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A Monte-Carlo calculation of the size distribution of latent alpha-recoil tracks
Nuclear Instruments and Methods in Physics Research B 183 (2001) 347±357
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A Monte-Carlo calculation of the size distribution of latent alpha-recoil tracks Raymond Jonckheere b
a,*
, K ursßad G ogen
b
a Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Forschungsstelle Archaometrie, Heidelberger Akademie der Wissenschaften, Karlstrasse 4, D-69117 Heidelberg, Germany
Received 6 March 2001; received in revised form 26 April 2001
Abstract This paper proposes a Monte-Carlo method for calculating the time-dependent size distribution of alpha-recoil tracks in minerals. The results show that, in the case of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, the size distribution comprises two distinct track-populations. The ®rst has a mean size of 30 nm and standard deviation of 5 nm and consists of tracks that are the result of a single alpha decay. The second has a broad range of sizes with a mean of 125 nm and standard deviation 50 nm and consists of tracks that, for the most part, result from complete decay of 238 U and 234 U to stable 206 Pb. The ®rst population saturates at around 1 Ma, whereas the second shows approximately linear growth. On the basis of the present results, it becomes possible to calculate both the number of recoil tracks intersecting a unit surface of natural minerals and the interconnectedness of recoil damage as a function of time, which has direct implications for alpha-recoil track dating of mica and for the prediction of the long-term behaviour of mineral host phases for the disposal of high-level nuclear waste. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 34.50.)s; 91.70.)c; 28.91.Kw; 23.60.+e; 02.70.Uu Keywords: Phlogopite; Recoil-track size; Recoil-track dating; Nuclear-waste disposal
1. Introduction Natural minerals contain trace amounts of uranium, thorium and their radioactive daughters. The primary mode of decay of several of these
* Corresponding author. Tel.: +49-6221-516-337; fax: +496221-516-633. E-mail address: [email protected] (R. Jonckheere).
isotopes is through emission of an alpha particle. Each alpha decay sets free several MeV, which are shared as kinetic energy by the alpha particle (103 keV) and the recoiling daughter nucleus (102 keV). The alpha particle loses its energy through electronic interactions with the lattice atoms over most of its 20 lm range, and produces 102 isolated atomic displacements through direct nuclear interactions at the end. The recoil nucleus loses its energy through nuclear interactions over a range 30±40 nm, producing a highly localised
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 6 9 1 - 7
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R. Jonckheere, K. Gogen / Nucl. Instr. and Meth. in Phys. Res. B 183 (2001) 347±357
displacement cascade resulting in 103 lattice defects, which, together, constitute an alpha-recoil track. Alpha-recoil tracks are thus the products of radioactive decay. Their number per unit volume increases with time, and is thus a measure for the age of the mineral [1±7]. In minerals of the mica group, the recoil tracks intersecting the cleavage planes can be revealed by etching (Fig. 1). The etch ®gures can be observed with a microscope, and the number of tracks intersecting a unit surface can thus be measured. One of the problems in recoil-track dating, like in ®ssion-track dating, is to relate the areal densities of etched, surface-intersecting tracks to their volumetric densities. The size of the recoil tracks is a key parameter in this relationship. This paper proposes a method for calculating the time-dependent size distribution of alpha-recoil tracks resulting from 238 U-decay, using a Monte-Carlo technique. The accumulation of alpha-recoil tracks is also an important factor aecting the long-term stability of mineral host phases for the disposal of high-level nuclear waste. It is well known that the leach rates of these minerals increase with their degree of metamictisation, i.e. with the number of alpha-recoil tracks per unit volume [8]. The integrity of these minerals is not much aected as long as alpha-recoil tracks constitute only isolated areas of lattice damage within the crystal. How-
Fig. 1. Triangular recoil tracks in a cleaved internal surface of phlogopite mica, etched for 6 min in 40% HF, at 25°C.
ever, as the number of recoil tracks per unit volume increases with time, the interconnectedness of the damaged areas and their connections with the surface will also increase, thus increasing the macroscopic leach rate. To estimate this interconnectedness, it is necessary to know both the recoil-track density and the recoil-track size distribution as a function of time. Thus the method for calculating the recoil-track size distribution proposed here also oers the prospect of theoretically predicting the behaviour of mineral host phases for the disposal of high-level nuclear waste. It is clear that the recoil-track size distribution will depend on the composition of the mineral. The present calculations were made for the mineral phlogopite KMg3 j F; OH2 jAlSi3 O10 . Phlogopite is not a candidate host phase for nuclear waste disposal, but it has proved the most suitable mineral for recoil-track dating [6,7]. This choice does not constitute a severe restriction because the recoil-track size distribution in other minerals is obtained by scaling of the size-axis. In accordance with the choice of mineral, the recoil-track size distribution is calculated for the uranium-series isotopes, i.e. the principal recoil-track producing isotopes in natural minerals. 2. Assumptions The most abundant Uranium isotope in natural minerals is 238 Uh 0:992743; 9. The alpha decay of 238 U is the start of the uranium-series decay chain, involving 8 alpha and 6 beta decays, and ending in stable 206 Pb (Fig. 2). The energy released in beta decay is of the order of 1 MeV, most of which is shared by the beta particle and an antineutrino. Because of their low mass, beta particles do not eciently transfer energy to the lattice atoms; the daughter nuclei are also almost never energetic enough to recoil permanently from their lattice positions. In general, beta decay produces of the order of 0.1 atomic displacements per event [8]. The energy released in alpha decay of the uranium-series isotopes ranges from 5 to 8 MeV. Between 70 and 100 keV is communicated as kinetic energy to the daughter nucleus, the remainder is carried away by the alpha particle
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Fig. 2. The uranium-series isotopes (after 9). Neglecting branches with branching ratios < 0:001 (grey) reduces the uranium-series to a single, non-branching chain of 14 successive alpha and beta decays (black). Decay constants in a 1 .
(Table 1). The alpha particles dissipate most of their energy in ionisation processes but undergo enough elastic collisions, in particular towards the end of their trajectories, to produce several hundred isolated atomic displacements. The alpha particles thus contribute to a diuse background of isolated lattice defects, but do not lead to the
formation of localised areas of etchable lattice damage. The recoil of the less energetic but far more massive (atomic mass: 206±234) daughter nuclei accounts for most of the total number of atomic displacements. Elastic collisions with the lattice atoms produce highly localised cascades, resulting in 1000±2000 displacements [8] or some-
Table 1 The range and straggling of the recoiling daughter nuclei resulting from alpha decay of the principal uranium-series isotopes, calculated using the binary collision approximation SRIMa Parent ! Daughter
k a 1
Ealpha (MeV)
Erecoil (keV)
Rrecoil (nm)
Srecoil (nm)
Vacancies
238
1:54 10 10 2:81 10 06 8:65 10 06 4:33 10 04 6:60 1001 1:19 1005 1:33 1011 1:82 1000
4.20 4.77 4.68 4.78 5.49 6.00 7.69 5.31
71.7 82.9 82.8 98.9 110.1 143.7 101.1 103.1
29.5 32.3 32.2 36.4 39.1 46.5 36.5 37.2
6.0 6.6 6.8 7.6 8.3 9.8 7.8 8.1
1243 1423 1417 1660 1823 2316 2316 1715
U ! 234 Th U ! 230 Th 230 Th ! 226 Ra 226 Ra ! 222 Rn 222 Rn ! 218 Po 218 Po ! 214 Bi 214 Po ! 210 Pb 210 Po ! 206 Pb 234
a k: decay constant; Ealpha : kinetic energy of the alpha particle; Erecoil : kinetic energy of the recoil nucleus; Rrecoil : range of the recoil nucleus; Srecoil : range-straggling of the recoil nucleus; vacancies: lattice vacancies per recoil ion.
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what less than 0.1 dpa (displacements per atom) in a volume of 10 5 lm3 . It should be stressed that the number of displacements depends on the threshold displacement energies of the lattice atoms, and that these are not well known for a complex mineral like phlogopite. The number of displaced atoms thus does not provide more than order of magnitude estimates. The threshold displacement energy does not aect the calculated ranges, however. The kinetic energies of alpha particles and recoil nuclei in Table 1 are those for transitions to the ground state of the daughter nucleus. In all cases these are by far the most probable transitions [10]. For those isotopes 238 U; 234 U; 230 Th; where transitions to excited states are also frequent (20±30%), these states are less than 250 keV above the ground state. In these cases, the recoil energy is overestimated by less than 5%. The isotopes 226 Ra; 222 Rn; 218 Po; 214 Po; 210 Po that can decay to relatively more excited states (500 to 800 keV above the ground state; overestimate up to 15%) do so with probabilities of less than 0.1%, except 226 Ra (6%). The data in Table 1 are thus accurate enough for a ®rst-order calculation of the recoil-track size distribution. The decay of excited states resulting from both beta-decay and alpha-decay events is accompanied by the emission of gamma radiation. The ensuing gamma recoil is not energetic enough to permanently displace the isotope from its lattice position; neither do the gamma particles produce lattice defects [8]. In a calculation of the size of alpha-recoil tracks, recoils from intervening beta-decay events (beta recoils) or de-excitation of the daughter nuclei (gamma recoils) can thus be neglected. In particular, after the two successive beta decays from 234 Th to 234 U (Fig. 2), the daughter 234 U occupies the same lattice position as the parent 234 Th. The rate of these processes is also several orders of magnitude higher than those of the following alpha decay. In consequence, these beta decays neither aect recoil-track size nor its rate of increase as a function of time. A further apparent complication arises from the fact that the uranium series shows complex branching at 218 Po and beyond (Fig. 2). But, as it is acceptable for the present calculation to neglect branching ratios <0:1%
(grey in Fig. 2), the entire uranium series reduces to a single non-branching chain (black in Fig. 2). In principle, the beta decays beyond 218 Po cannot be ignored from the point of view of the rate of increase of the alpha-recoil track size, because the decay constants are of a comparable order of magnitude as those for alpha decay. The aim of this calculation is, however, to determine the growth of recoil-track size in the time range 0±1 Ma. On this timescale both beta and alpha decay beyond 218 Po is quasi-instantaneous, and the exact values of the decay constants are of no consequence. Finally, 238 U not only disintegrates by alpha decay but also by spontaneous nuclear ®ssion. The energy released in uranium ®ssion amounts to 200 MeV, the larger part of which (170 MeV) is transferred as kinetic energy to the two nuclear fragments. The ®ssion fragments produce local damaged regions along their trajectories, through both electronic and elastic interactions with the lattice atoms. The formation of ®ssion tracks involves in the order of 105 displacements due to elastic collisions, over a volume of 10 3 lm3 , or somewhat less than 0.1 dpa. Ionisation along the particles' trajectories leads to additional atomic displacements through an ion-explosion spike or related mechanism [11], resulting in a much higher overall damage density. Uranium ®ssion can however be ignored here because the decay constant for ®ssion of 238 U [kF 8:46 10 17 a 1 ; [12]] is 106 times less than that for alpha decay [ka 1:551 10 10 a 1 ; [13]]. Etched ®ssion tracks can also be distinguished from etched recoil tracks under the microscope, and thus have no eect on recoil track dating. In conclusion, it is sucient to consider just a single chain of 8 successive alpha decays to calculate the size distribution of recoil tracks resulting from the decay of the uranium series isotopes to a good approximation. 3. Calculation The calculation assumes that the direction in which an alpha particle is expelled from a parent nucleus is entirely determined by intra-nuclear
R. Jonckheere, K. Gogen / Nucl. Instr. and Meth. in Phys. Res. B 183 (2001) 347±357
factors, and thus is random with respect to crystallographic axes of the host mineral. In this case the direction in which the daughter nucleus recoils is also random. Further, it is assumed that the range of the recoiling nucleus does not depend on the direction of recoil. This neglects possible eects due to anisotropy of the host crystal, but allows to use a simple binary collision approximation method to calculate the recoil ranges. Thus, when 238 U emits an alpha particle, the remaining nucleus 234 Th] recoils in an arbitrary direction over a distance determined by its kinetic energy, the composition of the crystal and by random collision eects. The same happens when 234 U emits an alpha particle, etc. After a number of decays the nucleus has moved in a zigzag pattern through the lattice (Fig. 3). The ®rst decay forms the recoil track [14±16]. Subsequent decays increase its size but do not add to the number of tracks. For the present purpose, the size of a latent track is de®ned as the maximum distance between the positions taken in by the decaying nucleus and depends on the starting isotope and on the decay-chain length.
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For each recoil, the mean vector range Ri;m and straggling Si were calculated with the computer code SRIM [17]. The isotope masses and energies were taken from Table 1. The speci®c density of phlogopite was set at 2:8 g cm 3 [18]; the threshold displacement energy of all its lattice atoms was set at 15 eV. The results are reported in Table 1. In order to calculate the recoil track size for successive recoils, an orthogonal reference frame oxyz is introduced, centred on the position of the 238 U nucleus (Fig. 3). Plane angles b1 and c1 are generated at random such that they de®ne a random orientation in space. The actual vector range of the ®rst recoil R1 is generated at random in such a way that it has a normal probability distribution with mean R1;m and standard deviation S1 . The co-ordinates of the position where the recoil nucleus has come to rest are calculated from R1 ; b1 and c1 . This de®nes the starting position of the second recoil. The position where the nucleus comes to rest after the second recoil is calculated in the same way from R2 ; b2 and c2 , and so on for following recoils. Finally the recoil track size is calculated from these co-ordinates.
Fig. 3. The zigzag motion of successive recoil nuclei through the mineral lattice. As shown on the right, the actual position of the recoil nucleus after each recoil is assumed to be normally distributed about its mean vector range.
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Other decay chains than that starting with 238 U need to be considered. At the time of phlogopite crystallisation from a melt, i.e. at the time 238 U is built into the crystal lattice, the other uraniumseries isotopes are also incorporated into the lattice at dierent positions. Each of them, except for stable 206 Pb, is at the head of a partial decay chain that produces recoil tracks. It is thus necessary to add together the recoil tracks resulting from all these chains. The partial chains need not be modelled separately. A representation of a partial chain is obtained by a translation of the reference frame oxyz to the position of its starting isotope in Fig. 3, illustrating the fact that the recoil-track size distributions for all partial chains can be calculated from the co-ordinates obtained from the ®rst Monte-Carlo simulation. Fig. 4 shows the results
Fig. 4. The recoil-track size distributions as a function of the number of recoils for each of the partial decay chains, i.e. starting with each of the alpha-decaying uranium-series isotopes. Each distribution is the result of modelling 10.000 tracks.
for the chains starting with 238 U; 234 U; 230 Th; 226 Ra; 222 Rn; 218 Po; 214 Po and 210 Po. The relative number of recoil tracks produced by the dierent partial chains depends on their relative initial concentrations. It is reasonable to assume that the uranium-series isotopes are in secular equilibrium in the melt and that they are built into the phlogopite lattice with equal probability, i.e. that no fractional crystallisation takes place. In this case, their relative initial concentrations are ®xed, and the relative number of recoil tracks of any length contributed by each partial decay chain at each point in time follows from the law of radio-active decay. The result in Fig. 5 shows that only the four longest living members of the uranium-series contribute signi®cantly to the total number of recoil tracks in the time range 0±1 Ma. This implies that the necessary condition for the result to be valid is somewhat less strict than that assumed before. Secular equilibrium and absence of fractionation during phlogopite crystallisation must only apply to 238 U; 234 U; 230 Th and 226 Ra, on condition that 222 Rn; 218 Po; 214 Po and 210 Po are not enriched by an order of magnitude or more. Importantly, this means that it makes no dierence
Fig. 5. The number of recoil tracks of any length contributed by each of the partial decay chains, indicated by its starting isotope, as a function of time. The vertical scale is in arbitrary units. The partial decay chains that are not shown contribute a negligible number of recoil tracks.
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whether the noble gas 222 Rn remains mixed with the other uranium-series isotopes in the melt or escapes to a volatile phase. The relative number of recoil tracks with chain lengths of 1; 2; . . . decays, produced by a particular starting isotope, i.e. within a particular partial decay chain, at a given point in time, is given by the relative concentrations of the daughters of this isotope. For example, for the decay chain starting with 238 U, the concentration of 234 U determines the number of recoil tracks with chain length 1, the concentration of 230 Th that of recoil tracks with
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chain length 2, etc. It has been shown above that, to a good approximation, the uranium series can be reduced to a single, non-branching decay chain (Fig. 2). In this case, the Bateman equations [9,19] can be used to calculate the time-dependent concentration of each daughter isotope. These concentrations, expressed as a function of decay-chain length, are shown in Fig. 6 for the decay chains starting with 238 U; 234 U; 230 Th and 226 Ra. The products of the ratios in Figs. 5 and 6 give the relative number of recoil tracks of a particular length produced by a particular starting isotope as
Fig. 6. The relative number of recoil tracks resulting from 1; 2; . . ., etc. recoil events, shown as a function of time for the decay chains starting with the four most stable uranium series isotopes, 238 U; 234 U; 230 Th; 226 Ra. Recoil tracks resulting from a number of successive decays that is not indicated are negligible over the entire time interval.
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Fig. 7. The size distribution of alpha-recoil tracks within a volume of phlogopite mica (volume tracks), produced by the uranium-series isotopes in the time interval between 0 and 1 Ma. The vertical scale is arbitrary.
a function of time. The time-dependent size distribution of recoil tracks is therefore obtained by adding together the size distributions in Fig. 4 in the time-dependent ratios given by these products. The result is shown in Fig. 7.
4. Results Fig. 7 shows that the time-dependent size distribution of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, is built up of two sub-populations. The ®rst population consists of tracks that are the result of a single alpha decay. They have a relatively narrow range of sizes with a mean of 30 nm. The second population for the most part consists of tracks that are the result of the complete decay of 238 U and 234 U to stable 206 Pb. They have a broad range of sizes with a mean of 125 nm. The ®rst population increases non-linearly with time and saturates around 1 Ma, whereas the second shows approximately linear growth in the entire time interval between 100 ka and 1 Ma (Fig. 8).
Fig. 8. The total number of recoil tracks (all tracks), the number of tracks resulting from a single recoil (size < 65 nm) and the number of tracks resulting from multiple recoils (size > 65 nm), shown as a function of time. The vertical scale is arbitrary.
The probability of a recoil track intersecting the surface increases in proportion to its size. The relationship between the size distribution of volume tracks and that of surface tracks is however not straightforward, because this probability also depends on the track shape. In the case of linear defects, such as ®ssion tracks, the areal density
R. Jonckheere, K. Gogen / Nucl. Instr. and Meth. in Phys. Res. B 183 (2001) 347±357
q R and volume density N R of tracks with length R are related through q R 12R N R [11]. For spherical defects it is q R R N R. The general formula for recoil tracks may be written as q R S RR N R, wherein S R is a factor between 1/2 and 1. In ®rst approximation, most recoil tracks will more closely resemble spherical than linear defects because the probability of two or more successive recoils in the same direction is small. Fig. 9 shows the size distribution of surfaceintersecting tracks in phlogopite assuming that S R 1 for all R. The remaining small peak at 30 nm will be less important than that which appears in Fig. 9, because it corresponds to recoil tracks that are the result of a single decay. The tail at >150 nm will be also less important because tracks of this size involve at least four recoils in the same direction. On the other hand, the trajectory of a recoil nucleus is, in general, not a straight line and a considerable part of the damage is created by secondary cascades far from the particle's trajectory. It is thus to be expected that at 30 nm and >150 nm: 1=2 < S R < 1, whereas S R 1 in the intervening interval. Fig. 9 is thus an acceptable ®rst-order approximation of the size distribution of recoil tracks intersecting the surface of phlogopite minerals. Its numerical integration is
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expected to yield the number of recoil tracks per unit area that can be etched and counted under a microscope as a function of the age of the mineral [20]. The etched recoil tracks in Fig. 1 also show a range of sizes. This is not a re¯ection of the sizedistribution of the latent tracks, but, instead, results from the continual addition of new tracks to the population of etched tracks during etching. Because of the small size of latent recoil tracks, even a small amount of surface etching causes a signi®cant number of tracks situated below the original surface to give rise to etch pits, that are identical to those of tracks intersecting the original surface except for their smaller diameter. In this case, the size distribution of etched tracks is uniform instead of bimodal, in agreement with experiment [20]. If the size distribution of etched recoil tracks bears no relation to that of the latent tracks, it is not possible to distinguish under the microscope etched tracks resulting from a single decay from those resulting from successive decays. It would indeed be advantageous for recoil-track dating to count only the latter, because their density is a quasi-linear function of time whereas that of tracks resulting from a single decay is a sublinear function of time, but this is probably not
Fig. 9. The size distribution of alpha-recoil tracks intersecting a cleaved surface of phlogopite mica (surface tracks), produced by the uranium-series isotopes in the time interval between 0 and 1 Ma. The vertical scale is arbitrary.
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possible with existing techniques. Even if the different segments of tracks resulting from successive decays gave rise to separate etch pits, which is not considered to be the case [2,3], these would be grouped so closely together as to be indistinguishable under the microscope. The continual addition of new tracks during etching has two further consequences. First, irrespective of the total etching time, part of the tracks will always escape observation as a result of the small initial diameters of the newly added tracks. Second, after prolonged etching, due to both their increase in size and number, the etched tracks will overlap so that distinguishing and counting them becomes dicult. These problems illustrate the importance of a reliable etching model that allows to relate the observed areal densities of etched tracks to the volumetric densities of latent tracks, and thus to the age of the mineral. A model is being developed, and will be discussed in a future paper [20]. The question of the interconnectedness of recoil tracks is not addressed at this stage, because, contrary to the probability of recoil tracks intersecting a ¯at surface, the probability of mutually intersecting recoil tracks is not amenable to a simple analytic approximation. There also remains the question of the speci®c isotopes and minerals to be considered [20]. The present results predict, however, that the interconnectedness of recoil damage in mineral host phases for the disposal of nuclear waste will be a complex function of time, because both the volume density and the size distribution of recoil tracks are complex, non-linear functions of time. 5. Conclusion The present paper proposes a Monte-Carlo method for calculating the time-dependent size distribution of alpha recoil tracks in minerals. In the case of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, the size distribution comprises two distinct track-populations. The ®rst has a mean size of 30 nm and standard deviation of 5 nm, and consists of tracks that are the result of a single alpha decay. The second has a broad range
of sizes with a mean of 125 nm and standard deviation 50 nm, and consists of tracks that are the result of the complete decay of 238 U and 234 U to stable 206 Pb. The ®rst population saturates at around 1 Ma, whereas the second shows approximately linear growth. The size distribution of recoil tracks produced by these isotopes in other minerals will be the same, except for a scaling of the size-axis by a factor equal to the ratio of the vector ranges of the recoil nuclei in phlogopite and the mineral considered. The size distribution of recoil tracks produced by other isotopes can be calculated on a case-by-case basis. The present method applies to all cases where the initial isotopic composition is known. The proposed method oers some de®nite prospects. First, it enables us to calculate the number of recoil tracks intersecting a unit surface of natural minerals as a function of their age. In mica, these tracks can be etched and counted under an optical microscope. This allows to determine the age of the mineral. As stressed in the discussion, a reliable etching model is however a precondition for recoil-track dating. Second, it provides a basis for calculating the interconnectedness of recoil damage of mineral host phases for the disposal of high-level nuclear waste, and thus for developing theoretical models to predict their long-term leaching behaviour. Third, the interconnectedness of recoil tracks, and its evolution as a function of time, is expected to aect the retention of radiogenic noble gases in natural minerals dated with the K±Ar, Ar±Ar and U/Th±He methods. These applications do, however, require further development of the present method. Acknowledgements We are indebted to an unknown referee for the valuable comments, which enabled us to improve the manuscript before publication. References [1] W.H. Huang, R.M. Walker, Science 155 (1967) 1103. [2] S.R. Hashemi-Nezhad, S.A. Durrani, Nucl. Tracks 5 (1981) 189.
R. Jonckheere, K. Gogen / Nucl. Instr. and Meth. in Phys. Res. B 183 (2001) 347±357 [3] S.R. Hashemi-Nezhad, S.A. Durrani, Nucl. Tracks 7 (1983) 141. [4] G.A. Wagner, Altersbestimmung von jungen Gesteinen und Artefakten, Enke Verlag, Stuttgart, 1995. [5] S.R. Hashemi-Nezhad, Nucl. Instr. and Meth. B 142 (1998) 98. [6] K. G ogen, Ph.D. Thesis, Heidelberg, 1999. [7] K. G ogen, G.A. Wagner, Chem. Geol. 166 (2000) 127. [8] W.J. Weber, R.C. Ewing, C.R.A. Catlow, T. Diaz de la Rubia, L.W. Hobbs, C. Kinoshita, Hj. Matzke, A.T. Motta, M. Nastasi, E.K.H. Salje, E.R. Vance, S.J. Zinkle, J. Mater. Res. 13 (1998) 1434. [9] G. Faure, Principles of Isotope Geology, Wiley, New York, 1987. [10] N.E. Holden, CRC Handbook of Chemistry and Physics, 1996, pp. 11±38. [11] G.A. Wagner, P. Van den haute, Fission-Track Dating, Kluwer Academic Publishers, Dordrecht, 1992.
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[12] D. Galliker, E. Hugentobler, B. Hahn, Helv. Phys. Acta 43 (1970) 593. [13] A.H. Jaey, K.F. Flynn, L.E. Glendenin, W.C. Bentley, A.M. Essling, Phys. Rev. C 4 (1971) 1889. [14] S. Katco, Science 166 (1969) 382. [15] T. Hashimoto, K. Kido, H. Sugiyama, T. Sotobayashi, Nucl. Instr. and Meth. 150 (1978) 509. [16] T. Hashimoto, H. Sugiyama, T. Sotobayashi, Nucl. Tracks 4 (1980) 263. [17] J.F. Ziegler, J. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [18] F. Klockmann, Lehrbuch der Mineralogie (revised and extended by P. Ramdohr and H. Strunz), Enke Verlag, Stuttgart, 1978. [19] H. Bateman, Proc. Cambr. Phil. Soc. 15 (1910) 423. [20] K. G ogen, R. Jonckheere, In preparation.
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A Monte-Carlo calculation of the size distribution of latent alpha-recoil tracks Raymond Jonckheere b
a,*
, K ursßad G ogen
b
a Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany Forschungsstelle Archaometrie, Heidelberger Akademie der Wissenschaften, Karlstrasse 4, D-69117 Heidelberg, Germany
Received 6 March 2001; received in revised form 26 April 2001
Abstract This paper proposes a Monte-Carlo method for calculating the time-dependent size distribution of alpha-recoil tracks in minerals. The results show that, in the case of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, the size distribution comprises two distinct track-populations. The ®rst has a mean size of 30 nm and standard deviation of 5 nm and consists of tracks that are the result of a single alpha decay. The second has a broad range of sizes with a mean of 125 nm and standard deviation 50 nm and consists of tracks that, for the most part, result from complete decay of 238 U and 234 U to stable 206 Pb. The ®rst population saturates at around 1 Ma, whereas the second shows approximately linear growth. On the basis of the present results, it becomes possible to calculate both the number of recoil tracks intersecting a unit surface of natural minerals and the interconnectedness of recoil damage as a function of time, which has direct implications for alpha-recoil track dating of mica and for the prediction of the long-term behaviour of mineral host phases for the disposal of high-level nuclear waste. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 34.50.)s; 91.70.)c; 28.91.Kw; 23.60.+e; 02.70.Uu Keywords: Phlogopite; Recoil-track size; Recoil-track dating; Nuclear-waste disposal
1. Introduction Natural minerals contain trace amounts of uranium, thorium and their radioactive daughters. The primary mode of decay of several of these
* Corresponding author. Tel.: +49-6221-516-337; fax: +496221-516-633. E-mail address: [email protected] (R. Jonckheere).
isotopes is through emission of an alpha particle. Each alpha decay sets free several MeV, which are shared as kinetic energy by the alpha particle (103 keV) and the recoiling daughter nucleus (102 keV). The alpha particle loses its energy through electronic interactions with the lattice atoms over most of its 20 lm range, and produces 102 isolated atomic displacements through direct nuclear interactions at the end. The recoil nucleus loses its energy through nuclear interactions over a range 30±40 nm, producing a highly localised
0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 0 6 9 1 - 7
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displacement cascade resulting in 103 lattice defects, which, together, constitute an alpha-recoil track. Alpha-recoil tracks are thus the products of radioactive decay. Their number per unit volume increases with time, and is thus a measure for the age of the mineral [1±7]. In minerals of the mica group, the recoil tracks intersecting the cleavage planes can be revealed by etching (Fig. 1). The etch ®gures can be observed with a microscope, and the number of tracks intersecting a unit surface can thus be measured. One of the problems in recoil-track dating, like in ®ssion-track dating, is to relate the areal densities of etched, surface-intersecting tracks to their volumetric densities. The size of the recoil tracks is a key parameter in this relationship. This paper proposes a method for calculating the time-dependent size distribution of alpha-recoil tracks resulting from 238 U-decay, using a Monte-Carlo technique. The accumulation of alpha-recoil tracks is also an important factor aecting the long-term stability of mineral host phases for the disposal of high-level nuclear waste. It is well known that the leach rates of these minerals increase with their degree of metamictisation, i.e. with the number of alpha-recoil tracks per unit volume [8]. The integrity of these minerals is not much aected as long as alpha-recoil tracks constitute only isolated areas of lattice damage within the crystal. How-
Fig. 1. Triangular recoil tracks in a cleaved internal surface of phlogopite mica, etched for 6 min in 40% HF, at 25°C.
ever, as the number of recoil tracks per unit volume increases with time, the interconnectedness of the damaged areas and their connections with the surface will also increase, thus increasing the macroscopic leach rate. To estimate this interconnectedness, it is necessary to know both the recoil-track density and the recoil-track size distribution as a function of time. Thus the method for calculating the recoil-track size distribution proposed here also oers the prospect of theoretically predicting the behaviour of mineral host phases for the disposal of high-level nuclear waste. It is clear that the recoil-track size distribution will depend on the composition of the mineral. The present calculations were made for the mineral phlogopite KMg3 j F; OH2 jAlSi3 O10 . Phlogopite is not a candidate host phase for nuclear waste disposal, but it has proved the most suitable mineral for recoil-track dating [6,7]. This choice does not constitute a severe restriction because the recoil-track size distribution in other minerals is obtained by scaling of the size-axis. In accordance with the choice of mineral, the recoil-track size distribution is calculated for the uranium-series isotopes, i.e. the principal recoil-track producing isotopes in natural minerals. 2. Assumptions The most abundant Uranium isotope in natural minerals is 238 Uh 0:992743; 9. The alpha decay of 238 U is the start of the uranium-series decay chain, involving 8 alpha and 6 beta decays, and ending in stable 206 Pb (Fig. 2). The energy released in beta decay is of the order of 1 MeV, most of which is shared by the beta particle and an antineutrino. Because of their low mass, beta particles do not eciently transfer energy to the lattice atoms; the daughter nuclei are also almost never energetic enough to recoil permanently from their lattice positions. In general, beta decay produces of the order of 0.1 atomic displacements per event [8]. The energy released in alpha decay of the uranium-series isotopes ranges from 5 to 8 MeV. Between 70 and 100 keV is communicated as kinetic energy to the daughter nucleus, the remainder is carried away by the alpha particle
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Fig. 2. The uranium-series isotopes (after 9). Neglecting branches with branching ratios < 0:001 (grey) reduces the uranium-series to a single, non-branching chain of 14 successive alpha and beta decays (black). Decay constants in a 1 .
(Table 1). The alpha particles dissipate most of their energy in ionisation processes but undergo enough elastic collisions, in particular towards the end of their trajectories, to produce several hundred isolated atomic displacements. The alpha particles thus contribute to a diuse background of isolated lattice defects, but do not lead to the
formation of localised areas of etchable lattice damage. The recoil of the less energetic but far more massive (atomic mass: 206±234) daughter nuclei accounts for most of the total number of atomic displacements. Elastic collisions with the lattice atoms produce highly localised cascades, resulting in 1000±2000 displacements [8] or some-
Table 1 The range and straggling of the recoiling daughter nuclei resulting from alpha decay of the principal uranium-series isotopes, calculated using the binary collision approximation SRIMa Parent ! Daughter
k a 1
Ealpha (MeV)
Erecoil (keV)
Rrecoil (nm)
Srecoil (nm)
Vacancies
238
1:54 10 10 2:81 10 06 8:65 10 06 4:33 10 04 6:60 1001 1:19 1005 1:33 1011 1:82 1000
4.20 4.77 4.68 4.78 5.49 6.00 7.69 5.31
71.7 82.9 82.8 98.9 110.1 143.7 101.1 103.1
29.5 32.3 32.2 36.4 39.1 46.5 36.5 37.2
6.0 6.6 6.8 7.6 8.3 9.8 7.8 8.1
1243 1423 1417 1660 1823 2316 2316 1715
U ! 234 Th U ! 230 Th 230 Th ! 226 Ra 226 Ra ! 222 Rn 222 Rn ! 218 Po 218 Po ! 214 Bi 214 Po ! 210 Pb 210 Po ! 206 Pb 234
a k: decay constant; Ealpha : kinetic energy of the alpha particle; Erecoil : kinetic energy of the recoil nucleus; Rrecoil : range of the recoil nucleus; Srecoil : range-straggling of the recoil nucleus; vacancies: lattice vacancies per recoil ion.
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what less than 0.1 dpa (displacements per atom) in a volume of 10 5 lm3 . It should be stressed that the number of displacements depends on the threshold displacement energies of the lattice atoms, and that these are not well known for a complex mineral like phlogopite. The number of displaced atoms thus does not provide more than order of magnitude estimates. The threshold displacement energy does not aect the calculated ranges, however. The kinetic energies of alpha particles and recoil nuclei in Table 1 are those for transitions to the ground state of the daughter nucleus. In all cases these are by far the most probable transitions [10]. For those isotopes 238 U; 234 U; 230 Th; where transitions to excited states are also frequent (20±30%), these states are less than 250 keV above the ground state. In these cases, the recoil energy is overestimated by less than 5%. The isotopes 226 Ra; 222 Rn; 218 Po; 214 Po; 210 Po that can decay to relatively more excited states (500 to 800 keV above the ground state; overestimate up to 15%) do so with probabilities of less than 0.1%, except 226 Ra (6%). The data in Table 1 are thus accurate enough for a ®rst-order calculation of the recoil-track size distribution. The decay of excited states resulting from both beta-decay and alpha-decay events is accompanied by the emission of gamma radiation. The ensuing gamma recoil is not energetic enough to permanently displace the isotope from its lattice position; neither do the gamma particles produce lattice defects [8]. In a calculation of the size of alpha-recoil tracks, recoils from intervening beta-decay events (beta recoils) or de-excitation of the daughter nuclei (gamma recoils) can thus be neglected. In particular, after the two successive beta decays from 234 Th to 234 U (Fig. 2), the daughter 234 U occupies the same lattice position as the parent 234 Th. The rate of these processes is also several orders of magnitude higher than those of the following alpha decay. In consequence, these beta decays neither aect recoil-track size nor its rate of increase as a function of time. A further apparent complication arises from the fact that the uranium series shows complex branching at 218 Po and beyond (Fig. 2). But, as it is acceptable for the present calculation to neglect branching ratios <0:1%
(grey in Fig. 2), the entire uranium series reduces to a single non-branching chain (black in Fig. 2). In principle, the beta decays beyond 218 Po cannot be ignored from the point of view of the rate of increase of the alpha-recoil track size, because the decay constants are of a comparable order of magnitude as those for alpha decay. The aim of this calculation is, however, to determine the growth of recoil-track size in the time range 0±1 Ma. On this timescale both beta and alpha decay beyond 218 Po is quasi-instantaneous, and the exact values of the decay constants are of no consequence. Finally, 238 U not only disintegrates by alpha decay but also by spontaneous nuclear ®ssion. The energy released in uranium ®ssion amounts to 200 MeV, the larger part of which (170 MeV) is transferred as kinetic energy to the two nuclear fragments. The ®ssion fragments produce local damaged regions along their trajectories, through both electronic and elastic interactions with the lattice atoms. The formation of ®ssion tracks involves in the order of 105 displacements due to elastic collisions, over a volume of 10 3 lm3 , or somewhat less than 0.1 dpa. Ionisation along the particles' trajectories leads to additional atomic displacements through an ion-explosion spike or related mechanism [11], resulting in a much higher overall damage density. Uranium ®ssion can however be ignored here because the decay constant for ®ssion of 238 U [kF 8:46 10 17 a 1 ; [12]] is 106 times less than that for alpha decay [ka 1:551 10 10 a 1 ; [13]]. Etched ®ssion tracks can also be distinguished from etched recoil tracks under the microscope, and thus have no eect on recoil track dating. In conclusion, it is sucient to consider just a single chain of 8 successive alpha decays to calculate the size distribution of recoil tracks resulting from the decay of the uranium series isotopes to a good approximation. 3. Calculation The calculation assumes that the direction in which an alpha particle is expelled from a parent nucleus is entirely determined by intra-nuclear
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factors, and thus is random with respect to crystallographic axes of the host mineral. In this case the direction in which the daughter nucleus recoils is also random. Further, it is assumed that the range of the recoiling nucleus does not depend on the direction of recoil. This neglects possible eects due to anisotropy of the host crystal, but allows to use a simple binary collision approximation method to calculate the recoil ranges. Thus, when 238 U emits an alpha particle, the remaining nucleus 234 Th] recoils in an arbitrary direction over a distance determined by its kinetic energy, the composition of the crystal and by random collision eects. The same happens when 234 U emits an alpha particle, etc. After a number of decays the nucleus has moved in a zigzag pattern through the lattice (Fig. 3). The ®rst decay forms the recoil track [14±16]. Subsequent decays increase its size but do not add to the number of tracks. For the present purpose, the size of a latent track is de®ned as the maximum distance between the positions taken in by the decaying nucleus and depends on the starting isotope and on the decay-chain length.
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For each recoil, the mean vector range Ri;m and straggling Si were calculated with the computer code SRIM [17]. The isotope masses and energies were taken from Table 1. The speci®c density of phlogopite was set at 2:8 g cm 3 [18]; the threshold displacement energy of all its lattice atoms was set at 15 eV. The results are reported in Table 1. In order to calculate the recoil track size for successive recoils, an orthogonal reference frame oxyz is introduced, centred on the position of the 238 U nucleus (Fig. 3). Plane angles b1 and c1 are generated at random such that they de®ne a random orientation in space. The actual vector range of the ®rst recoil R1 is generated at random in such a way that it has a normal probability distribution with mean R1;m and standard deviation S1 . The co-ordinates of the position where the recoil nucleus has come to rest are calculated from R1 ; b1 and c1 . This de®nes the starting position of the second recoil. The position where the nucleus comes to rest after the second recoil is calculated in the same way from R2 ; b2 and c2 , and so on for following recoils. Finally the recoil track size is calculated from these co-ordinates.
Fig. 3. The zigzag motion of successive recoil nuclei through the mineral lattice. As shown on the right, the actual position of the recoil nucleus after each recoil is assumed to be normally distributed about its mean vector range.
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Other decay chains than that starting with 238 U need to be considered. At the time of phlogopite crystallisation from a melt, i.e. at the time 238 U is built into the crystal lattice, the other uraniumseries isotopes are also incorporated into the lattice at dierent positions. Each of them, except for stable 206 Pb, is at the head of a partial decay chain that produces recoil tracks. It is thus necessary to add together the recoil tracks resulting from all these chains. The partial chains need not be modelled separately. A representation of a partial chain is obtained by a translation of the reference frame oxyz to the position of its starting isotope in Fig. 3, illustrating the fact that the recoil-track size distributions for all partial chains can be calculated from the co-ordinates obtained from the ®rst Monte-Carlo simulation. Fig. 4 shows the results
Fig. 4. The recoil-track size distributions as a function of the number of recoils for each of the partial decay chains, i.e. starting with each of the alpha-decaying uranium-series isotopes. Each distribution is the result of modelling 10.000 tracks.
for the chains starting with 238 U; 234 U; 230 Th; 226 Ra; 222 Rn; 218 Po; 214 Po and 210 Po. The relative number of recoil tracks produced by the dierent partial chains depends on their relative initial concentrations. It is reasonable to assume that the uranium-series isotopes are in secular equilibrium in the melt and that they are built into the phlogopite lattice with equal probability, i.e. that no fractional crystallisation takes place. In this case, their relative initial concentrations are ®xed, and the relative number of recoil tracks of any length contributed by each partial decay chain at each point in time follows from the law of radio-active decay. The result in Fig. 5 shows that only the four longest living members of the uranium-series contribute signi®cantly to the total number of recoil tracks in the time range 0±1 Ma. This implies that the necessary condition for the result to be valid is somewhat less strict than that assumed before. Secular equilibrium and absence of fractionation during phlogopite crystallisation must only apply to 238 U; 234 U; 230 Th and 226 Ra, on condition that 222 Rn; 218 Po; 214 Po and 210 Po are not enriched by an order of magnitude or more. Importantly, this means that it makes no dierence
Fig. 5. The number of recoil tracks of any length contributed by each of the partial decay chains, indicated by its starting isotope, as a function of time. The vertical scale is in arbitrary units. The partial decay chains that are not shown contribute a negligible number of recoil tracks.
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whether the noble gas 222 Rn remains mixed with the other uranium-series isotopes in the melt or escapes to a volatile phase. The relative number of recoil tracks with chain lengths of 1; 2; . . . decays, produced by a particular starting isotope, i.e. within a particular partial decay chain, at a given point in time, is given by the relative concentrations of the daughters of this isotope. For example, for the decay chain starting with 238 U, the concentration of 234 U determines the number of recoil tracks with chain length 1, the concentration of 230 Th that of recoil tracks with
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chain length 2, etc. It has been shown above that, to a good approximation, the uranium series can be reduced to a single, non-branching decay chain (Fig. 2). In this case, the Bateman equations [9,19] can be used to calculate the time-dependent concentration of each daughter isotope. These concentrations, expressed as a function of decay-chain length, are shown in Fig. 6 for the decay chains starting with 238 U; 234 U; 230 Th and 226 Ra. The products of the ratios in Figs. 5 and 6 give the relative number of recoil tracks of a particular length produced by a particular starting isotope as
Fig. 6. The relative number of recoil tracks resulting from 1; 2; . . ., etc. recoil events, shown as a function of time for the decay chains starting with the four most stable uranium series isotopes, 238 U; 234 U; 230 Th; 226 Ra. Recoil tracks resulting from a number of successive decays that is not indicated are negligible over the entire time interval.
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Fig. 7. The size distribution of alpha-recoil tracks within a volume of phlogopite mica (volume tracks), produced by the uranium-series isotopes in the time interval between 0 and 1 Ma. The vertical scale is arbitrary.
a function of time. The time-dependent size distribution of recoil tracks is therefore obtained by adding together the size distributions in Fig. 4 in the time-dependent ratios given by these products. The result is shown in Fig. 7.
4. Results Fig. 7 shows that the time-dependent size distribution of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, is built up of two sub-populations. The ®rst population consists of tracks that are the result of a single alpha decay. They have a relatively narrow range of sizes with a mean of 30 nm. The second population for the most part consists of tracks that are the result of the complete decay of 238 U and 234 U to stable 206 Pb. They have a broad range of sizes with a mean of 125 nm. The ®rst population increases non-linearly with time and saturates around 1 Ma, whereas the second shows approximately linear growth in the entire time interval between 100 ka and 1 Ma (Fig. 8).
Fig. 8. The total number of recoil tracks (all tracks), the number of tracks resulting from a single recoil (size < 65 nm) and the number of tracks resulting from multiple recoils (size > 65 nm), shown as a function of time. The vertical scale is arbitrary.
The probability of a recoil track intersecting the surface increases in proportion to its size. The relationship between the size distribution of volume tracks and that of surface tracks is however not straightforward, because this probability also depends on the track shape. In the case of linear defects, such as ®ssion tracks, the areal density
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q R and volume density N R of tracks with length R are related through q R 12R N R [11]. For spherical defects it is q R R N R. The general formula for recoil tracks may be written as q R S RR N R, wherein S R is a factor between 1/2 and 1. In ®rst approximation, most recoil tracks will more closely resemble spherical than linear defects because the probability of two or more successive recoils in the same direction is small. Fig. 9 shows the size distribution of surfaceintersecting tracks in phlogopite assuming that S R 1 for all R. The remaining small peak at 30 nm will be less important than that which appears in Fig. 9, because it corresponds to recoil tracks that are the result of a single decay. The tail at >150 nm will be also less important because tracks of this size involve at least four recoils in the same direction. On the other hand, the trajectory of a recoil nucleus is, in general, not a straight line and a considerable part of the damage is created by secondary cascades far from the particle's trajectory. It is thus to be expected that at 30 nm and >150 nm: 1=2 < S R < 1, whereas S R 1 in the intervening interval. Fig. 9 is thus an acceptable ®rst-order approximation of the size distribution of recoil tracks intersecting the surface of phlogopite minerals. Its numerical integration is
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expected to yield the number of recoil tracks per unit area that can be etched and counted under a microscope as a function of the age of the mineral [20]. The etched recoil tracks in Fig. 1 also show a range of sizes. This is not a re¯ection of the sizedistribution of the latent tracks, but, instead, results from the continual addition of new tracks to the population of etched tracks during etching. Because of the small size of latent recoil tracks, even a small amount of surface etching causes a signi®cant number of tracks situated below the original surface to give rise to etch pits, that are identical to those of tracks intersecting the original surface except for their smaller diameter. In this case, the size distribution of etched tracks is uniform instead of bimodal, in agreement with experiment [20]. If the size distribution of etched recoil tracks bears no relation to that of the latent tracks, it is not possible to distinguish under the microscope etched tracks resulting from a single decay from those resulting from successive decays. It would indeed be advantageous for recoil-track dating to count only the latter, because their density is a quasi-linear function of time whereas that of tracks resulting from a single decay is a sublinear function of time, but this is probably not
Fig. 9. The size distribution of alpha-recoil tracks intersecting a cleaved surface of phlogopite mica (surface tracks), produced by the uranium-series isotopes in the time interval between 0 and 1 Ma. The vertical scale is arbitrary.
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possible with existing techniques. Even if the different segments of tracks resulting from successive decays gave rise to separate etch pits, which is not considered to be the case [2,3], these would be grouped so closely together as to be indistinguishable under the microscope. The continual addition of new tracks during etching has two further consequences. First, irrespective of the total etching time, part of the tracks will always escape observation as a result of the small initial diameters of the newly added tracks. Second, after prolonged etching, due to both their increase in size and number, the etched tracks will overlap so that distinguishing and counting them becomes dicult. These problems illustrate the importance of a reliable etching model that allows to relate the observed areal densities of etched tracks to the volumetric densities of latent tracks, and thus to the age of the mineral. A model is being developed, and will be discussed in a future paper [20]. The question of the interconnectedness of recoil tracks is not addressed at this stage, because, contrary to the probability of recoil tracks intersecting a ¯at surface, the probability of mutually intersecting recoil tracks is not amenable to a simple analytic approximation. There also remains the question of the speci®c isotopes and minerals to be considered [20]. The present results predict, however, that the interconnectedness of recoil damage in mineral host phases for the disposal of nuclear waste will be a complex function of time, because both the volume density and the size distribution of recoil tracks are complex, non-linear functions of time. 5. Conclusion The present paper proposes a Monte-Carlo method for calculating the time-dependent size distribution of alpha recoil tracks in minerals. In the case of recoil tracks in phlogopite, produced by the uranium-series isotopes in the time interval between 0 and 1 Ma, the size distribution comprises two distinct track-populations. The ®rst has a mean size of 30 nm and standard deviation of 5 nm, and consists of tracks that are the result of a single alpha decay. The second has a broad range
of sizes with a mean of 125 nm and standard deviation 50 nm, and consists of tracks that are the result of the complete decay of 238 U and 234 U to stable 206 Pb. The ®rst population saturates at around 1 Ma, whereas the second shows approximately linear growth. The size distribution of recoil tracks produced by these isotopes in other minerals will be the same, except for a scaling of the size-axis by a factor equal to the ratio of the vector ranges of the recoil nuclei in phlogopite and the mineral considered. The size distribution of recoil tracks produced by other isotopes can be calculated on a case-by-case basis. The present method applies to all cases where the initial isotopic composition is known. The proposed method oers some de®nite prospects. First, it enables us to calculate the number of recoil tracks intersecting a unit surface of natural minerals as a function of their age. In mica, these tracks can be etched and counted under an optical microscope. This allows to determine the age of the mineral. As stressed in the discussion, a reliable etching model is however a precondition for recoil-track dating. Second, it provides a basis for calculating the interconnectedness of recoil damage of mineral host phases for the disposal of high-level nuclear waste, and thus for developing theoretical models to predict their long-term leaching behaviour. Third, the interconnectedness of recoil tracks, and its evolution as a function of time, is expected to aect the retention of radiogenic noble gases in natural minerals dated with the K±Ar, Ar±Ar and U/Th±He methods. These applications do, however, require further development of the present method. Acknowledgements We are indebted to an unknown referee for the valuable comments, which enabled us to improve the manuscript before publication. References [1] W.H. Huang, R.M. Walker, Science 155 (1967) 1103. [2] S.R. Hashemi-Nezhad, S.A. Durrani, Nucl. Tracks 5 (1981) 189.
R. Jonckheere, K. Gogen / Nucl. Instr. and Meth. in Phys. Res. B 183 (2001) 347±357 [3] S.R. Hashemi-Nezhad, S.A. Durrani, Nucl. Tracks 7 (1983) 141. [4] G.A. Wagner, Altersbestimmung von jungen Gesteinen und Artefakten, Enke Verlag, Stuttgart, 1995. [5] S.R. Hashemi-Nezhad, Nucl. Instr. and Meth. B 142 (1998) 98. [6] K. G ogen, Ph.D. Thesis, Heidelberg, 1999. [7] K. G ogen, G.A. Wagner, Chem. Geol. 166 (2000) 127. [8] W.J. Weber, R.C. Ewing, C.R.A. Catlow, T. Diaz de la Rubia, L.W. Hobbs, C. Kinoshita, Hj. Matzke, A.T. Motta, M. Nastasi, E.K.H. Salje, E.R. Vance, S.J. Zinkle, J. Mater. Res. 13 (1998) 1434. [9] G. Faure, Principles of Isotope Geology, Wiley, New York, 1987. [10] N.E. Holden, CRC Handbook of Chemistry and Physics, 1996, pp. 11±38. [11] G.A. Wagner, P. Van den haute, Fission-Track Dating, Kluwer Academic Publishers, Dordrecht, 1992.
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[12] D. Galliker, E. Hugentobler, B. Hahn, Helv. Phys. Acta 43 (1970) 593. [13] A.H. Jaey, K.F. Flynn, L.E. Glendenin, W.C. Bentley, A.M. Essling, Phys. Rev. C 4 (1971) 1889. [14] S. Katco, Science 166 (1969) 382. [15] T. Hashimoto, K. Kido, H. Sugiyama, T. Sotobayashi, Nucl. Instr. and Meth. 150 (1978) 509. [16] T. Hashimoto, H. Sugiyama, T. Sotobayashi, Nucl. Tracks 4 (1980) 263. [17] J.F. Ziegler, J. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [18] F. Klockmann, Lehrbuch der Mineralogie (revised and extended by P. Ramdohr and H. Strunz), Enke Verlag, Stuttgart, 1978. [19] H. Bateman, Proc. Cambr. Phil. Soc. 15 (1910) 423. [20] K. G ogen, R. Jonckheere, In preparation.