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On the efficiency of fission-track counts in an internal and external apatite surface and in a muscovite external detector
Radiation Measurements 35 (2002) 29–40
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On the e ciency of "ssion-track counts in an internal and external apatite surface and in a muscovite external detector R. Jonckheerea; ∗ , P. Van den hauteb a Max-Planck-Institut b Geologisch
fur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Instituut, Universiteit Gent, Krijgslaan 281, B-9000 Gent, Belgium
Received 9 August 2000; received in revised form 15 June 2001; accepted 23 June 2001
Abstract The frequency distributions per unit area of the projected lengths (p-distributions) of "ssion tracks intersecting an internal and external apatite surface and the surface of a muscovite external detector have been established by measurement. Deviations from the ideal distributions on a number of points can be tied to the e4ects of track revelation and observation. The e4ect of track revelation, in particular, masks the e4ect of variations of true track length, and precludes temperature–time path modelling based on length measurements of surface tracks. These e4ects do not prevent calculation of the track counting e ciencies (q) in the track registration geometries of interest to "ssion-track analysis: q = 1:01 ± 0:01 for induced tracks revealed in an external surface of Durango apatite, q = 0:91 ± 0:01 for both fossil and induced tracks revealed in an internal surface of Durango apatite, and q = 0:91 ± 0:01 for induced tracks revealed in a muscovite external detector. The fact that the latter are signi"cantly less than unity is not due to an etching e4ect (critical angle c ) but to an observation threshold, best described by a critical depth zc . For tracks revealed in an internal surface, q decreases rapidly with decreasing track length. As a result, the apparent age of strongly annealed apatite samples may be underestimated by as much as 5%, irrespective of whether the c 2002 Elsevier Science Ltd. All rights reserved. absolute method, the Z-method, or the -method is used for dating. Keywords: Fission-track; Apatite; Muscovite; External detector
1. Introduction Not all the "ssion tracks intersecting the surface of a track detector can be identi"ed under an optical microscope. The fraction of tracks that is counted has been represented by q, wherein stands for etching e ciency and q for observation e ciency. It has been shown that and q cannot be separated and that their combined e4ect, q, cannot, at present, be calculated on the basis of existing models of track revelation and observation (Jonckheere and Van den haute, 1996); q must therefore be determined by experiment. It ∗ Corresponding author. Tel.: +49-6221-516-337; fax: +496221-516-633. E-mail address: [email protected] (R. Jonckheere).
follows from its de"nition that q is the ratio of the experimental ( m ; measured) to the theoretical ( t ; true) density of etchable "ssion tracks intersecting the detector surface. The de"nition of the experimental track density is operational and unambiguous: the result of a track density measurement with the aid of an optical microscope. The theoretical track density is that de/ned by t = 1=2glt N , wherein g is the geometry factor (g = 1=2 for track revealed in an external surface and external detector and g = 1 for track revealed in an internal surface), lt is the mean true etchable track length (Laslett et al., 1982), and N is the volumetric track density. Thus, t includes tracks that do not produce an etch "gure at all (critical angle) as well as tracks that give rise to an etch "gure that is not su ciently distinctive to be identi"ed as a track under the microscope (observation threshold).
c 2002 Elsevier Science Ltd. All rights reserved. 1350-4487/02/$ - see front matter PII: S 1 3 5 0 - 4 4 8 7 ( 0 1 ) 0 0 2 6 2 - 1
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Contrary to the experimental track density, estimating the theoretical track density is not straightforward. Gold et al. (1968), Wall (1970, 1986), Khan and Durrani (1972) and Roberts et al. (1984) have determined the e ciency of "ssion-track counts in muscovite detectors by measuring the number of tracks produced by a calibrated "ssion source, and calculating their true number from its activity. In these experiments, the material for which q is determined must function as an external detector with respect to the source, and the source itself is deposited as an ultra-thin "lm on the surface of the detector, to ensure that a negligible number of "ssion fragments are stopped within it. The track registration geometry is thus di4erent from that encountered in routine "ssion-track counts, where the tracks are either revealed in an internal or external mineral surface or in the surface of an external detector, irradiated in contact with a thick "ssion source. The q values obtained using calibrated "ssion sources are thus not applicable to the conditions of routine "ssion-track analysis. Jonckheere and Van den haute (1998, 1999) have proposed that it is possible to estimate the true as well as the experimental track density from the frequency distributions of projected track lengths (p-distributions) of surface-intersecting tracks. The "rst is calculated by integrating its equation, provided that it can be "tted to the data with con"dence; the second is calculated from the number of tracks in the distribution. Our present aim is to show that this provides more reliable estimates of q in each of the three geometries of interest to "ssion-track analysis than have been hitherto available. In addition, q is directly determined for the track registration geometries used in "ssion-track analysis and for the same etching and observation conditions as those of routine "ssion-track counts. With suitable software, such as that developed for this study, it is even possible to determine q (and the true track density) for each sample, simultaneously with the "ssion-track counts. 2. Experiments The "rst experiment was aimed at establishing the p-distributions of induced "ssion tracks in the three track registration geometries of interest: an internal and external apatite surface and a muscovite external detector. To this end, two sections were cut from separate cm-sized crystals of Durango apatite. The "rst section was cut parallel and the second perpendicular to the c-axis; their crystallographic indices are, respectively, {1 1 2I 0} and {0 0 0 1}. Both sections were annealed for 24 h at 450◦ C to erase the fossil tracks, and mounted in an epoxy resin. The mounts were ground on 600-, 800-, and 1000-mesh corundum powders and polished on 6-, 3-, and 1-m diamond paste. The polished mounts were covered with muscovite external detectors and irradiated in channel 8 of the Thetis research reactor at the Institute for Nuclear Sciences in
Gent (Belgium), to produce induced "ssion tracks. The thermal neutron Kuence was measured with Au and Co metal-activation monitors (Van den haute et al., 1988), and amounted to [1:31 ± 0:03] × 1015 cm−2 . The induced tracks in the apatite sections were etched for 60 s in 2.5% HNO3 at 25◦ C; the induced tracks in the muscovite external detectors were etched in 40% HF for 20 min, also at 25◦ C. At this point, the registration geometry for the induced tracks in apatite is that of an external surface, because the sections were ground and polished before the neutron irradiation. In order to obtain the p-distributions of induced tracks in an internal surface, the two sections were re-polished and re-etched, using the same procedure as before, after the length measurements for the external surface had been completed. The measurements were performed with an Olympus BH-2 microscope equipped with a drawing tube, that projects a point-like light source, mounted on the cursor of a Kontron MOP-3 digitiser, positioned next to the microscope, onto the microscope image. The co-ordinates of the track extremities, indicated with the cursor, were sent on-line to an Apple computer, where a dedicated program calculated the track densities, lengths and orientations. The microscopic analyses were carried out in transmitted light, using a 100 × dry objective and 10 × oculars. The true overall magni"cation of the microscope includes a drawing tube factor of 1:25×, and amounted to 1274×, as calibrated against a certi"ed stage micrometer. The p-distributions of "ssion tracks revealed in the internal apatite surface are shown in Fig. 1a (basal section) and 1b (prism section), those of tracks revealed in the external surface in Fig. 1c (basal section) and 1d (prism section), and those of tracks revealed in the surface of the muscovite external detectors in Fig. 1e and f. The second experiment was aimed at measuring the p-distributions of fossil tracks in the internal surfaces of apatite with di4erent etching characteristics. To this end, ∼1 mm thick sections were cut parallel to the {0 0 0 1}, {1 0 1I 0}; {1 1 2I 0}; {1 0 1I 1} and {1 1 2I 1}-planes of a separate crystal of Durango apatite. These sections divide into three groups with respect to their etching characteristics: pitted surfaces ({0 0 0 1} and {1 1 2I 1}), scratched surfaces ({1 1 2I 0}), and textured surfaces ({1 0 1I 0} and {1 0 1I 1}) (Jonckheere and Van den haute, 1996). They were neither annealed nor irradiated, and thus retained their original fossil tracks. They were mounted in epoxy resin, ground and polished as described. The fossil tracks were etched for 60 s in 2.5% HNO3 at 25◦ C, and counted and measured in the same manner as for the sections with induced tracks. The p-distributions are shown in Fig. 2a–e. For evaluating the methods for calculating q discussed here, it is important that the distribution of the true etchable lengths of fossil and induced tracks is known. To this end, the lengths of horizontal con"ned track were measured in the prismatic faces after an additional etching step of 120 s. The frequency distributions of con"ned track length are length-biased (Laslett et al., 1982) and thus not
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
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Fig. 1. The p-distribution of induced "ssion tracks in internal (a, b) and external (c, d) surfaces of Durango apatite and in co-irradiated muscovite external detectors (e, f).
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Fig. 2. The p-distribution of spontaneous "ssion tracks in pitted (a, b), scratched (c) and textured (d, e) internal surfaces of Durango apatite.
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 3. Polar plots and length distributions of horizontal con"ned fossil (a) and induced (b) "ssion tracks measured in prismatic surfaces of Durango apatite.
exactly identical to the frequency distributions of true etchable length, but, for the present purposes, this bias can be neglected in the samples studied here (Jonckheere, 1995). The polar plots and length distributions of fossil and induced con"ned tracks are shown in Fig. 3a and b. The mean length of fossil con"ned tracks is 14:4 ± 0:1 m and that of induced tracks is 16:3 ± 0:1 m. The length of induced tracks does not depend on their azimuth angle to the c-axis, although Green et al. (1986) do report a slight but unquanti"ed anisotropy of induced track length in Durango apatite and Donelick (1991) reports a di4erence of 3% between the length of induced tracks parallel and perpendicular to the
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c-axis in the Tioga apatite. The maximum length di4erence due to anisotropy of the fossil tracks amounts to 5 ± 2%. These anisotropies are thus small enough not to have a signi"cant e4ect on the p-distributions (Jonckheere and Van den haute, 1999). The con"ned track length distributions have standard deviations of 1:0 m, con"rming that both the fossil and induced tracks in Durango apatite possess a range of true etchable lengths (Fig. 3). It is well known that this produces a tail to the p-distribution of tracks revealed in an internal surface (inverted J-shape; Laslett et al., 1982, 1994; Jonckheere and Van den haute, 1999). However, this only a4ects the interval of projected lengths bracketed by the minimum (lt; min ) and maximum (lt; max ) true etchable length: [lt; min = p = lt; max ]. The remaining part of the p-distribution [0 6 p 6 lt; min ] remains linear and entirely determined by the mean true etchable length and the true volumetric track density (Jonckheere and Van den haute, 1999). For both fossil and induced tracks in Durango apatite, the coe cient of variation of the con"ned track length distribution is less than 0.10 (fossil tracks: 1 m=14:4 m = 0:07; induced tracks: 1 m=16:3 m = 0:06). It has been shown that in this case the tail is insigni"cant (Jonckheere and Van den haute, 1999). It has also been shown that track revelation has a more important e4ect on the p-distribution, opposite to that due to the range of etchable track lengths. As a consequence, the p-distributions of track revealed in an internal surface never show the inverted J-shape, but, instead, a track de"cit at p-values approaching the mean true etchable track length (Jonckheere and Van den haute, 1999). In the two methods for calculating q proposed here, this de"cit is either accounted for or shown to have a negligible e4ect (Section 3). In conclusion, the tracks in the studied samples do possess a range of true etchable lengths, but this has no e4ect on the experimental determination of the q-values, and the results can be interpreted to be valid for an idealised population of "ssion tracks with a constant etchable length identical to the mean true etchable length in the samples. This is important because it allows to apply the present results to more complex length distributions in geological samples. 3. Internal surface The p-distributions of "ssion tracks intersecting an internal mineral surface (Fig. 1a and b: induced tracks; Fig. 2: fossil tracks) are in fair but not in perfect agreement with the theoretical “triangular” distribution (Dakowski, 1978; Laslett et al., 1994; Laslett and Galbraith, 1996; Jonckheere and Van den haute, 1998, 1999). In this respect, the distributions divide into three length intervals: (1) 0 –3 m, in which there is a marked track de"cit compared to the theoretical distribution; (2) 3–13 m, in which the distributions show a linear decreasing trend, in agreement with the theoretical distribution; and (3) the length interval ¿ 13 m, which again shows a distinct track de"cit.
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R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
The deviation from the theoretical distribution in the interval 0 –3 m can result from limited precision of the length measurements or from an observation threshold. Limited precision without loss of tracks would, however, result in a surplus of tracks with lengths just in excess of 3 m (Jonckheere and Van den haute, 1999; Fig. 10a). Given that this is not the case, the track de"cit below 3 m is interpreted as a real loss due to an observation threshold. By itself, the trend in the interval 0 –3 m is not characteristic enough to identify this threshold with con"dence. The fact that a part of the tracks with projected lengths up to 3 m is not observed suggests an observation threshold described by a critical depth zc , because values of zc ¡ 1 m can produce a signi"cant loss of tracks with projected lengths up to 3 m, whereas higher threshold values are required if the observation threshold is described by a critical projected length (pc ) or a critical etchable length (ec ) (Jonckheere and Van den haute, 1999; Figs. 6a, e and i and corresponding equations in Table 1). An observation threshold also a4ects the frequency distribution per unit area of the etchable length (e-distribution), depth (z-distribution) and inclination (-distribution) of surface-intersecting "ssion tracks; e-, z- and -distributions of induced and fossil tracks in apatite have been published by Dakowski (1978) and Al-Khalifa and Major (1987). A comparison with the corresponding theoretical distributions (Jonckheere and Van den haute, 1999) shows that: (1) as in the p-distribution, there are less than the expected number of tracks with small values of e and z (¡ 3 m); (2) the monotonous increase in the interval 0 –3 m is more abrupt in the z-distribution and more gradual in the p- and e-distributions; (3) the maximum of the -distribution is shifted to the right compared with the theoretical distribution, i.e. to -values ¿ =4. It follows from these observations that (1) there is no agreement at all between the experimental and theoretical distributions for a threshold described by ¿ c (critical angle) and (2) there is better agreement between the experimental and theoretical distributions for the threshold criterion z ¿ zc (critical depth) than either for the criterion p ¿ pc (critical projected length) or e ¿ ec (critical etchable length). The agreement between the experimental and theoretical distributions is less than perfect, but, considering the di culty of measuring short tracks precisely and earlier theoretical arguments (Jonckheere and Van den haute, 1999), the evidence is su cient to conclude that "ssion tracks are observed under an optical microscope on condition that their depth z below the detector surface exceeds a critical value zc , and that they escape observation if z ¡ zc . A pit in a perfect surface can indeed only be observed with an optical microscope if its depth is greater than half the wavelength of the light source. For visible light with a wavelength of ∼400 nm, this implies a minimum value of zc of ∼0:2 m. The value of zc derived from "tting the equation of the theoretical distribution (Jonckheere and Van den haute, 1999) to the p-distributions in Figs. 1a, b and 2 amounts to ∼0:8 m.
This di4erence is not excessive when one considers that an etched apatite surface is not perfectly smooth and that, in order to identify and count it, the analyst must not only see the track but must also be able to distinguish it from other features. The interval 3–13 m of the p-distributions is quasi-linear. If una4ected by the e4ects of track revelation, observation and measurement, extrapolation of this interval allows to estimate the mean true etchable length (lt ) of the tracks (Jonckheere and Van den haute, 1999). If the result agrees with the mean length of con"ned tracks (lc ), it is reasonable to conclude that the central interval of the p-distributions is indeed not a4ected by these factors, and that linear extrapolation also provides an accurate estimate of the true track density ( t ). This extrapolation was performed by "tting a regression line to the 10 points in Figs. 1a, b and 2, one for each 1 m interval between 3 and 13 m. If the regression equation is written as N (p) = A − Bp, then lr , i.e. the intersection of the regression line with the p-axis, is given by lr = A=B (the subscript r refers to the regression method). The true track density r is obtained by integrating the regression equation between 0 and lr : r = A2 =2BS, wherein S is the counted surface area. The correlation coe cients of the regression lines (r), and the values of lr and r are shown in Tables 1 and 2. The r-values do not di4er much from −1, indicating a good "t. The lr =lc -ratio depends on the type of surface. For pitted surfaces it is equal to 1 (mean: 0:997 ± 0:010). For scratched surfaces lr =lc it is somewhat less, but still close to 1 (mean: 0:980 ± 0:020). The lower value for the textured surfaces (mean: 0:855 ± 0:045) indicates a signi"cant underestimation of the true track length (∼2 m). This di4erence must result from errors in indicating the track intersection with the surface. Thus calculating
r from the regression equation is warranted for pitted and scratched surfaces, but not for textured surfaces. The interval ¿ 13 m of the p-distributions for the internal surface is characterised by a small track de"cit. Calculation shows that several factors related to the range of true etchable lengths (Laslett et al., 1982, 1994; Laslett and Galbraith, 1996) and to track revelation, observation and measurement (Jonckheere and Van den haute, 1999) a4ect this length interval. Surface etching is, however, the only factor that produces a track de"cit. Imprecision of the length measurements, statistical length variation and anisotropic length produce a convex trend that entails a track surplus in the interval ¿ 13 m (Jonckheere and Van den haute, 1999). Since tracks with p ¿ 13 m constitute a small fraction (2–3%) of the tracks in the p-distribution, the actual de"cit has but a minor e4ect on the mean projected track length. The de"cit of tracks with p ¡ 3 m, on the other hand, has a signi"cant e4ect. The true etchable track length (la ; the subscript a refers to the second method for estimating the mean true length) can thus be obtained by choosing an arbitrary value a ¿ 3 m, and calculating the mean projected length (ma ) of tracks longer than a: ma = (la + 2a)=3 (Fig. 4). This provides a more precise if somewhat less accurate method
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
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Table 1 Measured and theoretical areal densities and lengths of induced "ssion tracks revealed in an internal and external apatite surface and in the surface of a co-irradiated muscovite external detector and calculation of the track counting e ciencies (q)a Orientation Surface
N
n
m
r
lr
r
a
la
la =lc
lr =lc [q]a
[q]r
Internal surface (apatite) {0 0 0 1} P 4265 219 3:16 ± 0:05 3.51 16.5 −0:996 3:50 ± 0:07 15:9 ± 0:2 0:98 ± 0:01 1.01 {1 1 2I 0} S 4610 190 3:94 ± 0:06 4.34 16.3 −0:995 4:32 ± 0:08 15:6 ± 0:2 0:96 ± 0:01 1.00
0:90 ± 0:02 0.90 0:91 ± 0:02 0.91
External surface (apatite) {0 0 0 1} P 4646 420 1:80 ± 0:03 1.76 16.5 −0:996 1:75 ± 0:04 15:9 ± 0:2 0:98 ± 0:01 1.01 {1 1 2I 0} S 2907 219 2:15 ± 0:04 2.17 16.3 −0:995 2:16 ± 0:04 15:6 ± 0:2 0:96 ± 0:01 1.00
1:03 ± 0:02 1.03 1:00 ± 0:03 0.99
External detector (muscovite) {0 0 0 1} P 2990 229 1:90 ± 0:03 2.07 10.0 −0:997 2:09 ± 0:06 10:5 ± 0:1 0:51 ± 0:01 0.49 {1 1 2I 0} S 3860 237 2:37 ± 0:04 2.62 10.0 −0:992 2:65 ± 0:07 10:5 ± 0:1 0:51 ± 0:01 0.49
0:91 ± 0:03 0.92 0:90 ± 0:03 0.90
a P—pitted surface; S—scratched surface; N —number of tracks; n—number of unit "elds counted (1 unit "eld = 6:16 × 10−5 cm 2 for the internal and external surface, and 6:87 × 10−5 cm2 for the external detector); m (105 cm−2 )—measured track density; r (105 cm−2 )— theoretical track density calculated with the regression method; lr (m)—track length calculated with the regression method; r—correlation coe cient of the regression line to the projected track length distribution; a (105 cm−2 )—track density calculated on the basis of the number of tracks with projected length. 3 m; la (m)—track length calculated on the basis of the mean projected length of tracks longer than 3 m; [q]a —track counting e ciency calculated on the basis of tracks with projected length. 3 m; [q]r —track counting e ciency calculated with the regression method. All errors are 1; no errors are given for the regression method because the non-homoscedasticity of the projected track length distribution prevents their simple calculation.
Table 2 Measured and theoretical areal densities and lengths of fossil "ssion tracks revealed in internal pitted, scratched and textured surfaces of apatite, and calculation of the corresponding track counting e ciencies (q)a Orientation N
n
m
r
lr
r
a
la
la =lc
lr =lc
[q]a
[q]r
Pitted surfaces {0 0 0 1} 2611 {1 1 2I 1} 1344
201 105
2:21 ± 0:05 2:08 ± 0:05
2.31 2.29
14.3 14.2
−0:994 −0:983
2:32 ± 0:06 2:30 ± 0:08
13:9 ± 0:2 14:0 ± 0:2
0:97 ± 0:02 0:97 ± 0:02
0.99 0.99
0.91 ± 0.03 0:91 ± 0:04
0.91 0.91
Scratched surfaces {1 1 2I 0} 2413 200
1:96 ± 0:04
2.10
14.0
−0:996
2:10 ± 0:05
13:8 ± 0:2
0:96 ± 0:02
0.93
0:93 ± 0:03
0.93
Textured surfaces {1 0 1I 0} 1279 {1 0 1I 1} 1095
2:08 ± 0:06 1.80 ± 0.06
2.31 2.21
12.7 12.6
−0:982 −0:991
2:32 ± 0:08 2:22 ± 0:08
12:5 ± 0:2 12:4 ± 0:2
0:87 ± 0:02 0:86 ± 0:02
0.90 0.81
0:90 ± 0:04 0:81 ± 0:04
0.90 0.81
a Symbols
100 99
have the same meaning as in Table 1; m ; r and a are in 105 tracks=cm2 ; lr and la in m; errors are 1.
for calculating both the true etchable track length (la ) and true track density ( a ), since (Fig. 4): la = 3ma − 2a;
(1)
a = [la =[la − a]]2 a ;
(2)
wherein a is the measured density of tracks longer than 3 m. Tables 1 and 2 show the results for a = 3 m. As expected, la is somewhat less than lr . The di4erence is more marked for the induced tracks because their p-distributions show a larger de"cit of tracks ¿ 13 m. For the fossil tracks, the di4erence between la and lr is less than 1%. The la =lc -ratios con"rm the conclusions based on the lr =lc -ratios, in particular, that the track length can be derived from mea-
surements of p in pitted and scratched-surfaces, but that these lead to an underestimation in textured surfaces. 4. External surface The p-distribution of induced tracks in an external apatite surface (Fig. 1c and d) agrees well with the theoretical distribution (Jonckheere and Van den haute, 1998). For p ¿ lr =2, the latter is derived from the regression line to the p-distribution for the internal surface, since both p-distributions are identical in this length interval. In the interval p ¡ lr =2, the theoretical distribution is obtained by mirroring the regression line about lr =2, which value is itself
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Fig. 4. Schematic diagram of the p-distribution of tracks revealed in an internal surface, indicating a; ma ; la , a and a (see text), and showing the e4ects of an observation threshold between A and E and of track revelation at D.
derived from the regression line. There is no track de"cit in the interval 0 –3 m, consistent with an observation threshold described by z ¿ zc . As for the internal surface, the track de"cit in the interval ¿ 13 m is the result of surface etching. The maximum at lr =2 is not sharp: there is a small but systematic surplus of tracks with projected lengths just short of lr =2. Two factors that can account for this (Jonckheere and Van den haute, 1999) are (1) surface etching, and (2) an o4-centre and variable locus of the "ssioned uranium atom with respect to the track end-points (track asymmetry). The distortion of the maximum at lr =2 is not more pronounced in the p-distribution of tracks revealed in the prismatic face (Fig. 1d) than in that of tracks revealed in the basal face (Fig. 1c), despite the fact that the etch rate of the surface in the former exceeds that in the latter by almost a factor of 5 (Jonckheere and Van den haute, 1996). This implies that track asymmetry is the more important factor. The mean projected track length (lm ), calculated from the distributions in Fig. 1c and d, is 8:0 ± 0:1 m. In "rst approximation, lm is half the mean true etchable track length. The ratio lm =lc (0:49 ± 0:01) is close to this value. The small di4erence results from the track de"cit in the interval ¿ 13 m and to a lesser extent from the track surplus just below lm . 5. External detector The p-distributions of induced tracks revealed in a muscovite external detector are shown in Fig. 1e and f. These were summed with 12 more identical distributions; the result is a distribution with just short of 60.000 tracks, shown in Fig. 5. Like the p-distribution of tracks revealed in internal surface, it divides into three length intervals: (1)
Fig. 5. The p-distribution of induced "ssion tracks revealed in 16 muscovite external detectors irradiated against Durango apatite.
0 –2 m, in which there is a marked track de"cit compared to the theoretical distribution (Jonckheere and Van den haute, 1998), (2) 2–9 m, in which the distribution shows a linear decreasing trend in agreement with the theoretical distribution and (3) the length interval ¿ 9 m, in which, in this case, the distribution is markedly convex. As in the internal surface, the track de"cit in the interval 0 –2 m is the result of an observation threshold described by z ¿ zc . Fitting the theoretical equation (Jonckheere and Van den haute, 1999) to the graph in Fig. 5 gives a "rst-order estimate of zc : zc ≈ 0:5 m. This agrees with the general experience that "ssion tracks in muscovite are easier to identify and count than tracks in apatite. The etched muscovite surface is, however, not free of other defects such as alpha-recoil tracks, so that, even in this case, the threshold for track identi"cation is higher than the optical limit (0:2 m). A regression line was "tted to the linear interval 2–9 m; lr and
r were calculated from the regression equation; la and a were also calculated from the number and mean projected length of tracks longer than 2 m. All are shown in Table 1. The la -values are systematically somewhat higher than the lr -values as a result of the surplus of tracks in the interval ¿ 9 m. On the other hand, both la and lr are close to half the length of con"ned tracks in muscovite (20:5 ± 0:1 m; Bigazzi, 1967). We conclude that the interval 2–9 m is not a4ected by factors related to track revelation, observation and measurement and that r and a are accurate estimates of the actual track density. Numerical modelling has shown that the convex trend in the length interval ¿ 9 m is the result of the o4-centre and variable position of the "ssioned uranium nucleus with respect to the track extremities (track asymmetry; Jonckheere and Van den haute, 1999).
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
6. q-Factor The e ciency of "ssion-track counts under a microscope (q) is de"ned as the ratio of the measured ( m ) to the true ( t ) areal densities of tracks intersecting the detector surface. As described, m was estimated by summing the tracks in the p-distributions for each of the three track registration geometries relevant to "ssion-track analysis. For the internal surface and external detector, t was estimated in two ways: (1) from the regression line to the linear part of the p-distribution ( r ), and (2) from the mean projected length of tracks longer than an appropriate limit ( a ). For the external surface, neither r nor a could be determined directly from the p-distribution. They were, instead, derived from the r and a -values for the internal surface, which are twice that for the external surface. This gives two independent estimates of q for the three registration geometries: [q]r = m = r and [q]a = m = a , reported in Table 1 for induced tracks revealed in the three track registration geometries and in Table 2 for fossil tracks revealed in pitted, scratched and textured internal surfaces. The notion of a critical angle c = arcsin[vb =vt ] (vb = etching velocity perpendicular to the etched surface; vt = etching velocity along the "ssion track) implies that q is the same for an internal and external surface with the same crystallographic orientation (q = 1 − sin2 c ; Jonckheere and Van den haute, 1999), but di4erent for internal surfaces with di4erent crystallographic orientation (di4erent vb ). The experimental results demonstrate the exact opposite: Table 1 shows that q is much lower for tracks revealed in an internal surface (0:91 ± 0:01) than for tracks revealed in an external surface (1:01 ± 0:01) of Durango apatite. On the other hand, there is no signi"cant di4erence between the q values for tracks revealed in pitted and scratched internal surfaces (textured surfaces are discussed below). This proves that q is not in the "rst place a mineral=etchant property. It is thus meaningless to talk about “the etching e ciency of a mineral” in connection with the accuracy of "ssion-track age determinations. The results in Table 1 also show that q is the same for internal surfaces of apatite and for a muscovite external detector (0:91 ± 0:01). This is no doubt a coincidence, but it con"rms again that q is unrelated to the etching velocities of the surface, which di4er two orders of magnitude for apatite and muscovite under the etching conditions used here (Blok et al., 1972; Jonckheere and Van den haute, 1996). The notion of a critical depth for track identi"cation implies that q varies as q = 1 − [zc =R] + 1=4[zc =R]2 for the internal surface and as q = 1 − 1=2[zc =R]2 for the external surface, wherein zc is the critical depth for track observation and 2R the true etchable track length (Jonckheere and Van den haute, 1999). For an induced track length of 16:3 m and a critical depth of 0:8 m, the calculated q-values are q = 0:91 for the internal surface and 1.00 for the external surface, in agreement with the values in Table 1. The value zc = 0:8 m corresponds to that obtained by "tting the the-
37
oretical equations to the distributions in Fig. 1a and b, with zc as a free parameter (Section 3). For the external detector, q varies as q = 1 − 2[zc =R] + [zc =R]2 (Jonckheere and Van den haute, 1999). For 2R = 20:5 m (Bigazzi, 1967), and zc = 0:5 m (Section 5), we calculate that q = 0:91, in agreement with the experimental value. The fact that zc is less than the value for apatite agrees with the general experience that "ssion tracks are easier to identify in muscovite than in apatite. That this lesser value nevertheless causes the same loss of tracks as in the internal apatite surface is due to the fact that, in "rst approximation, the etchable length of surface-intersecting tracks in an external detector varies from 0 to R, and that in an internal surface it varies from 0 to 2R. There are thus proportionally more tracks with depths less than a given value. The same reasoning also makes it plain that q must be close to unity for tracks revealed in an external apatite surface. In this geometry, the etchable length of surface-intersecting tracks ranges from R to 2R. Thus only part of the tracks at an angle to the surface less than arcsin [zc =R] can escape observation. For zc = 0:8 m and R = 8 m, arcsin [zc =R] is small, and the loss of tracks due to an observation threshold is minimal. As described above, the exact value of q is: q = 1 − 1=2[zc =R]2 = 0:995: In the above formulae, R represents the true etchable track length. The dependence of q on etchable track length implies that q is less for partially annealed (shortened) tracks than for unannealed tracks. In particular, it is expected that the q-values for fossil (mean length: 14:4 ± 0:1 m) and induced tracks (mean length: 16:3 ± 0:1 m) in an internal apatite surface will also be di4erent. The results in Table 1 (induced tracks) and Table 2 (fossil tracks) show that this is not the case. Fig. 6 shows how q varies with true etchable track length for the internal surface, assuming zc = 0:8 m. It is seen that q varies little as a function of track length between 14.4 and 16:3 m, explaining the identical q-values for the fossil and induced tracks. However, the fact that q drops o4 rapidly at shorter lengths has important implications for "ssion-track dating. It implies that the fossil tracks in a strongly annealed sample are counted with less e ciency than either the induced tracks in an annealed and irradiated aliquot of the same sample or the fossil tracks in a set of age standards. This means that the "ssion track age of such a sample, whether it is dated with the absolute method, the Z- or the -method, will be underestimated by an amount exceeding that resulting from the reduction of track length alone. The exact age underestimation will depend on the track length distribution. In speci"c cases where the track length distribution is known from con"ned track length measurements, this amount can be calculated by integrating q over the range of lengths in the sample. The q-values for the internal surface in Table 1 (induced tracks) and Table 2 (fossil tracks) reveal no signi"cant di4erence between scratched and pitted surfaces, despite the fact that the etch rate of the surface, ranges from 0:08 ± 0:01 m=min to 0:37 ± 0:01 m=min (Jonckheere and Van den haute, 1996). The results for the
38
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 6. The track counting e ciency (q) in an internal surface of apatite as a function of true etchable track length for an observation threshold described by a critical depth zc ≈ 0:8 m.
textured surfaces are less straightforward. The fact that the la =lc and lr =lc -ratios are less than 1 indicates that their q-factors are not reliable. On the other hand, the q-value for {1 0 1I 0} (0:90±0:04) is identical to that of the scratched and pitted surfaces and the measured track density (2:08 ± 0:06 × 105 cm−2 ) is close to that for {0 0 0 1}; {1 1 2I 0} and {1 1 2I 1} (mean: 2:08 ± 0:07 × 105 cm2 ). The track density in {1 0 1I 1} (1:80 ± 0:06 × 105 cm−2 ) is much lower and so is its q-factor (0:81 ± 0:04). Although the etch rates perpendicular to {1 0 1I 0} (0:64 ± 0:01 m=min) and {1 0 1I 1} (0:61 ± 0:01 m=min) are similar, {1 0 1I 0} is characterised by a "ne and {1 0 1I 1} by a coarse texture (Jonckheere and Van den haute, 1996). The lower track density for {1 0 1I 1} thus probably results from the greater di culty of identifying short tracks against a more irregular background. The fact that this is reKected in a proportionally lower q-value seems to suggest that the methods used for calculating q are fairly robust, and still give a reasonable value even when the projected track lengths cannot be measured accurately. Perhaps the most important result is that q is much less than unity for internal surfaces of apatite (mean for scratched and pitted surfaces: q = 0:91 ± 0:01) and for muscovite external detectors (q = 0:91 ± 0:01), in direct contradic-
tion with earlier estimates based on the etching characteristics (etch rates) of these surfaces (Fleischer et al., 1975; Gleadow, 1978, 1981; Hurford and Green, 1983). We have argued before at length that the etching models on which these estimates are based are inappropriate for describing track revelation in minerals (Jonckheere and Van den haute, 1996). Seitz et al. (1973) measured the track density in muscovite external detectors with an electron microscope and an optical microscope, and obtained a 10% lower track density with the latter. This con"rms that the counting ef"ciency of an optical microscope is limited by an observation threshold. Seitz et al. (1973) assumed that the track counting e ciency of their electron microscope is close to 1. The counting e ciency for their optical microscope then amounts to ∼0:90, in agreement with the q-value reported here. Measurements of the track counting e ciency in muscovite (Gold et al., 1968; Roberts et al., 1968, 1984; Khan and Durrani, 1972; Khan, 1980; Wall, 1970, 1986) have also produced higher values than those obtained here. Their results are, however, not directly comparable with ours, because the track registration geometries are not the same. In these experiments, ultra-thin "lms containing a "ssionable isotope were vacuum-deposited on the muscovite surface.
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40 Table 3 Track counting e ciencies (q) for muscovite external detectors determined using a thin source (TS) geometry and their conversion to a thick source (ED) geometrya Ref.
[q]TS
[q]ED
[1] [2] [3] [4]
0:948 ± 0:005 0.918 0:988 ± 0:009 0:938 ± 0:020
0:899 ± 0:009 0.843 0:975 ± 0:018 0:880 ± 0:038
Mean
0:948 ± 0:015
0:899 ± 0:028
a [1]—Gold
et al. (1968); [2]—Khan and Durrani (1972); [3]— Roberts et al. (1984); [4]—Wall (1970, 1986). Quoted errors are 1.
In this thin source geometry, all tracks in the external detector have an etchable length R. Tracks with a depth below the observation threshold thus constitute a smaller fraction of the total number of tracks than in an external detector irradiated in contact with a thick source, in which the etchable length of surface-intersecting tracks varies from 0 to R. The track counting e ciency for the thin source geometry has been calculated by Khan (1980): [q]TS = 1 − sin c . For a thin source, we have that: sin c = zc =R; there is thus no distinction between the notion of a critical angle c for track revelation and a critical depth zc for track observation, and zc [q]TS = 1 − : (3) R For an external detector irradiated in contact with a thick source, we have (Jonckheere and Van den haute, 1999) zc 2 : (4) [q]ED = 1 − R Thus [q]ED = [[q]TS ]2 :
(5)
Table 3 summarises the q-values for muscovite external detectors determined using thin sources and their conversion to thick sources. The recalculated results of Roberts et al. (1968) and Wall (1970, 1986) agree with ours. Despite the scatter, these values con"rm that the track counting e ciency in muscovite external detectors used in "ssion-track dating is considerably less than 1. The overall unweighted mean also agrees with our result. Unless this lower e ciency is accounted for, absolute measurements of the uranium concentration in minerals using muscovite external detectors will underestimate the true concentration by as much as 10%. 7. Conclusions The p-distributions of "ssion tracks intersecting an internal and external apatite surface and the surface of a muscovite external detector in Figs. 1 and 2 are in broad
39
agreement with the corresponding theoretical distributions (Dakowski, 1978; Laslett et al., 1994; Laslett and Galbraith, 1996; Jonckheere and Van den haute, 1998). However, the real distributions deviate from the ideal distributions on a number of points. These deviations can be tied to the e4ects of track revelation and observation (Jonckheere and Van den haute, 1999): (1) an observation threshold produces a de"cit of tracks with short projected lengths in the p-distributions of tracks revealed in an internal apatite surface and in a muscovite external detector; (2) track asymmetry shifts and broadens the maximum of the p-distribution of tracks revealed in an external apatite surface and causes the p-distribution of tracks revealed in a muscovite external detector to be concave at high p-values (inverted J-shape); (3) surface etching produces a small de"cit of tracks with long projected lengths in the p-distributions of tracks revealed in an internal and external apatite surface. This masks the inverted J-shape resulting from variations in true track length (Laslett et al., 1982, 1994) and precludes temperature–time path modelling based on length measurements of surface-intersecting tracks (Laslett et al., 1994; Laslett and Galbraith, 1996). These e4ects do not prevent a "rst-order calculation of the track counting e ciencies (q) in the track revelation geometries relevant to "ssion-track analysis; q = 1:01 ± 0:01 for an external apatite surface, and 0:91 ± 0:01 for both an internal apatite surface and a muscovite external detector. The fact that the latter are signi"cantly less than 1 is not the result of an etching e4ect, but due to an observation threshold, best described by a critical depth zc . For the etching and observation conditions used here, zc ≈ 0:8 m for apatite and zc ≈ 0:5 m for muscovite. This is higher than the optical limit of the microscope (0:2 m), but not excessive when one considers that an etched surface is not smooth and that the analyst must not only see the track but must also be able to distinguish it from other features. A critical depth of the order of the track diameter is thus reasonable. An observation threshold described by a critical depth implies that q depends on the track length distribution. Since q decreases with decreasing track length, the apparent age of a sample containing short tracks may be underestimated by as much as 5%, irrespective of whether it is dated with the absolute, the Z- or the -method.
Acknowledgements The results reported here are part of the Ph.D. research of R.J., who is indebted to the Belgian Institute for Science and Technology for a grant. P.V. expresses his gratitude to the Fund for Scienti"c Research–Flanders. We are grateful to G.A. Wagner for allowing R.J. to invest time and material in the preparation of the manuscript. We are indebted to two unknown referees for the valuable comments which much improved our manuscript.
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References Al-Khalifa, I.J.M., Major, J.V., 1987. Comparison of Monte Carlo simulations with "ssion track observations in apatite. Nucl. Tracks Radiat. Meas. 13, 185–195. Bigazzi, G., 1967. Length of "ssion tracks and age of muscovite samples. Earth Planet. Sci. Lett. 3, 434–438. Blok, H., Kiely, F.M., Pate, B.D., 1972. The track lengths of heavy ions in mica. Nucl. Instrum. Methods 100, 403–411. Dakowski, M., 1978. Length distributions of "ssion tracks in thick crystals. Nucl. Track Detection 2, 181–189. Donelick, R.A., 1991. Crystallographic orientation dependence of mean etchable "ssion track length in apatite: an empirical modal and experimental observations. Am. Mineralogist 76, 83–91. Fleischer, R.L., Price, P.B., Walker, R.M., 1975. Nuclear tracks in solids. Principles and Applications. University of California Press, Berkeley, CA, pp. 604. Gleadow, A.J.W., 1978. Anisotropic and variable track etching characteristics in natural sphenes. Nucl. Track Detection 2, 105–117. Gleadow, A.J.W., 1981. Fission track dating methods: what are the real alternatives? Nucl. Tracks 5, 3–14. Gold, R.A., Armani, R.J., Roberts, J.H., 1968. Absolute "ssion rate measurements with solid-state track recorders. Nucl. Sci. Eng. 34, 13–32. Green, P.F., Duddy, I.R., Gleadow, A.J.W., Tingate, P.R., Laslett, G.M., 1986. Thermal annealing of "ssion tracks in apatite. 1. A qualitative description. Chem. Geol. (Isot. Geosci. Sect.) 59, 237–253. Hurford, A.J., Green, P.F., 1983. The zeta age calibration of "ssion track dating. Isot. Geosci. 1, 285–317. Jonckheere, R., 1995. Absolute age determinations of apatite based on uranium "ssion tracks: a methodical investigation. Ph.D. Thesis, University of Gent, Belgium, pp. 504 (in Flemish). Jonckheere, R., Van den haute, P., 1996. Observations on the geometry of etched "ssion tracks in apatite: implications for models of track revelation. Am. Mineralogist 81, 1476–1493. Jonckheere, R., Van den haute, P., 1998. On the frequency distributions per unit area of the dimensions of "ssion tracks
revealed in an internal and external mineral surface and in the surface of an external detector. Radiat. Meas. 29, 135–143. Jonckheere, R., Van den haute, P., 1999. On the frequency distributions per unit area of the projected and etchable lengths of surface-intersecting "ssion tracks: inKuences of track revelation, observation and measurement. Radiat. Meas. 30, 155–179. Khan, H.A., 1980. Track registration and development e ciencies of solid state nuclear track detectors. Nucl. Instrum. Methods 173, 43–54. Khan, H.A., Durrani, S.A., 1972. E ciency calibration of solid state nuclear track detectors. Nucl. Instrum. Methods 98, 229–236. Laslett, G.M., Galbraith, R.F., 1996. Statistical properties of semi-tracks in "ssion track analysis. Radiat. Meas. 26, 565–576. Laslett, G.M., Kendall, W.S., Gleadow, A.J.W., Duddy, I.R., 1982. Bias in measurement of "ssion track length distributions. Nucl. Tracks 6, 79–85. Laslett, G.M., Galbraith, R.F., Green, P.F., 1994. The analysis of projected "ssion track lengths. Radiat. Meas. 23, 103–123. Roberts, J.H., Gold, R., Armani, R.J., 1968. Spontaneous "ssion decay-constant of 238 U. Phys. Rev. 174, 1482–1484. Roberts, J.H., Ruddy, F.H., Gold, R., 1984. Optical e ciency for "ssion fragment track counting in muscovite solid state track recorders. Nucl. Tracks Radiat. Meas. 8, 365–369. Seitz, M.G., Walker, R.M., Carpenter, B.S., 1973. Improved methods for measurement of thermal neutron dose by the "ssion track technique. J. Appl. Phys. 44, 510–512. Van den haute, P., Jonckheere, R., De Corte, F., 1988. Thermal neutron Kuence determination for "ssion-track dating with metal activation monitors: a re-investigation. Chem. Geol. (Isot. Geosci. Sect.) 73, 233–244. Wall, T., 1970. A measurement of the "ssion track recording e ciency of mica. Australian Atomic Energy Commission Report AAEC=TM541. Wall, T., 1986. Use of an alternative neutron dosimetry standard for "ssion track dating. Nucl. Tracks Radiat. Meas. 12, 887–890.
www.elsevier.com/locate/radmeas
On the e ciency of "ssion-track counts in an internal and external apatite surface and in a muscovite external detector R. Jonckheerea; ∗ , P. Van den hauteb a Max-Planck-Institut b Geologisch
fur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Instituut, Universiteit Gent, Krijgslaan 281, B-9000 Gent, Belgium
Received 9 August 2000; received in revised form 15 June 2001; accepted 23 June 2001
Abstract The frequency distributions per unit area of the projected lengths (p-distributions) of "ssion tracks intersecting an internal and external apatite surface and the surface of a muscovite external detector have been established by measurement. Deviations from the ideal distributions on a number of points can be tied to the e4ects of track revelation and observation. The e4ect of track revelation, in particular, masks the e4ect of variations of true track length, and precludes temperature–time path modelling based on length measurements of surface tracks. These e4ects do not prevent calculation of the track counting e ciencies (q) in the track registration geometries of interest to "ssion-track analysis: q = 1:01 ± 0:01 for induced tracks revealed in an external surface of Durango apatite, q = 0:91 ± 0:01 for both fossil and induced tracks revealed in an internal surface of Durango apatite, and q = 0:91 ± 0:01 for induced tracks revealed in a muscovite external detector. The fact that the latter are signi"cantly less than unity is not due to an etching e4ect (critical angle c ) but to an observation threshold, best described by a critical depth zc . For tracks revealed in an internal surface, q decreases rapidly with decreasing track length. As a result, the apparent age of strongly annealed apatite samples may be underestimated by as much as 5%, irrespective of whether the c 2002 Elsevier Science Ltd. All rights reserved. absolute method, the Z-method, or the -method is used for dating. Keywords: Fission-track; Apatite; Muscovite; External detector
1. Introduction Not all the "ssion tracks intersecting the surface of a track detector can be identi"ed under an optical microscope. The fraction of tracks that is counted has been represented by q, wherein stands for etching e ciency and q for observation e ciency. It has been shown that and q cannot be separated and that their combined e4ect, q, cannot, at present, be calculated on the basis of existing models of track revelation and observation (Jonckheere and Van den haute, 1996); q must therefore be determined by experiment. It ∗ Corresponding author. Tel.: +49-6221-516-337; fax: +496221-516-633. E-mail address: [email protected] (R. Jonckheere).
follows from its de"nition that q is the ratio of the experimental ( m ; measured) to the theoretical ( t ; true) density of etchable "ssion tracks intersecting the detector surface. The de"nition of the experimental track density is operational and unambiguous: the result of a track density measurement with the aid of an optical microscope. The theoretical track density is that de/ned by t = 1=2glt N , wherein g is the geometry factor (g = 1=2 for track revealed in an external surface and external detector and g = 1 for track revealed in an internal surface), lt is the mean true etchable track length (Laslett et al., 1982), and N is the volumetric track density. Thus, t includes tracks that do not produce an etch "gure at all (critical angle) as well as tracks that give rise to an etch "gure that is not su ciently distinctive to be identi"ed as a track under the microscope (observation threshold).
c 2002 Elsevier Science Ltd. All rights reserved. 1350-4487/02/$ - see front matter PII: S 1 3 5 0 - 4 4 8 7 ( 0 1 ) 0 0 2 6 2 - 1
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R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Contrary to the experimental track density, estimating the theoretical track density is not straightforward. Gold et al. (1968), Wall (1970, 1986), Khan and Durrani (1972) and Roberts et al. (1984) have determined the e ciency of "ssion-track counts in muscovite detectors by measuring the number of tracks produced by a calibrated "ssion source, and calculating their true number from its activity. In these experiments, the material for which q is determined must function as an external detector with respect to the source, and the source itself is deposited as an ultra-thin "lm on the surface of the detector, to ensure that a negligible number of "ssion fragments are stopped within it. The track registration geometry is thus di4erent from that encountered in routine "ssion-track counts, where the tracks are either revealed in an internal or external mineral surface or in the surface of an external detector, irradiated in contact with a thick "ssion source. The q values obtained using calibrated "ssion sources are thus not applicable to the conditions of routine "ssion-track analysis. Jonckheere and Van den haute (1998, 1999) have proposed that it is possible to estimate the true as well as the experimental track density from the frequency distributions of projected track lengths (p-distributions) of surface-intersecting tracks. The "rst is calculated by integrating its equation, provided that it can be "tted to the data with con"dence; the second is calculated from the number of tracks in the distribution. Our present aim is to show that this provides more reliable estimates of q in each of the three geometries of interest to "ssion-track analysis than have been hitherto available. In addition, q is directly determined for the track registration geometries used in "ssion-track analysis and for the same etching and observation conditions as those of routine "ssion-track counts. With suitable software, such as that developed for this study, it is even possible to determine q (and the true track density) for each sample, simultaneously with the "ssion-track counts. 2. Experiments The "rst experiment was aimed at establishing the p-distributions of induced "ssion tracks in the three track registration geometries of interest: an internal and external apatite surface and a muscovite external detector. To this end, two sections were cut from separate cm-sized crystals of Durango apatite. The "rst section was cut parallel and the second perpendicular to the c-axis; their crystallographic indices are, respectively, {1 1 2I 0} and {0 0 0 1}. Both sections were annealed for 24 h at 450◦ C to erase the fossil tracks, and mounted in an epoxy resin. The mounts were ground on 600-, 800-, and 1000-mesh corundum powders and polished on 6-, 3-, and 1-m diamond paste. The polished mounts were covered with muscovite external detectors and irradiated in channel 8 of the Thetis research reactor at the Institute for Nuclear Sciences in
Gent (Belgium), to produce induced "ssion tracks. The thermal neutron Kuence was measured with Au and Co metal-activation monitors (Van den haute et al., 1988), and amounted to [1:31 ± 0:03] × 1015 cm−2 . The induced tracks in the apatite sections were etched for 60 s in 2.5% HNO3 at 25◦ C; the induced tracks in the muscovite external detectors were etched in 40% HF for 20 min, also at 25◦ C. At this point, the registration geometry for the induced tracks in apatite is that of an external surface, because the sections were ground and polished before the neutron irradiation. In order to obtain the p-distributions of induced tracks in an internal surface, the two sections were re-polished and re-etched, using the same procedure as before, after the length measurements for the external surface had been completed. The measurements were performed with an Olympus BH-2 microscope equipped with a drawing tube, that projects a point-like light source, mounted on the cursor of a Kontron MOP-3 digitiser, positioned next to the microscope, onto the microscope image. The co-ordinates of the track extremities, indicated with the cursor, were sent on-line to an Apple computer, where a dedicated program calculated the track densities, lengths and orientations. The microscopic analyses were carried out in transmitted light, using a 100 × dry objective and 10 × oculars. The true overall magni"cation of the microscope includes a drawing tube factor of 1:25×, and amounted to 1274×, as calibrated against a certi"ed stage micrometer. The p-distributions of "ssion tracks revealed in the internal apatite surface are shown in Fig. 1a (basal section) and 1b (prism section), those of tracks revealed in the external surface in Fig. 1c (basal section) and 1d (prism section), and those of tracks revealed in the surface of the muscovite external detectors in Fig. 1e and f. The second experiment was aimed at measuring the p-distributions of fossil tracks in the internal surfaces of apatite with di4erent etching characteristics. To this end, ∼1 mm thick sections were cut parallel to the {0 0 0 1}, {1 0 1I 0}; {1 1 2I 0}; {1 0 1I 1} and {1 1 2I 1}-planes of a separate crystal of Durango apatite. These sections divide into three groups with respect to their etching characteristics: pitted surfaces ({0 0 0 1} and {1 1 2I 1}), scratched surfaces ({1 1 2I 0}), and textured surfaces ({1 0 1I 0} and {1 0 1I 1}) (Jonckheere and Van den haute, 1996). They were neither annealed nor irradiated, and thus retained their original fossil tracks. They were mounted in epoxy resin, ground and polished as described. The fossil tracks were etched for 60 s in 2.5% HNO3 at 25◦ C, and counted and measured in the same manner as for the sections with induced tracks. The p-distributions are shown in Fig. 2a–e. For evaluating the methods for calculating q discussed here, it is important that the distribution of the true etchable lengths of fossil and induced tracks is known. To this end, the lengths of horizontal con"ned track were measured in the prismatic faces after an additional etching step of 120 s. The frequency distributions of con"ned track length are length-biased (Laslett et al., 1982) and thus not
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
31
Fig. 1. The p-distribution of induced "ssion tracks in internal (a, b) and external (c, d) surfaces of Durango apatite and in co-irradiated muscovite external detectors (e, f).
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R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 2. The p-distribution of spontaneous "ssion tracks in pitted (a, b), scratched (c) and textured (d, e) internal surfaces of Durango apatite.
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 3. Polar plots and length distributions of horizontal con"ned fossil (a) and induced (b) "ssion tracks measured in prismatic surfaces of Durango apatite.
exactly identical to the frequency distributions of true etchable length, but, for the present purposes, this bias can be neglected in the samples studied here (Jonckheere, 1995). The polar plots and length distributions of fossil and induced con"ned tracks are shown in Fig. 3a and b. The mean length of fossil con"ned tracks is 14:4 ± 0:1 m and that of induced tracks is 16:3 ± 0:1 m. The length of induced tracks does not depend on their azimuth angle to the c-axis, although Green et al. (1986) do report a slight but unquanti"ed anisotropy of induced track length in Durango apatite and Donelick (1991) reports a di4erence of 3% between the length of induced tracks parallel and perpendicular to the
33
c-axis in the Tioga apatite. The maximum length di4erence due to anisotropy of the fossil tracks amounts to 5 ± 2%. These anisotropies are thus small enough not to have a signi"cant e4ect on the p-distributions (Jonckheere and Van den haute, 1999). The con"ned track length distributions have standard deviations of 1:0 m, con"rming that both the fossil and induced tracks in Durango apatite possess a range of true etchable lengths (Fig. 3). It is well known that this produces a tail to the p-distribution of tracks revealed in an internal surface (inverted J-shape; Laslett et al., 1982, 1994; Jonckheere and Van den haute, 1999). However, this only a4ects the interval of projected lengths bracketed by the minimum (lt; min ) and maximum (lt; max ) true etchable length: [lt; min = p = lt; max ]. The remaining part of the p-distribution [0 6 p 6 lt; min ] remains linear and entirely determined by the mean true etchable length and the true volumetric track density (Jonckheere and Van den haute, 1999). For both fossil and induced tracks in Durango apatite, the coe cient of variation of the con"ned track length distribution is less than 0.10 (fossil tracks: 1 m=14:4 m = 0:07; induced tracks: 1 m=16:3 m = 0:06). It has been shown that in this case the tail is insigni"cant (Jonckheere and Van den haute, 1999). It has also been shown that track revelation has a more important e4ect on the p-distribution, opposite to that due to the range of etchable track lengths. As a consequence, the p-distributions of track revealed in an internal surface never show the inverted J-shape, but, instead, a track de"cit at p-values approaching the mean true etchable track length (Jonckheere and Van den haute, 1999). In the two methods for calculating q proposed here, this de"cit is either accounted for or shown to have a negligible e4ect (Section 3). In conclusion, the tracks in the studied samples do possess a range of true etchable lengths, but this has no e4ect on the experimental determination of the q-values, and the results can be interpreted to be valid for an idealised population of "ssion tracks with a constant etchable length identical to the mean true etchable length in the samples. This is important because it allows to apply the present results to more complex length distributions in geological samples. 3. Internal surface The p-distributions of "ssion tracks intersecting an internal mineral surface (Fig. 1a and b: induced tracks; Fig. 2: fossil tracks) are in fair but not in perfect agreement with the theoretical “triangular” distribution (Dakowski, 1978; Laslett et al., 1994; Laslett and Galbraith, 1996; Jonckheere and Van den haute, 1998, 1999). In this respect, the distributions divide into three length intervals: (1) 0 –3 m, in which there is a marked track de"cit compared to the theoretical distribution; (2) 3–13 m, in which the distributions show a linear decreasing trend, in agreement with the theoretical distribution; and (3) the length interval ¿ 13 m, which again shows a distinct track de"cit.
34
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
The deviation from the theoretical distribution in the interval 0 –3 m can result from limited precision of the length measurements or from an observation threshold. Limited precision without loss of tracks would, however, result in a surplus of tracks with lengths just in excess of 3 m (Jonckheere and Van den haute, 1999; Fig. 10a). Given that this is not the case, the track de"cit below 3 m is interpreted as a real loss due to an observation threshold. By itself, the trend in the interval 0 –3 m is not characteristic enough to identify this threshold with con"dence. The fact that a part of the tracks with projected lengths up to 3 m is not observed suggests an observation threshold described by a critical depth zc , because values of zc ¡ 1 m can produce a signi"cant loss of tracks with projected lengths up to 3 m, whereas higher threshold values are required if the observation threshold is described by a critical projected length (pc ) or a critical etchable length (ec ) (Jonckheere and Van den haute, 1999; Figs. 6a, e and i and corresponding equations in Table 1). An observation threshold also a4ects the frequency distribution per unit area of the etchable length (e-distribution), depth (z-distribution) and inclination (-distribution) of surface-intersecting "ssion tracks; e-, z- and -distributions of induced and fossil tracks in apatite have been published by Dakowski (1978) and Al-Khalifa and Major (1987). A comparison with the corresponding theoretical distributions (Jonckheere and Van den haute, 1999) shows that: (1) as in the p-distribution, there are less than the expected number of tracks with small values of e and z (¡ 3 m); (2) the monotonous increase in the interval 0 –3 m is more abrupt in the z-distribution and more gradual in the p- and e-distributions; (3) the maximum of the -distribution is shifted to the right compared with the theoretical distribution, i.e. to -values ¿ =4. It follows from these observations that (1) there is no agreement at all between the experimental and theoretical distributions for a threshold described by ¿ c (critical angle) and (2) there is better agreement between the experimental and theoretical distributions for the threshold criterion z ¿ zc (critical depth) than either for the criterion p ¿ pc (critical projected length) or e ¿ ec (critical etchable length). The agreement between the experimental and theoretical distributions is less than perfect, but, considering the di culty of measuring short tracks precisely and earlier theoretical arguments (Jonckheere and Van den haute, 1999), the evidence is su cient to conclude that "ssion tracks are observed under an optical microscope on condition that their depth z below the detector surface exceeds a critical value zc , and that they escape observation if z ¡ zc . A pit in a perfect surface can indeed only be observed with an optical microscope if its depth is greater than half the wavelength of the light source. For visible light with a wavelength of ∼400 nm, this implies a minimum value of zc of ∼0:2 m. The value of zc derived from "tting the equation of the theoretical distribution (Jonckheere and Van den haute, 1999) to the p-distributions in Figs. 1a, b and 2 amounts to ∼0:8 m.
This di4erence is not excessive when one considers that an etched apatite surface is not perfectly smooth and that, in order to identify and count it, the analyst must not only see the track but must also be able to distinguish it from other features. The interval 3–13 m of the p-distributions is quasi-linear. If una4ected by the e4ects of track revelation, observation and measurement, extrapolation of this interval allows to estimate the mean true etchable length (lt ) of the tracks (Jonckheere and Van den haute, 1999). If the result agrees with the mean length of con"ned tracks (lc ), it is reasonable to conclude that the central interval of the p-distributions is indeed not a4ected by these factors, and that linear extrapolation also provides an accurate estimate of the true track density ( t ). This extrapolation was performed by "tting a regression line to the 10 points in Figs. 1a, b and 2, one for each 1 m interval between 3 and 13 m. If the regression equation is written as N (p) = A − Bp, then lr , i.e. the intersection of the regression line with the p-axis, is given by lr = A=B (the subscript r refers to the regression method). The true track density r is obtained by integrating the regression equation between 0 and lr : r = A2 =2BS, wherein S is the counted surface area. The correlation coe cients of the regression lines (r), and the values of lr and r are shown in Tables 1 and 2. The r-values do not di4er much from −1, indicating a good "t. The lr =lc -ratio depends on the type of surface. For pitted surfaces it is equal to 1 (mean: 0:997 ± 0:010). For scratched surfaces lr =lc it is somewhat less, but still close to 1 (mean: 0:980 ± 0:020). The lower value for the textured surfaces (mean: 0:855 ± 0:045) indicates a signi"cant underestimation of the true track length (∼2 m). This di4erence must result from errors in indicating the track intersection with the surface. Thus calculating
r from the regression equation is warranted for pitted and scratched surfaces, but not for textured surfaces. The interval ¿ 13 m of the p-distributions for the internal surface is characterised by a small track de"cit. Calculation shows that several factors related to the range of true etchable lengths (Laslett et al., 1982, 1994; Laslett and Galbraith, 1996) and to track revelation, observation and measurement (Jonckheere and Van den haute, 1999) a4ect this length interval. Surface etching is, however, the only factor that produces a track de"cit. Imprecision of the length measurements, statistical length variation and anisotropic length produce a convex trend that entails a track surplus in the interval ¿ 13 m (Jonckheere and Van den haute, 1999). Since tracks with p ¿ 13 m constitute a small fraction (2–3%) of the tracks in the p-distribution, the actual de"cit has but a minor e4ect on the mean projected track length. The de"cit of tracks with p ¡ 3 m, on the other hand, has a signi"cant e4ect. The true etchable track length (la ; the subscript a refers to the second method for estimating the mean true length) can thus be obtained by choosing an arbitrary value a ¿ 3 m, and calculating the mean projected length (ma ) of tracks longer than a: ma = (la + 2a)=3 (Fig. 4). This provides a more precise if somewhat less accurate method
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
35
Table 1 Measured and theoretical areal densities and lengths of induced "ssion tracks revealed in an internal and external apatite surface and in the surface of a co-irradiated muscovite external detector and calculation of the track counting e ciencies (q)a Orientation Surface
N
n
m
r
lr
r
a
la
la =lc
lr =lc [q]a
[q]r
Internal surface (apatite) {0 0 0 1} P 4265 219 3:16 ± 0:05 3.51 16.5 −0:996 3:50 ± 0:07 15:9 ± 0:2 0:98 ± 0:01 1.01 {1 1 2I 0} S 4610 190 3:94 ± 0:06 4.34 16.3 −0:995 4:32 ± 0:08 15:6 ± 0:2 0:96 ± 0:01 1.00
0:90 ± 0:02 0.90 0:91 ± 0:02 0.91
External surface (apatite) {0 0 0 1} P 4646 420 1:80 ± 0:03 1.76 16.5 −0:996 1:75 ± 0:04 15:9 ± 0:2 0:98 ± 0:01 1.01 {1 1 2I 0} S 2907 219 2:15 ± 0:04 2.17 16.3 −0:995 2:16 ± 0:04 15:6 ± 0:2 0:96 ± 0:01 1.00
1:03 ± 0:02 1.03 1:00 ± 0:03 0.99
External detector (muscovite) {0 0 0 1} P 2990 229 1:90 ± 0:03 2.07 10.0 −0:997 2:09 ± 0:06 10:5 ± 0:1 0:51 ± 0:01 0.49 {1 1 2I 0} S 3860 237 2:37 ± 0:04 2.62 10.0 −0:992 2:65 ± 0:07 10:5 ± 0:1 0:51 ± 0:01 0.49
0:91 ± 0:03 0.92 0:90 ± 0:03 0.90
a P—pitted surface; S—scratched surface; N —number of tracks; n—number of unit "elds counted (1 unit "eld = 6:16 × 10−5 cm 2 for the internal and external surface, and 6:87 × 10−5 cm2 for the external detector); m (105 cm−2 )—measured track density; r (105 cm−2 )— theoretical track density calculated with the regression method; lr (m)—track length calculated with the regression method; r—correlation coe cient of the regression line to the projected track length distribution; a (105 cm−2 )—track density calculated on the basis of the number of tracks with projected length. 3 m; la (m)—track length calculated on the basis of the mean projected length of tracks longer than 3 m; [q]a —track counting e ciency calculated on the basis of tracks with projected length. 3 m; [q]r —track counting e ciency calculated with the regression method. All errors are 1; no errors are given for the regression method because the non-homoscedasticity of the projected track length distribution prevents their simple calculation.
Table 2 Measured and theoretical areal densities and lengths of fossil "ssion tracks revealed in internal pitted, scratched and textured surfaces of apatite, and calculation of the corresponding track counting e ciencies (q)a Orientation N
n
m
r
lr
r
a
la
la =lc
lr =lc
[q]a
[q]r
Pitted surfaces {0 0 0 1} 2611 {1 1 2I 1} 1344
201 105
2:21 ± 0:05 2:08 ± 0:05
2.31 2.29
14.3 14.2
−0:994 −0:983
2:32 ± 0:06 2:30 ± 0:08
13:9 ± 0:2 14:0 ± 0:2
0:97 ± 0:02 0:97 ± 0:02
0.99 0.99
0.91 ± 0.03 0:91 ± 0:04
0.91 0.91
Scratched surfaces {1 1 2I 0} 2413 200
1:96 ± 0:04
2.10
14.0
−0:996
2:10 ± 0:05
13:8 ± 0:2
0:96 ± 0:02
0.93
0:93 ± 0:03
0.93
Textured surfaces {1 0 1I 0} 1279 {1 0 1I 1} 1095
2:08 ± 0:06 1.80 ± 0.06
2.31 2.21
12.7 12.6
−0:982 −0:991
2:32 ± 0:08 2:22 ± 0:08
12:5 ± 0:2 12:4 ± 0:2
0:87 ± 0:02 0:86 ± 0:02
0.90 0.81
0:90 ± 0:04 0:81 ± 0:04
0.90 0.81
a Symbols
100 99
have the same meaning as in Table 1; m ; r and a are in 105 tracks=cm2 ; lr and la in m; errors are 1.
for calculating both the true etchable track length (la ) and true track density ( a ), since (Fig. 4): la = 3ma − 2a;
(1)
a = [la =[la − a]]2 a ;
(2)
wherein a is the measured density of tracks longer than 3 m. Tables 1 and 2 show the results for a = 3 m. As expected, la is somewhat less than lr . The di4erence is more marked for the induced tracks because their p-distributions show a larger de"cit of tracks ¿ 13 m. For the fossil tracks, the di4erence between la and lr is less than 1%. The la =lc -ratios con"rm the conclusions based on the lr =lc -ratios, in particular, that the track length can be derived from mea-
surements of p in pitted and scratched-surfaces, but that these lead to an underestimation in textured surfaces. 4. External surface The p-distribution of induced tracks in an external apatite surface (Fig. 1c and d) agrees well with the theoretical distribution (Jonckheere and Van den haute, 1998). For p ¿ lr =2, the latter is derived from the regression line to the p-distribution for the internal surface, since both p-distributions are identical in this length interval. In the interval p ¡ lr =2, the theoretical distribution is obtained by mirroring the regression line about lr =2, which value is itself
36
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 4. Schematic diagram of the p-distribution of tracks revealed in an internal surface, indicating a; ma ; la , a and a (see text), and showing the e4ects of an observation threshold between A and E and of track revelation at D.
derived from the regression line. There is no track de"cit in the interval 0 –3 m, consistent with an observation threshold described by z ¿ zc . As for the internal surface, the track de"cit in the interval ¿ 13 m is the result of surface etching. The maximum at lr =2 is not sharp: there is a small but systematic surplus of tracks with projected lengths just short of lr =2. Two factors that can account for this (Jonckheere and Van den haute, 1999) are (1) surface etching, and (2) an o4-centre and variable locus of the "ssioned uranium atom with respect to the track end-points (track asymmetry). The distortion of the maximum at lr =2 is not more pronounced in the p-distribution of tracks revealed in the prismatic face (Fig. 1d) than in that of tracks revealed in the basal face (Fig. 1c), despite the fact that the etch rate of the surface in the former exceeds that in the latter by almost a factor of 5 (Jonckheere and Van den haute, 1996). This implies that track asymmetry is the more important factor. The mean projected track length (lm ), calculated from the distributions in Fig. 1c and d, is 8:0 ± 0:1 m. In "rst approximation, lm is half the mean true etchable track length. The ratio lm =lc (0:49 ± 0:01) is close to this value. The small di4erence results from the track de"cit in the interval ¿ 13 m and to a lesser extent from the track surplus just below lm . 5. External detector The p-distributions of induced tracks revealed in a muscovite external detector are shown in Fig. 1e and f. These were summed with 12 more identical distributions; the result is a distribution with just short of 60.000 tracks, shown in Fig. 5. Like the p-distribution of tracks revealed in internal surface, it divides into three length intervals: (1)
Fig. 5. The p-distribution of induced "ssion tracks revealed in 16 muscovite external detectors irradiated against Durango apatite.
0 –2 m, in which there is a marked track de"cit compared to the theoretical distribution (Jonckheere and Van den haute, 1998), (2) 2–9 m, in which the distribution shows a linear decreasing trend in agreement with the theoretical distribution and (3) the length interval ¿ 9 m, in which, in this case, the distribution is markedly convex. As in the internal surface, the track de"cit in the interval 0 –2 m is the result of an observation threshold described by z ¿ zc . Fitting the theoretical equation (Jonckheere and Van den haute, 1999) to the graph in Fig. 5 gives a "rst-order estimate of zc : zc ≈ 0:5 m. This agrees with the general experience that "ssion tracks in muscovite are easier to identify and count than tracks in apatite. The etched muscovite surface is, however, not free of other defects such as alpha-recoil tracks, so that, even in this case, the threshold for track identi"cation is higher than the optical limit (0:2 m). A regression line was "tted to the linear interval 2–9 m; lr and
r were calculated from the regression equation; la and a were also calculated from the number and mean projected length of tracks longer than 2 m. All are shown in Table 1. The la -values are systematically somewhat higher than the lr -values as a result of the surplus of tracks in the interval ¿ 9 m. On the other hand, both la and lr are close to half the length of con"ned tracks in muscovite (20:5 ± 0:1 m; Bigazzi, 1967). We conclude that the interval 2–9 m is not a4ected by factors related to track revelation, observation and measurement and that r and a are accurate estimates of the actual track density. Numerical modelling has shown that the convex trend in the length interval ¿ 9 m is the result of the o4-centre and variable position of the "ssioned uranium nucleus with respect to the track extremities (track asymmetry; Jonckheere and Van den haute, 1999).
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
6. q-Factor The e ciency of "ssion-track counts under a microscope (q) is de"ned as the ratio of the measured ( m ) to the true ( t ) areal densities of tracks intersecting the detector surface. As described, m was estimated by summing the tracks in the p-distributions for each of the three track registration geometries relevant to "ssion-track analysis. For the internal surface and external detector, t was estimated in two ways: (1) from the regression line to the linear part of the p-distribution ( r ), and (2) from the mean projected length of tracks longer than an appropriate limit ( a ). For the external surface, neither r nor a could be determined directly from the p-distribution. They were, instead, derived from the r and a -values for the internal surface, which are twice that for the external surface. This gives two independent estimates of q for the three registration geometries: [q]r = m = r and [q]a = m = a , reported in Table 1 for induced tracks revealed in the three track registration geometries and in Table 2 for fossil tracks revealed in pitted, scratched and textured internal surfaces. The notion of a critical angle c = arcsin[vb =vt ] (vb = etching velocity perpendicular to the etched surface; vt = etching velocity along the "ssion track) implies that q is the same for an internal and external surface with the same crystallographic orientation (q = 1 − sin2 c ; Jonckheere and Van den haute, 1999), but di4erent for internal surfaces with di4erent crystallographic orientation (di4erent vb ). The experimental results demonstrate the exact opposite: Table 1 shows that q is much lower for tracks revealed in an internal surface (0:91 ± 0:01) than for tracks revealed in an external surface (1:01 ± 0:01) of Durango apatite. On the other hand, there is no signi"cant di4erence between the q values for tracks revealed in pitted and scratched internal surfaces (textured surfaces are discussed below). This proves that q is not in the "rst place a mineral=etchant property. It is thus meaningless to talk about “the etching e ciency of a mineral” in connection with the accuracy of "ssion-track age determinations. The results in Table 1 also show that q is the same for internal surfaces of apatite and for a muscovite external detector (0:91 ± 0:01). This is no doubt a coincidence, but it con"rms again that q is unrelated to the etching velocities of the surface, which di4er two orders of magnitude for apatite and muscovite under the etching conditions used here (Blok et al., 1972; Jonckheere and Van den haute, 1996). The notion of a critical depth for track identi"cation implies that q varies as q = 1 − [zc =R] + 1=4[zc =R]2 for the internal surface and as q = 1 − 1=2[zc =R]2 for the external surface, wherein zc is the critical depth for track observation and 2R the true etchable track length (Jonckheere and Van den haute, 1999). For an induced track length of 16:3 m and a critical depth of 0:8 m, the calculated q-values are q = 0:91 for the internal surface and 1.00 for the external surface, in agreement with the values in Table 1. The value zc = 0:8 m corresponds to that obtained by "tting the the-
37
oretical equations to the distributions in Fig. 1a and b, with zc as a free parameter (Section 3). For the external detector, q varies as q = 1 − 2[zc =R] + [zc =R]2 (Jonckheere and Van den haute, 1999). For 2R = 20:5 m (Bigazzi, 1967), and zc = 0:5 m (Section 5), we calculate that q = 0:91, in agreement with the experimental value. The fact that zc is less than the value for apatite agrees with the general experience that "ssion tracks are easier to identify in muscovite than in apatite. That this lesser value nevertheless causes the same loss of tracks as in the internal apatite surface is due to the fact that, in "rst approximation, the etchable length of surface-intersecting tracks in an external detector varies from 0 to R, and that in an internal surface it varies from 0 to 2R. There are thus proportionally more tracks with depths less than a given value. The same reasoning also makes it plain that q must be close to unity for tracks revealed in an external apatite surface. In this geometry, the etchable length of surface-intersecting tracks ranges from R to 2R. Thus only part of the tracks at an angle to the surface less than arcsin [zc =R] can escape observation. For zc = 0:8 m and R = 8 m, arcsin [zc =R] is small, and the loss of tracks due to an observation threshold is minimal. As described above, the exact value of q is: q = 1 − 1=2[zc =R]2 = 0:995: In the above formulae, R represents the true etchable track length. The dependence of q on etchable track length implies that q is less for partially annealed (shortened) tracks than for unannealed tracks. In particular, it is expected that the q-values for fossil (mean length: 14:4 ± 0:1 m) and induced tracks (mean length: 16:3 ± 0:1 m) in an internal apatite surface will also be di4erent. The results in Table 1 (induced tracks) and Table 2 (fossil tracks) show that this is not the case. Fig. 6 shows how q varies with true etchable track length for the internal surface, assuming zc = 0:8 m. It is seen that q varies little as a function of track length between 14.4 and 16:3 m, explaining the identical q-values for the fossil and induced tracks. However, the fact that q drops o4 rapidly at shorter lengths has important implications for "ssion-track dating. It implies that the fossil tracks in a strongly annealed sample are counted with less e ciency than either the induced tracks in an annealed and irradiated aliquot of the same sample or the fossil tracks in a set of age standards. This means that the "ssion track age of such a sample, whether it is dated with the absolute method, the Z- or the -method, will be underestimated by an amount exceeding that resulting from the reduction of track length alone. The exact age underestimation will depend on the track length distribution. In speci"c cases where the track length distribution is known from con"ned track length measurements, this amount can be calculated by integrating q over the range of lengths in the sample. The q-values for the internal surface in Table 1 (induced tracks) and Table 2 (fossil tracks) reveal no signi"cant di4erence between scratched and pitted surfaces, despite the fact that the etch rate of the surface, ranges from 0:08 ± 0:01 m=min to 0:37 ± 0:01 m=min (Jonckheere and Van den haute, 1996). The results for the
38
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40
Fig. 6. The track counting e ciency (q) in an internal surface of apatite as a function of true etchable track length for an observation threshold described by a critical depth zc ≈ 0:8 m.
textured surfaces are less straightforward. The fact that the la =lc and lr =lc -ratios are less than 1 indicates that their q-factors are not reliable. On the other hand, the q-value for {1 0 1I 0} (0:90±0:04) is identical to that of the scratched and pitted surfaces and the measured track density (2:08 ± 0:06 × 105 cm−2 ) is close to that for {0 0 0 1}; {1 1 2I 0} and {1 1 2I 1} (mean: 2:08 ± 0:07 × 105 cm2 ). The track density in {1 0 1I 1} (1:80 ± 0:06 × 105 cm−2 ) is much lower and so is its q-factor (0:81 ± 0:04). Although the etch rates perpendicular to {1 0 1I 0} (0:64 ± 0:01 m=min) and {1 0 1I 1} (0:61 ± 0:01 m=min) are similar, {1 0 1I 0} is characterised by a "ne and {1 0 1I 1} by a coarse texture (Jonckheere and Van den haute, 1996). The lower track density for {1 0 1I 1} thus probably results from the greater di culty of identifying short tracks against a more irregular background. The fact that this is reKected in a proportionally lower q-value seems to suggest that the methods used for calculating q are fairly robust, and still give a reasonable value even when the projected track lengths cannot be measured accurately. Perhaps the most important result is that q is much less than unity for internal surfaces of apatite (mean for scratched and pitted surfaces: q = 0:91 ± 0:01) and for muscovite external detectors (q = 0:91 ± 0:01), in direct contradic-
tion with earlier estimates based on the etching characteristics (etch rates) of these surfaces (Fleischer et al., 1975; Gleadow, 1978, 1981; Hurford and Green, 1983). We have argued before at length that the etching models on which these estimates are based are inappropriate for describing track revelation in minerals (Jonckheere and Van den haute, 1996). Seitz et al. (1973) measured the track density in muscovite external detectors with an electron microscope and an optical microscope, and obtained a 10% lower track density with the latter. This con"rms that the counting ef"ciency of an optical microscope is limited by an observation threshold. Seitz et al. (1973) assumed that the track counting e ciency of their electron microscope is close to 1. The counting e ciency for their optical microscope then amounts to ∼0:90, in agreement with the q-value reported here. Measurements of the track counting e ciency in muscovite (Gold et al., 1968; Roberts et al., 1968, 1984; Khan and Durrani, 1972; Khan, 1980; Wall, 1970, 1986) have also produced higher values than those obtained here. Their results are, however, not directly comparable with ours, because the track registration geometries are not the same. In these experiments, ultra-thin "lms containing a "ssionable isotope were vacuum-deposited on the muscovite surface.
R. Jonckheere, P. Van den haute / Radiation Measurements 35 (2002) 29–40 Table 3 Track counting e ciencies (q) for muscovite external detectors determined using a thin source (TS) geometry and their conversion to a thick source (ED) geometrya Ref.
[q]TS
[q]ED
[1] [2] [3] [4]
0:948 ± 0:005 0.918 0:988 ± 0:009 0:938 ± 0:020
0:899 ± 0:009 0.843 0:975 ± 0:018 0:880 ± 0:038
Mean
0:948 ± 0:015
0:899 ± 0:028
a [1]—Gold
et al. (1968); [2]—Khan and Durrani (1972); [3]— Roberts et al. (1984); [4]—Wall (1970, 1986). Quoted errors are 1.
In this thin source geometry, all tracks in the external detector have an etchable length R. Tracks with a depth below the observation threshold thus constitute a smaller fraction of the total number of tracks than in an external detector irradiated in contact with a thick source, in which the etchable length of surface-intersecting tracks varies from 0 to R. The track counting e ciency for the thin source geometry has been calculated by Khan (1980): [q]TS = 1 − sin c . For a thin source, we have that: sin c = zc =R; there is thus no distinction between the notion of a critical angle c for track revelation and a critical depth zc for track observation, and zc [q]TS = 1 − : (3) R For an external detector irradiated in contact with a thick source, we have (Jonckheere and Van den haute, 1999) zc 2 : (4) [q]ED = 1 − R Thus [q]ED = [[q]TS ]2 :
(5)
Table 3 summarises the q-values for muscovite external detectors determined using thin sources and their conversion to thick sources. The recalculated results of Roberts et al. (1968) and Wall (1970, 1986) agree with ours. Despite the scatter, these values con"rm that the track counting e ciency in muscovite external detectors used in "ssion-track dating is considerably less than 1. The overall unweighted mean also agrees with our result. Unless this lower e ciency is accounted for, absolute measurements of the uranium concentration in minerals using muscovite external detectors will underestimate the true concentration by as much as 10%. 7. Conclusions The p-distributions of "ssion tracks intersecting an internal and external apatite surface and the surface of a muscovite external detector in Figs. 1 and 2 are in broad
39
agreement with the corresponding theoretical distributions (Dakowski, 1978; Laslett et al., 1994; Laslett and Galbraith, 1996; Jonckheere and Van den haute, 1998). However, the real distributions deviate from the ideal distributions on a number of points. These deviations can be tied to the e4ects of track revelation and observation (Jonckheere and Van den haute, 1999): (1) an observation threshold produces a de"cit of tracks with short projected lengths in the p-distributions of tracks revealed in an internal apatite surface and in a muscovite external detector; (2) track asymmetry shifts and broadens the maximum of the p-distribution of tracks revealed in an external apatite surface and causes the p-distribution of tracks revealed in a muscovite external detector to be concave at high p-values (inverted J-shape); (3) surface etching produces a small de"cit of tracks with long projected lengths in the p-distributions of tracks revealed in an internal and external apatite surface. This masks the inverted J-shape resulting from variations in true track length (Laslett et al., 1982, 1994) and precludes temperature–time path modelling based on length measurements of surface-intersecting tracks (Laslett et al., 1994; Laslett and Galbraith, 1996). These e4ects do not prevent a "rst-order calculation of the track counting e ciencies (q) in the track revelation geometries relevant to "ssion-track analysis; q = 1:01 ± 0:01 for an external apatite surface, and 0:91 ± 0:01 for both an internal apatite surface and a muscovite external detector. The fact that the latter are signi"cantly less than 1 is not the result of an etching e4ect, but due to an observation threshold, best described by a critical depth zc . For the etching and observation conditions used here, zc ≈ 0:8 m for apatite and zc ≈ 0:5 m for muscovite. This is higher than the optical limit of the microscope (0:2 m), but not excessive when one considers that an etched surface is not smooth and that the analyst must not only see the track but must also be able to distinguish it from other features. A critical depth of the order of the track diameter is thus reasonable. An observation threshold described by a critical depth implies that q depends on the track length distribution. Since q decreases with decreasing track length, the apparent age of a sample containing short tracks may be underestimated by as much as 5%, irrespective of whether it is dated with the absolute, the Z- or the -method.
Acknowledgements The results reported here are part of the Ph.D. research of R.J., who is indebted to the Belgian Institute for Science and Technology for a grant. P.V. expresses his gratitude to the Fund for Scienti"c Research–Flanders. We are grateful to G.A. Wagner for allowing R.J. to invest time and material in the preparation of the manuscript. We are indebted to two unknown referees for the valuable comments which much improved our manuscript.
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