the two coupled mechanisms: (1) Insofar as the amount of vacant sites diminishes, the rate this happens also diminishes (dependence on N ); (2) As the amount of vacant sites diminishes, the energy barrier diminishes and the transmission coefficient is increased (dependence on (N, T )). The resulting rate of the amount of vacant sites variation is the interplay between these two mechanisms. If, as a first approximation, the potential were taken to be explicitly dependent on
(15)
which is similar in its general format to Eq. (7). Another matter is that the format for the spatial part of the barrier energy (square) was arbitrarily chosen. If, on one hand, it is a simplification, it has been chosen because its solutions are well understood. In addition, as the exact fission-track structure is unknown, there is no reason to use more complicated potentials. Finally, the transformation expressed in Eq. (2) assumes just that the fission track has cylindrical symmetry and that etching obeys rate laws. As it has been previously discussed, the first is justified because the track is formed around the fragment trajectory and the latter by the fact that most chemical reactions including mineral dissolution (Brady and House, 1996) follow rate laws. Then, it seems reasonable to make the same assumption for track etching. The discussion above makes possible to reanalyze the time variation of the barrier energy, given by Eq. (12), in terms of self-consistency. Suppose a simple format for the dependence of U on N: U = CN, in which C is a proportionality constant. In addition, let, as a rough approximation, the transmission coefficient be constant in Eq. (15), in such a way that U = CN0 exp{−t exp[−w (U − kB T )1/2 ]},
(16)
Eq. (16) explicitly relates the heating duration with the barrier energy. Solving this equation to isolate the variable U in the argument of the exponential 2 CN0 1 U = kB T + 2 ln(t) − ln ln . (17) U w The term ln(ln(CN0 /U )) will vary much slower than the term ln(t), so that it can be replaced (as a still rough approximation) by a constant B. Then, after some more algebra U = kB T + +
1 w 2
(B − ln())2 w 2
−
2(B − ln()) w 2
ln(t)
[ln(t)]2 .
(18)
Identifying the first and second terms with U0 , the third with A1 and the fourth with A2 , the general format of Eq. (12) is obtained. However, given that so many approximations were employed to construct Eq. (18), further interpretation should be avoided. 3. Comparison with experimental results 3.1. Evaluation of the model parameters The most complete and carefully carried out annealing experiment of neutron-induced fission tracks has been presented by
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interpretation to be fully applicable. The used index was the temperature of 100% fading in a 30 Ma annealing time, TF,30 , which for Durango-like apatite is in the range 95–100 ◦ C, for Otway Basin data. Following the interpretation by Ketcham and coworkers (1999), TF,30 , was calculated as being the temperature needed to reduce the mean track length to 0.41 (the lowest observed value for this quantity). In Fig. 1, it can also be observed that there is a threshold temperature (for each isochronal curve), T = U/kB , at which the reduction curve breaks. At temperatures higher than this, the displaced atom energy is higher than the barrier energy. In this way, displaced atoms can easily cross the barrier and the track vanishes very quickly.
mean barrier energy, U (meV)
57 56 55 54 53 52 51 50 49 48 47 1xe7
1xe9
1xe11 1xe13 ln(heating duration)
1xe15
Fig. 2. Time dependence of the mean energy barrier, U, on the duration of the heating experiments, t (given in s), for Durango apatite. The scatter plot shows the U values estimated by fitting individual (constant t) heating curves. The continuous line represents the second order polynomial in ln(t) used to obtain the model fit in Fig. 1.
Carlson and coworkers (1999). Etched track lengths have been measured for 15 compositionally diverse apatites. In that study, 1, 10, 100 and 1000 h isochronal heating experiments have been performed for various temperatures. Results for the Durango apatite (a widely used fission-track standard) are shown in Fig. 1 (scatter plot). It can be observed that, just as stated above, reduced track lengths below ∼ 0.4 are not found. Let Eq. (11) be fitted to each individual curve in Fig. 1. Fittings in this work were performed using the 2 minimization scheme available in the software Origin䉸 6.0. The parameter n, which depends on etching and geometrical properties of the fission tracks, is expected to be constant for a given sample. Supposing additionally that the parameter w is constant, all the time dependence is let to the energy of the barrier, U. Initially, the w , n, and U values are found for the 1 h curve. Then, fixing w and n, the U values are found for the other time duration curves. These values are plotted in Fig. 2 as a function of ln(t). Although these data can be well fitted by a linear curve (in ln(t)), the systematic behavior found for all the tested apatites suggests that it is more appropriated to use a second order polynomial. The general format of the U − ln(t) curves show the same upward concavity noted in Fig. 2. In this way, Eqs. (11) and (12) can be joined to give L = exp{−n exp[−w (U0 − A1 ln(t) L0 + A2 ln2 (t) − kB T )1/2 ]}.
(19)
Eq. (19) can then be used to fit the experimental data. In case of the Durango apatite, the obtained parameter values are n = 2.4408, w =34.4618 eV−1/2 , U0 =63.77 meV, A1 =1.12 meV, and A2 = 6.5 × 10−3 meV. The fit curve is shown in Fig. 1 (continuous line). The presented parameters were calibrated also to yield reasonable geological predictions. At this point the Otway Basin version is used, since the KTB data needs further
3.2. Correlation with the apatite structure Barbarand and coworkers (2003) obtained another data set of interest. It was idealized to complement the one discussed above. For 13 compositionally diverse apatites, curves for 10 h heating have been obtained. Curves for 100 and 1000 h were not so regularly measured. However, the principal advantage of this data set is that the unit-cell parameters a and c have been measured for 11 of these samples, making possible a discussion on the influence of the apatite structure in annealing. The authors have already shown that the cell parameters are the best control in fission-track annealing. Next, this will be analyzed from the standpoint of the presented model. Let, for sake of comparison, the value of n be kept constant as determined for the Durango apatite. Using Eq. (11), the curves for 10 h heating can be fitted to determine the values of w and U. The results are shown in Fig. 3, in which the width and the energy of the potential barrier are compared with the cell parameters of the studied apatites. No correlation is observed relating the barrier widths whose values are around 3.2 nm (Fig. 3c and d). On the other hand, clear correlations are noted relating the barrier energy (Fig. 3a and b), suggesting that this parameter is the principal model descriptor for annealing. The correlation found with the unit-cell parameter c is especially interesting because it is in c-axis that dislocations in ion positions due to changes in chemical composition take place (Calderín et al., 2003). Finally, let Fig. 4 be analyzed. In this figure, fitting U values are plotted as function of the heating times for the 10, 100 and 1000 h annealing curves for the different apatites. Some characteristic chlorine contents (in oxide weight percent) are shown near the curves as determined by Barbarand and coworkers (2003). Two features deserve to be pointed out. The first is that curves can be grouped by chlorine content. Higher the chlorine contents higher the energy barrier, agreeing with the early realization that higher chlorine content apatites are more resistant to annealing (Green et al., 1985). The second feature is that the barrier energies follow near parallel trends for all apatites indicating that the general mechanism controlling the annealing of fission tracks in apatite is not affected by chemical composition variation.
w´ for 10h heating (eV-1/2)
barrier energy for 10h heating (meV)
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
9.36 9.38 9.40 9.42 9.44 9.46 57
397
6.85 6.86 6.87 6.88 6.89 6.90 57
56
56
55
55
54
54
53
53
52
52
51
(b)
(a)
51
50 38
50 38
36
36
34
34
32
32
30
30
28 (c)
(d)
9.36 9.38 9.40 9.42 9.44 9.46 cell par ameter a (Å)
28
6.85 6.86 6.87 6.88 6.89 6.90 cell parameter c (Å)
barrier energy (meV)
Fig. 3. Correlation of the model parameters characterizing the potential barrier (w and energy) with the apatite unit-cell parameters as reported by Barbarand and coworkers (Barbarand et al., 2003) for compositionally diverse apatites.
58 57 56 55 54 53 52 51 50 49 48 47
4.2. Comment on the significance of L0
cl- content 1.54 to 4.67 0.71 to 0.83 0.02 to 0.39
< 0.031
1xe9
1xe10 1xe11 1xe12 1xe13 1xe14 ln (heating duration)
e15
Fig. 4. Time dependence of the mean energy barrier, U, for compositionally diverse apatites (Barbarand et al., 2003). Chlorine contents shown near curves are given in oxide weight percent. Straight lines connecting the points are just to distinguish the curves. Note that curves can be grouped by apatite chlorine content.
4. Discussion and conclusion 4.1. Comments on the model parameters After Eq. (19), there are five parameters to be considered. The most complicated to define is the parameter n, which contains the combined dependence on etching and track geometry. If the track could be considered a perfect cylinder, the parameter n would be the etching reaction order. Parameters w and U0 characterize the potential barrier, being its width and energy, respectively, and A1 and A2 control its variation. The barrier energy is the primary responsible for track stability, preventing atoms to immediately occupy the vacant sites.
Donelick and coworkers (1990), carried out an experiment in which tracks were induced in apatite samples by thermal neutron irradiation. Then, portions of these apatites were etched after different times. It was noted a progressive reduction of the etched fission-track lengths until around 3 weeks after irradiation. The authors suggest that reproducible measurements of the mean etchable fission-track lengths in apatite may be made on tracks that are aged for at least one month below the room temperature. In this way, the “initial” mean track length, L0 value is the one measured after at least one month after the track creation. Note however, that this initial reduction in the track length is a still transient process compared with the million-year timescale of the annealing phenomenon. 4.3. Final remarks A model for annealing of fission tracks in apatite has been built up based on the transmission of displaced atoms through an energy barrier and general observations on track properties, which succeeded to fit in experimental data. Particularly, the role of the barrier energy in the annealing process could be well characterized by correlations discussed in Section 3.2. However, due to the several simplifications employed, it is still semi-empirical because the parameter values must be found by data inspection. Acknowledgements The State of São Paulo Research Foundation (FAPESP) through a Post Doctoral grant, process number 01/02805-9, and a Thematic Project, process number 00/03960-5, supported this work.
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References Barbarand, J., Carter, A., Wood, I., Hurford, T., 2003. Compositional and structural control of fission-track annealing in apatite. Chem. Geol. 198, 107–137. Brady, P.V., House, W.A., 1996. Surface-controlled dissolution and growth of minerals. In: Brady, P.V. (Ed.), Physics and Chemistry of Mineral Surfaces. CRC Press, Boca Raton, pp. 225–305. Calderín, L., Stott, M.J., Rubio, A., 2003. Electronic and crystallographic structure of apatites. Phys. Rev. B 67, 134106. Carlson, W.D., 1990. Mechanisms and kinetics of apatite fission-track annealing. Am. Mineral. 75, 1120–1139. Carlson, W.D., 1993. Mechanisms and kinetics of fission-track annealing—replay. Am. Mineral. 78, 446–449. Carlson, W.D., Donelick, R.A., Ketcham, R.A., 1999. Variability of apatite fission-track annealing kinetics: I. Experimental results. Am. Mineral. 84, 1213–1223. Crowley, K.D., Cameron, M., Schaefer, R.L., 1991. Experimental studies of annealing of etched fission tracks in fluorapatite. Geochim. Cosmoch. Acta 55, 1449–1465. Dartyge, E., Duraud, J.P., Langevin, Y., Maurette, M., 1981. New model of nuclear particle tracks in dielectric minerals. Phys. Rev. B 23, 5213–5229. Donelick, R.A., Roden, M.K., Mooers, J.D., Stephen Carpenter, B., Miller, D.S., 1990. Etchable length reduction of induced fission tracks in apatite at room temperature (∼ 23◦ C): crystallographic orientation effects and “initial” mean lengths. Nucl. Tracks. Radiat. Meas. 17, 261–265. Fleischer, R.L., Price, P.B., Walker, R.M., 1975. Nuclear Tracks in Solids: Principles and Applications. University of California Press, Berkeley, 605pp. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1985. Fission-track annealing in apatite: track length measurements and the form of the Arrhenius plot. Nucl. Tracks Radiat. Meas. 10, 323–328. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1986. Thermal annealing of fission tracks in apatite 1. Qualitative description. Chem. Geol. (Isot. Geosci. Sect.) 59, 237–253. Green, P.F., Duddy, I.R., Laslett, G.M., Hegarty, K.A., Gleadow, A.J.W., Lovering, J.F., 1989. Thermal annealing of fission tracks in apatite 4. Quantitative modelling techniques and extension to geological timescales. Chem. Geol. (Isot. Geosci. Sect.) 79, 155–182. Green, P.F., Laslett, G.W., Duddy, I.R., 1993. Mechanisms and kinetics of fission-track annealing—discussion. Am. Mineral. 78, 441–445. Guedes, S., Hadler N, J.C., Iunes, P.J., Zuñiga, A., Tello S, C.A., Paulo, S.R., 2003a. The use of the U (n, f ) dosimetry in the determination of the f through fission-track techniques. Nucl. Instr. and Meth. A 496, 215–221.
Guedes, S., Hadler N, J.C., Sarkis, J.E.S., Oliveira, K.M.G., Kakazu, M.H., Iunes, P.J., Saiki, M., Tello S, C.A., Paulo, S.R., 2003b. Spontaneous-fission decay constant of 238 U measured by nuclear track techniques without neutron irradiation. J. Radioanal. Nucl. Chem. 258, 117–122. Guedes, S., Hadler N, J.C., Iunes, P.J., Tello S, C.A., 2004. Kinetic model for the relationship between confined fission-track length shortening and fission-track age reduction in minerals. Nucl. Instr. and Meth. B 217, 627–636. Guedes, S., Iunes, P.J., Hadler N, J.C., Bigazzi, G., Tello S, C.A., Alencar, I., Palissari, R., Curvo, E.A.C., Moreira, P.A.F.P., 2005. Kinetic model for the relationship between mean diameter shortening and age reduction in glass samples. Radiat. Meas. 39, 647–652. Holden, N.E., Hoffman, D.C., 2000. Spontaneous fission half-lives ground-state nuclides (Technical report). Pure Appl. Chem. 72, 1525–1562. Jaskierowics, G., Dunlop, A., Jonckheere, R., 2004. Track formation in fluorapatite irradiated with energetic cluster ions. Nucl. Instr. and Meth. B 222, 213–227. Jonckheere, R., 2003. On methodical problems in estimating geological temperature and time from measurements of fission tracks in apatite. Radiat. Meas. 36, 43–55. Ketcham, R.A., Donelick, R.A., Carlson, W.D., 1999. Variability of fissiontrack annealing kinetics: III. Extrapolation to geological time scales. Am. Mineral. 84, 1235–1255. Laslett, G.M., Galbraith, R.F., 1996. Statistical modeling of thermal annealing of fission tracks in apatite. Geochim. Cosmochim. Acta 60, 5117–5131. Laslett, G.M., Green, P.F., Duddy, I.R., Gleadow, A.J.W., 1987. Thermal annealing of fission tracks in apatite. 2. A quantitative analysis. Chem. Geol. (Isot. Geosci. Sect.) 65, 1–13. Paul, T.A., 1993. Transmission electron microscopy investigation of unetched fission tracks in fluorapatite-physical process of annealing. Nucl. Tracks Radiat. Meas. 21, 507–511. Paul, T.A., Fitzgerald, P.G., 1992. Transmission electron microscopic investigation of fission tracks in fluorapatite. Am. Mineral. 77, 336–344. Sakurai, J.J., 1994. Modern Quantum Mechanics. Addison-Wesley, Menlo Park, CA, 500pp. Sood, N.K., Singh, T., Sandhu, A.S., Singh, L., 2000. A global model for annealing in solid state track detectors. Radiat. Meas. 32, 79–82. Yada, K., Tanji, T., Sunagawa, I., 1987. Radiation induced lattice defects in natural zircon (ZrSiO4 ) observed at atomic resolution. Phys. Chem. Minerals 14, 197–204.
Kinetic model for the annealing of fission tracks in minerals and its application to apatite S. Guedes a, ∗ , J.C. Hadler N a , K.M.G. Oliveira b , P.A.F.P. Moreira a , P.J. Iunes a , C.A. Tello S a a Departamento de Raios Cósmicos e Cronologia, Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, CP 6165, Campinas SP
13083-970, Brazil b Departamento de Físico-Química, Instituto de Química, Universidade Estadual de Campinas, CP 6154, Campinas SP 13083-970, Brazil
Received 11 October 2004; received in revised form 28 April 2005; accepted 10 June 2005
Abstract Fission tracks are formed in apatite and other minerals after the passage of fission fragments, which deliver locally intense amounts of energy to the crystal lattice. It is well known that the observable mean track lengths are reduced due to thermal treatment. If the annealing kinetics are known, it is sometimes possible to infer the thermal history a given sample experienced. Given the present lack of appropriate information on track formation, annealing and etching, researchers have used empirical models fitted to laboratory data on annealing to describe the annealing kinetics. In this work, a kinetic model is presented to describe the annealing process. It is based upon some experimental evidence. Instead of furnishing a complete and detailed description, it is intended to relate the observable quantities, namely, etched confined fission tracks, time and temperature based on simple hypotheses using a simplified view of the track. A kinetic model equation for the reduced mean track length, L/L0 , as a function of temperature, T, and heating duration, t, which fits quite well the available literature, has been derived and is given by (L/L0 ) = exp{−n exp[−w (U0 − A1 ln(t) + A2 ln2 (t) − kB T )1/2 ]} in which n is a parameter related to etching and track geometry, w and U0 are the width and the energy of a newly hypothesized potential barrier, respectively. A1 and A2 account for the dependence of the energy barrier on the duration of heating. Correlations with cell parameters of compositionally different apatites show that the barrier energy is the principal model descriptor for annealing. © 2005 Elsevier Ltd. All rights reserved. PACS: 29.40.Gx; 61.82.Ms; 61.85.+p; 93.85.+q Keywords: Apatite; Fission track; Annealing; Potential barrier; Model
1. Introduction Fission fragments released during fission (spontaneous or neutron-induced) of uranium atoms deliver high specific damage to the lattice of host minerals. The net result is the creation of a region, with cylindrical symmetry around the fragment trajectory, permeated with vacant atomic sites and interstitial atoms. The combined range of the two fragments is the length of the so-called latent fission track. Transmission ∗ Corresponding author. Tel.: +55 19 3788 5362; fax: +55 19 3788 5512.
E-mail address: guedes@ifi.unicamp.br (S. Guedes). 1350-4487/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2005.06.040
electron microscope (TEM) studies (Paul and Fitzgerald, 1992) show that the track diameters are in the magnitude of several nanometers while they are about 20 m in length in apatite. It has been early realized that fission tracks that reach a polished mineral surface can be enlarged by chemical etching to sizes where they are observable in optical microscopes and that latent tracks are often stable for geological times. Thus, as spontaneous fission obeys the decay law with a decay constant of ∼ 8.5 × 10−17 a−1 (Holden and Hoffman, 2000; Guedes et al., 2003a, b), the number of tracks is a suitable measure of the time minerals have
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
been recording tracks in the geological timescale. This is the principle of the fission-track dating method (Fleischer et al., 1975). Another important feature is that fission tracks have their etchable lengths reduced by thermal treatment. This process is called annealing. The annealing of fission tracks can be used to infer the thermal history of the host mineral provided the annealing rates are known, i.e. it is necessary to know quantitatively how tracks shorten due to temperature and time. The best measurable length of fission tracks is the horizontal confined fission-track length, which is measured in tracks fully contained in the crystal, parallel to the observation surface. These tracks are etched through surface tracks or crystal cleavages. Typical values of fresh neutron-induced fission-track lengths in apatite are around 16 m, considerably shorter than the latent track length measured using TEM (Paul and Fitzgerald, 1992). For the purpose of studying annealing rates, laboratory data sets relating confined track lengths in apatite with time and temperature of thermal treatment have been obtained (e.g., Green et al., 1986; Crowley et al., 1991; Carlson et al., 1999; Barbarand et al., 2003) and fitted with empirical equations (e.g., Laslett et al., 1987; Crowley et al., 1991; Laslett and Galbraith, 1996; Ketcham et al., 1999). Strictly, any purely empirical equations hold only in the range of time of the laboratory experiments, which are no longer than 10,000 h. However, to obtain thermal histories, these equations are extrapolated to the geological timescale, which can reach ∼ 500 Ma in the case of apatite. This approach has been validated by borehole data from the Otway Basin (Green et al., 1989). Nevertheless, it has been proposed that results from another borehole, the KTB (Kontinetales Tiefbohrprogramm der Bundesrepublik Deutschland), do not corroborate the Otway predictions (Jonckheere, 2003). On the other hand, few attempts to propose mechanisms of annealing have been done. Carlson (1990) proposed a model for annealing of fission tracks based on the migration of lattice defects. Although it has succeeded in fit the laboratory data, it has not been well accepted mainly because of the ad hoc introduction of unetchable gaps for track length reduction (L/L0 ) < 0.65 (Green et al., 1993; Carlson, 1993), where L is the track length after annealing and L0 is the track length measured in fresh induced fission tracks, in order to make the fit possible. Sood et al. (2000) also proposed a model for migration of defects and were able to reproduce the empirical fanning Arrhenius equation proposed by Laslett et al. (1987). It is in this context that the present work is inserted. It is aimed to contribute to the efforts of understanding and interpreting the experimental data on mean etched fission-track length reduction due to time and temperature. In the following, a model is proposed, based on general experimental observations. It is not the objective of this work to describe the annealing process in detail, but propose a general mechanism. Because of this, the track is presented in a simplified fashion. The final goal is to relate the observable quantities, namely, time, temperature and the length of etched confined fission tracks. The resulting
393
equation is compared with the experimental data and the model implications are discussed.
2. The model 2.1. Conceptual picture Studies with small angle X-ray scattering methods (Dartyge et al., 1981) revealed the existence of extended defect zones, separated by gap zones loaded with point defects in the track. Yada et al. (1987) showed that fission tracks in zircon are composed of an almost amorphous inner region with a clear separation from the undamaged crystal lattice. Paul (1993) observed the same structure for fission tracks in apatite. Jaskierowics et al. (2004) found similar behavior for C60 cluster tracks in apatite. Based upon these observations, it is assumed that the track is composed of a region permeated with vacant atomic sites (Yada et al., 1987) and interstitial atoms, which cause the lattice distortion. The lattice regeneration occurs when displaced atoms occupy vacant sites. It is proposed, in this work, that the track stability is assured by the existence of a potential energy barrier, which prevents immediate track restoration. It is additionally supposed that the barrier characteristics (width and energy) depend on the amount of vacant sites and interstitial atoms. The justification for the hypothesized energy barrier is that this mechanism can account for an unexplained experimental result: when temperature is increased in fission-track annealing experiments, the tracks are gradually reduced until ∼ 40% (in case of apatite) of their initial lengths (Ketcham et al., 1999), when they suddenly cease to exist. According to the proposed model, displaced atoms will just recombine with vacant sites if they succeed in transiting the energy barrier. The barrier could be the energy to move an interstitial to a vacant site or that to move a vacancy to an interstitial—whichever is smaller. If every transmission event results in recombination, the number of vacant sites is diminished, leading to a new state of equilibrium with consistent decreasing in the energy of the potential barrier. In addition, if the apatite experiences higher temperatures, the displaced atom thermal energy is increased and consequently the transmission probability is also increased. Note that if the energy of the displaced atoms is higher than the energy barrier, they are free to recombine with the vacant sites. When this threshold energy is reached, the number of vacant sites in the track experiences an abrupt fall to near zero and the observable etched track ceases to exist. As mentioned above, the observable quantities are lengths of etched confined fission tracks. Thus, the amount of residual vacant sites in the track must be transformed into an etched confined track. Guedes and coworkers (2004, 2005) proposed that chemical etching obeys rate laws as most chemical reactions and were able to explain the experimental results for the relationship between surface track density reduction and length shortening of fission tracks in minerals, which shows a deviation from the behavior predicted by geometrical considerations.
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The principal implication of their model is that the etching rates depend on the residual amount of defects compounding the track. As tracks have cylindrical symmetry, if an axial track length shortening mechanism is accepted (Carlson, 1990), in which length shortening and diameter reduction occur at the same rate, the track length can be chosen as the main characteristic of the track volume. Taking the number of vacancy-interstitial pairs, N, to be proportional to the track volume and assuming that the defect etching velocity is much higher than that in the undamaged structure, it is proposed that L = kN
n
Applying the transformation given by Eq. (2) m−1 L = exp −n (Ni , T ) . L0
Note however that the only measurable quantities are L and L0 . There is no way to access the intermediary Ni ’s. For simplicity, we can define a new quantity given by (Ni s; T ) = (N0 , N1 , N2 , . . . , Nm−1 ; T ) =
(1)
with k and n being parameters related to the track volume. If the view of a chemical etching obeying rate laws is also adopted, the parameters k and n must also depend on etching conditions. Taking L0 and N0 to be the initial etched fission-track length and the initial number of vacancies, i.e. soon after the track attain its first equilibrium, the reduced confined fission-track length is n L N = . (2) L0 N0
N0 = (N0 , T )N0 .
(3)
In Eq. (3) it has been assumed that all displaced atoms have the same probability to transmit and that the transmission coefficient, , varies consistently with the number of vacant sites and is a function of temperature, mainly due to the displaced atom energy, which must be of the magnitude of kB T . Then, the amount of vacant sites is N1 = N0 − N0 = [1 − (N0 , T )]N0 .
(4)
Further variation in the amount of vacant sites is expressed by N1 = (N1 , T )N1
(5)
and N2 = N1 − N1 = [1 − (N1 , T )][1 − (N0 , T )]N0 .
i=0
(Ni , T ). (9)
i=0
1/2 ,
(10)
in which kB T is the energy of a displaced atom at temperature T and w is the width of the barrier. The constants mef and h¯ are the displaced atom effective mass and the Planck’s constant divided by 2. With energies in eV and w in nm, Eqs. (8)–(10) can be combined to obtain L = exp{−n exp[−w (U − kB T )1/2 ]}, L0
(11)
in which w is a constant embracing mef , h¯ and w. It is of interest to find how the barrier mean energy varies for the different times employed in annealing experiments. Let U0 be the barrier energy at the time the track attained the equilibrium. It is expected that insofar as the number of vacancies in the track diminishes, the barrier energy diminish consistently. Close inspection of the curves in Fig. 1 shows that annealing has logarithm dependence on the duration of the experiments, i.e. the distance between curves is approximately constant although the duration of the experiments varies in a logarithm scale. Thus, a suitable expression for the dependence of U on the duration of heating experiments is
(6)
Repeating this procedure for m steps during the heating experiment, it can be shown that m−1 m−1 N = [1 − (Ni , T )] exp − (Ni , T ) . (7) N0 i=0
2mef (U − kB T ) (U, T ) ≈ exp −2w h¯ 2
Consider now a fission track and its surroundings, initially in equilibrium at temperature T. The variation in N is given by the number of displaced atoms that succeeded in transmit through the potential barrier.
m−1
Suppose that each Ni is associated with an intermediary value of the barrier energy, Ui . If a new variable U can be defined to represent some association of Ui ’s, this new parameter can be associated with the Ni ’s. Thus, a transmission coefficient given in terms of a barrier energy and temperature can be used to represent the probability of displace atoms transmit through the potential barrier. In order to obtain an analytical expression, let the potential barrier be approximated by a square barrier and the approximate JWKB solution for the transmission coefficient (Sakurai, 1994, p. 449) be used.
2.2. Kinetic picture
(8)
i=0
U = U0 −
Aj [ln(t)]j .
(12)
j
In Eq. (12), t the duration of the annealing experiment (in seconds).The time t is a normalized time and the normalizing factor is set to 1 s. A better justification for Eq. (12) is provided
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
time, Eq. (14) would reduce to t N = exp − (t , T ) dt , N0 0
reduced mean track length, L/L0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
395
heating duration 1 hour 10 hours 100 hours 1,000 hours
0.0 250 300 350 400 450 500 550 600 650 temperature, T (K) Fig. 1. Results for the annealing of fission tracks in Durango apatite. The scatter plot represents the experimental data found in literature (Carlson et al., 1999) and the continuous line represents the model fit.
in the next topic. Then, it follows from Eqs. (11) and (12) that ⎧ ⎡ ⎛ ⎞1/2 ⎤⎫ ⎪ ⎪ ⎬ ⎨ L ⎥ ⎢ ⎝ j ⎠ = exp −n exp ⎣−w U0 − Aj [ln(t)] −kB T ⎦ . ⎪ ⎪ L0 ⎭ ⎩ j (13) As it will be shown in Section 3, Eq. (13) provides good fit to experimental results. 2.3. Further justification for the time variation of the barrier energy As mentioned before, it was preferred to construct a model based on the observations relating etched mean confined tracks and a few general observations on track structure. In this sense, Eq. (13) have been built up rather than demonstrated. The time dependence has been tacitly suppressed in Section 2.2, and then reintroduced as a relation for the dependence of the barrier energy on heating duration. Since the potential barrier is not constant in time, a more rigorous treatment would demand the self-consistent solution of the time dependent Schröndinger equation along with a semi-classical equation as dN = −(N, T )N, dt
(14)
in which is the frequency of collision of the displaced atoms on the barrier. Eq. (14) accounts for the dependence of the barrier energy on the amount of remaining vacant sites. This equation describes the two coupled mechanisms: (1) Insofar as the amount of vacant sites diminishes, the rate this happens also diminishes (dependence on N ); (2) As the amount of vacant sites diminishes, the energy barrier diminishes and the transmission coefficient is increased (dependence on (N, T )). The resulting rate of the amount of vacant sites variation is the interplay between these two mechanisms. If, as a first approximation, the potential were taken to be explicitly dependent on
(15)
which is similar in its general format to Eq. (7). Another matter is that the format for the spatial part of the barrier energy (square) was arbitrarily chosen. If, on one hand, it is a simplification, it has been chosen because its solutions are well understood. In addition, as the exact fission-track structure is unknown, there is no reason to use more complicated potentials. Finally, the transformation expressed in Eq. (2) assumes just that the fission track has cylindrical symmetry and that etching obeys rate laws. As it has been previously discussed, the first is justified because the track is formed around the fragment trajectory and the latter by the fact that most chemical reactions including mineral dissolution (Brady and House, 1996) follow rate laws. Then, it seems reasonable to make the same assumption for track etching. The discussion above makes possible to reanalyze the time variation of the barrier energy, given by Eq. (12), in terms of self-consistency. Suppose a simple format for the dependence of U on N: U = CN, in which C is a proportionality constant. In addition, let, as a rough approximation, the transmission coefficient be constant in Eq. (15), in such a way that U = CN0 exp{−t exp[−w (U − kB T )1/2 ]},
(16)
Eq. (16) explicitly relates the heating duration with the barrier energy. Solving this equation to isolate the variable U in the argument of the exponential 2 CN0 1 U = kB T + 2 ln(t) − ln ln . (17) U w The term ln(ln(CN0 /U )) will vary much slower than the term ln(t), so that it can be replaced (as a still rough approximation) by a constant B. Then, after some more algebra U = kB T + +
1 w 2
(B − ln())2 w 2
−
2(B − ln()) w 2
ln(t)
[ln(t)]2 .
(18)
Identifying the first and second terms with U0 , the third with A1 and the fourth with A2 , the general format of Eq. (12) is obtained. However, given that so many approximations were employed to construct Eq. (18), further interpretation should be avoided. 3. Comparison with experimental results 3.1. Evaluation of the model parameters The most complete and carefully carried out annealing experiment of neutron-induced fission tracks has been presented by
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interpretation to be fully applicable. The used index was the temperature of 100% fading in a 30 Ma annealing time, TF,30 , which for Durango-like apatite is in the range 95–100 ◦ C, for Otway Basin data. Following the interpretation by Ketcham and coworkers (1999), TF,30 , was calculated as being the temperature needed to reduce the mean track length to 0.41 (the lowest observed value for this quantity). In Fig. 1, it can also be observed that there is a threshold temperature (for each isochronal curve), T = U/kB , at which the reduction curve breaks. At temperatures higher than this, the displaced atom energy is higher than the barrier energy. In this way, displaced atoms can easily cross the barrier and the track vanishes very quickly.
mean barrier energy, U (meV)
57 56 55 54 53 52 51 50 49 48 47 1xe7
1xe9
1xe11 1xe13 ln(heating duration)
1xe15
Fig. 2. Time dependence of the mean energy barrier, U, on the duration of the heating experiments, t (given in s), for Durango apatite. The scatter plot shows the U values estimated by fitting individual (constant t) heating curves. The continuous line represents the second order polynomial in ln(t) used to obtain the model fit in Fig. 1.
Carlson and coworkers (1999). Etched track lengths have been measured for 15 compositionally diverse apatites. In that study, 1, 10, 100 and 1000 h isochronal heating experiments have been performed for various temperatures. Results for the Durango apatite (a widely used fission-track standard) are shown in Fig. 1 (scatter plot). It can be observed that, just as stated above, reduced track lengths below ∼ 0.4 are not found. Let Eq. (11) be fitted to each individual curve in Fig. 1. Fittings in this work were performed using the 2 minimization scheme available in the software Origin䉸 6.0. The parameter n, which depends on etching and geometrical properties of the fission tracks, is expected to be constant for a given sample. Supposing additionally that the parameter w is constant, all the time dependence is let to the energy of the barrier, U. Initially, the w , n, and U values are found for the 1 h curve. Then, fixing w and n, the U values are found for the other time duration curves. These values are plotted in Fig. 2 as a function of ln(t). Although these data can be well fitted by a linear curve (in ln(t)), the systematic behavior found for all the tested apatites suggests that it is more appropriated to use a second order polynomial. The general format of the U − ln(t) curves show the same upward concavity noted in Fig. 2. In this way, Eqs. (11) and (12) can be joined to give L = exp{−n exp[−w (U0 − A1 ln(t) L0 + A2 ln2 (t) − kB T )1/2 ]}.
(19)
Eq. (19) can then be used to fit the experimental data. In case of the Durango apatite, the obtained parameter values are n = 2.4408, w =34.4618 eV−1/2 , U0 =63.77 meV, A1 =1.12 meV, and A2 = 6.5 × 10−3 meV. The fit curve is shown in Fig. 1 (continuous line). The presented parameters were calibrated also to yield reasonable geological predictions. At this point the Otway Basin version is used, since the KTB data needs further
3.2. Correlation with the apatite structure Barbarand and coworkers (2003) obtained another data set of interest. It was idealized to complement the one discussed above. For 13 compositionally diverse apatites, curves for 10 h heating have been obtained. Curves for 100 and 1000 h were not so regularly measured. However, the principal advantage of this data set is that the unit-cell parameters a and c have been measured for 11 of these samples, making possible a discussion on the influence of the apatite structure in annealing. The authors have already shown that the cell parameters are the best control in fission-track annealing. Next, this will be analyzed from the standpoint of the presented model. Let, for sake of comparison, the value of n be kept constant as determined for the Durango apatite. Using Eq. (11), the curves for 10 h heating can be fitted to determine the values of w and U. The results are shown in Fig. 3, in which the width and the energy of the potential barrier are compared with the cell parameters of the studied apatites. No correlation is observed relating the barrier widths whose values are around 3.2 nm (Fig. 3c and d). On the other hand, clear correlations are noted relating the barrier energy (Fig. 3a and b), suggesting that this parameter is the principal model descriptor for annealing. The correlation found with the unit-cell parameter c is especially interesting because it is in c-axis that dislocations in ion positions due to changes in chemical composition take place (Calderín et al., 2003). Finally, let Fig. 4 be analyzed. In this figure, fitting U values are plotted as function of the heating times for the 10, 100 and 1000 h annealing curves for the different apatites. Some characteristic chlorine contents (in oxide weight percent) are shown near the curves as determined by Barbarand and coworkers (2003). Two features deserve to be pointed out. The first is that curves can be grouped by chlorine content. Higher the chlorine contents higher the energy barrier, agreeing with the early realization that higher chlorine content apatites are more resistant to annealing (Green et al., 1985). The second feature is that the barrier energies follow near parallel trends for all apatites indicating that the general mechanism controlling the annealing of fission tracks in apatite is not affected by chemical composition variation.
w´ for 10h heating (eV-1/2)
barrier energy for 10h heating (meV)
S. Guedes et al. / Radiation Measurements 41 (2006) 392 – 398
9.36 9.38 9.40 9.42 9.44 9.46 57
397
6.85 6.86 6.87 6.88 6.89 6.90 57
56
56
55
55
54
54
53
53
52
52
51
(b)
(a)
51
50 38
50 38
36
36
34
34
32
32
30
30
28 (c)
(d)
9.36 9.38 9.40 9.42 9.44 9.46 cell par ameter a (Å)
28
6.85 6.86 6.87 6.88 6.89 6.90 cell parameter c (Å)
barrier energy (meV)
Fig. 3. Correlation of the model parameters characterizing the potential barrier (w and energy) with the apatite unit-cell parameters as reported by Barbarand and coworkers (Barbarand et al., 2003) for compositionally diverse apatites.
58 57 56 55 54 53 52 51 50 49 48 47
4.2. Comment on the significance of L0
cl- content 1.54 to 4.67 0.71 to 0.83 0.02 to 0.39
< 0.031
1xe9
1xe10 1xe11 1xe12 1xe13 1xe14 ln (heating duration)
e15
Fig. 4. Time dependence of the mean energy barrier, U, for compositionally diverse apatites (Barbarand et al., 2003). Chlorine contents shown near curves are given in oxide weight percent. Straight lines connecting the points are just to distinguish the curves. Note that curves can be grouped by apatite chlorine content.
4. Discussion and conclusion 4.1. Comments on the model parameters After Eq. (19), there are five parameters to be considered. The most complicated to define is the parameter n, which contains the combined dependence on etching and track geometry. If the track could be considered a perfect cylinder, the parameter n would be the etching reaction order. Parameters w and U0 characterize the potential barrier, being its width and energy, respectively, and A1 and A2 control its variation. The barrier energy is the primary responsible for track stability, preventing atoms to immediately occupy the vacant sites.
Donelick and coworkers (1990), carried out an experiment in which tracks were induced in apatite samples by thermal neutron irradiation. Then, portions of these apatites were etched after different times. It was noted a progressive reduction of the etched fission-track lengths until around 3 weeks after irradiation. The authors suggest that reproducible measurements of the mean etchable fission-track lengths in apatite may be made on tracks that are aged for at least one month below the room temperature. In this way, the “initial” mean track length, L0 value is the one measured after at least one month after the track creation. Note however, that this initial reduction in the track length is a still transient process compared with the million-year timescale of the annealing phenomenon. 4.3. Final remarks A model for annealing of fission tracks in apatite has been built up based on the transmission of displaced atoms through an energy barrier and general observations on track properties, which succeeded to fit in experimental data. Particularly, the role of the barrier energy in the annealing process could be well characterized by correlations discussed in Section 3.2. However, due to the several simplifications employed, it is still semi-empirical because the parameter values must be found by data inspection. Acknowledgements The State of São Paulo Research Foundation (FAPESP) through a Post Doctoral grant, process number 01/02805-9, and a Thematic Project, process number 00/03960-5, supported this work.
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References Barbarand, J., Carter, A., Wood, I., Hurford, T., 2003. Compositional and structural control of fission-track annealing in apatite. Chem. Geol. 198, 107–137. Brady, P.V., House, W.A., 1996. Surface-controlled dissolution and growth of minerals. In: Brady, P.V. (Ed.), Physics and Chemistry of Mineral Surfaces. CRC Press, Boca Raton, pp. 225–305. Calderín, L., Stott, M.J., Rubio, A., 2003. Electronic and crystallographic structure of apatites. Phys. Rev. B 67, 134106. Carlson, W.D., 1990. Mechanisms and kinetics of apatite fission-track annealing. Am. Mineral. 75, 1120–1139. Carlson, W.D., 1993. Mechanisms and kinetics of fission-track annealing—replay. Am. Mineral. 78, 446–449. Carlson, W.D., Donelick, R.A., Ketcham, R.A., 1999. Variability of apatite fission-track annealing kinetics: I. Experimental results. Am. Mineral. 84, 1213–1223. Crowley, K.D., Cameron, M., Schaefer, R.L., 1991. Experimental studies of annealing of etched fission tracks in fluorapatite. Geochim. Cosmoch. Acta 55, 1449–1465. Dartyge, E., Duraud, J.P., Langevin, Y., Maurette, M., 1981. New model of nuclear particle tracks in dielectric minerals. Phys. Rev. B 23, 5213–5229. Donelick, R.A., Roden, M.K., Mooers, J.D., Stephen Carpenter, B., Miller, D.S., 1990. Etchable length reduction of induced fission tracks in apatite at room temperature (∼ 23◦ C): crystallographic orientation effects and “initial” mean lengths. Nucl. Tracks. Radiat. Meas. 17, 261–265. Fleischer, R.L., Price, P.B., Walker, R.M., 1975. Nuclear Tracks in Solids: Principles and Applications. University of California Press, Berkeley, 605pp. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1985. Fission-track annealing in apatite: track length measurements and the form of the Arrhenius plot. Nucl. Tracks Radiat. Meas. 10, 323–328. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1986. Thermal annealing of fission tracks in apatite 1. Qualitative description. Chem. Geol. (Isot. Geosci. Sect.) 59, 237–253. Green, P.F., Duddy, I.R., Laslett, G.M., Hegarty, K.A., Gleadow, A.J.W., Lovering, J.F., 1989. Thermal annealing of fission tracks in apatite 4. Quantitative modelling techniques and extension to geological timescales. Chem. Geol. (Isot. Geosci. Sect.) 79, 155–182. Green, P.F., Laslett, G.W., Duddy, I.R., 1993. Mechanisms and kinetics of fission-track annealing—discussion. Am. Mineral. 78, 441–445. Guedes, S., Hadler N, J.C., Iunes, P.J., Zuñiga, A., Tello S, C.A., Paulo, S.R., 2003a. The use of the U (n, f ) dosimetry in the determination of the f through fission-track techniques. Nucl. Instr. and Meth. A 496, 215–221.
Guedes, S., Hadler N, J.C., Sarkis, J.E.S., Oliveira, K.M.G., Kakazu, M.H., Iunes, P.J., Saiki, M., Tello S, C.A., Paulo, S.R., 2003b. Spontaneous-fission decay constant of 238 U measured by nuclear track techniques without neutron irradiation. J. Radioanal. Nucl. Chem. 258, 117–122. Guedes, S., Hadler N, J.C., Iunes, P.J., Tello S, C.A., 2004. Kinetic model for the relationship between confined fission-track length shortening and fission-track age reduction in minerals. Nucl. Instr. and Meth. B 217, 627–636. Guedes, S., Iunes, P.J., Hadler N, J.C., Bigazzi, G., Tello S, C.A., Alencar, I., Palissari, R., Curvo, E.A.C., Moreira, P.A.F.P., 2005. Kinetic model for the relationship between mean diameter shortening and age reduction in glass samples. Radiat. Meas. 39, 647–652. Holden, N.E., Hoffman, D.C., 2000. Spontaneous fission half-lives ground-state nuclides (Technical report). Pure Appl. Chem. 72, 1525–1562. Jaskierowics, G., Dunlop, A., Jonckheere, R., 2004. Track formation in fluorapatite irradiated with energetic cluster ions. Nucl. Instr. and Meth. B 222, 213–227. Jonckheere, R., 2003. On methodical problems in estimating geological temperature and time from measurements of fission tracks in apatite. Radiat. Meas. 36, 43–55. Ketcham, R.A., Donelick, R.A., Carlson, W.D., 1999. Variability of fissiontrack annealing kinetics: III. Extrapolation to geological time scales. Am. Mineral. 84, 1235–1255. Laslett, G.M., Galbraith, R.F., 1996. Statistical modeling of thermal annealing of fission tracks in apatite. Geochim. Cosmochim. Acta 60, 5117–5131. Laslett, G.M., Green, P.F., Duddy, I.R., Gleadow, A.J.W., 1987. Thermal annealing of fission tracks in apatite. 2. A quantitative analysis. Chem. Geol. (Isot. Geosci. Sect.) 65, 1–13. Paul, T.A., 1993. Transmission electron microscopy investigation of unetched fission tracks in fluorapatite-physical process of annealing. Nucl. Tracks Radiat. Meas. 21, 507–511. Paul, T.A., Fitzgerald, P.G., 1992. Transmission electron microscopic investigation of fission tracks in fluorapatite. Am. Mineral. 77, 336–344. Sakurai, J.J., 1994. Modern Quantum Mechanics. Addison-Wesley, Menlo Park, CA, 500pp. Sood, N.K., Singh, T., Sandhu, A.S., Singh, L., 2000. A global model for annealing in solid state track detectors. Radiat. Meas. 32, 79–82. Yada, K., Tanji, T., Sunagawa, I., 1987. Radiation induced lattice defects in natural zircon (ZrSiO4 ) observed at atomic resolution. Phys. Chem. Minerals 14, 197–204.