Length distributions of fission tracks in thick crystals

Nuclear Track Detection, Vol. 2, pp. 181-189

0145 224X/78/1201-0181 $02.00/0

I Pergamon Press Ltd. 1978. Printed in Great Britain

LENGTH DISTRIBUTIONS OF FISSION TRACKS IN THICK CRYSTALS M . DAKOWSKI*

Centre d'Etudes Nucl6aires, Saclay, 91190 Gif-sur-Yvette, France

(Received 3 January 1978; in revised form 17 March 1978)

Abstract-Distributions of projected lengths, full lengths, depth projections and inclination angles of fission tracks emerging from thick layers of dielectrics are calculated. Results are compared with experimental distributions for the (0001) plane of apatite crystals, for different annealing rates. Special attention is paid to error sources common to similar methods. These results seem to imply that certain errors in age correction methods based on the length distributions ought to be considered.

1. I N T R O D U C T I O N THE FULL recordable lengths of fission tracks generated in the bulk of the dielectric can be measured using the track-in-track (TINT) and/or track-incleavage (TINCLE) methods introduced by Lal et al. (1968). Using this technique Bhandari et al. (1971) were able to measure not only the mean track length of both fission fragments, but also the shapes of the length distributions. However, the method commonly used for density measurements and for some length estimations or measurements consists of measuring the characteristics of tracks appearing after etching on the polished surface of the thick dielectric (thick in comparison with the maximal track length). Such measurements are often quoted as "length distribution data", without specifying if one is really dealing with "projected length distribution" or "full length distribution". Published results on different parameters of track length distributions in thick dielectrics are scarce and will be discussed in Section 4. The characteristics of distributions of the surface tracks seem to be important, as one tries to estab-

lish the correction methods for the dating of annealed minerals on this basis. The aim of this work is to study systematically the different experimental track distributions (Section 3) and to compare the results with the predictions of a simple model presented in Section 2. Sources of different systematic errors are discussed along with their implications on the age correction methods (Section 4).

2. A N A L Y T I C A L F O R M U L A E F O R T H E DIFFERENT PARAMETERS OF TRACKS ORIGINATING IN "THICK CRYSTALS" 2.1 Assumptions A simple model will provide one with the following distributions: distribution distribution distribution distribution

of of of of

projected lengths, N(l), full lengths, N(r), inclination angle, N(0), depth lengths, N(Z).

*Permanent address: Institute of Nuclear Research, Swierk, 05-400 Otwock, Poland. The experimental part of this work was done there. 181

182

M. DAKOWSKI

These calculations are merely quoted to allow a critical examination of possible error sources and/or over-simplifications. Let us begin with stating explicitly some assumptions: For tracks before etching (latent tracks): (1) The distribution of tracks through the volume of the dielectric is homogeneous. (2) The distribution of tracks is isotropic in the crystal. (3) The total track length of both fission fragments is constant. For etching conditions: (4) Bulk etching velocity vB is negligible as compared to the track etching velocity vr (i.e. vB/v T ~ 1 ). (5) The full length of the latent track is revealed. In other words, the full recordable track length is equal to the range of two fission fragments, R. (6) Even the shortest tracks are measurable (i.e. optical microscopic limitations are neglected).

2.2 Presentation of model Let us consider all tracks before etching from a definite surface S, whose dimensions are much larger than R. Let us define a layer of thickness dZ at depth - Z , such that all the tracks whose lower ends are in this layer can be gathered in one point, as can be seen in Fig. 1. The lower ends can be treated as point-like objects. Any corrections (on the probability of finding and counting of these objects) caused by the different inclinations of the tracks are then unnecessary. The assumption of homogeneity of the tracks on the surface of Z =const. is used to justify this gathering.

~ i

n(tR "m")

FIG. 1. "'Porcupine" geometry for the model calculations. The first two assumptions are essential and selfexplanatory. The third is justified on the basis of relatively small dispersion of total kinetic energy in fission. The fourth assumption is justified experimentally for a carefully chosen etchant and time of etching, as explained in Section 3. The implications will be discussed later. The next assumption (5) together with assumption 3, replaces the relative dispersion of recordable tracks by zero. Measurement of Bhandari et al. (1971) for the neutron-induced tracks yields the dispersion ~ = l~tm for a mean track length of 16/~m. For the heavily annealed tracks, the relative dispersion is bigger. The final experimental distributions for surface-etched tracks (as in our case) should therefore be compared with the present calculations, after convolution of the calculated results by the appropriate dispersion. For the qualitative conclusions of this paper this seemed, however, to be unnecessary. The comparisons with experiment are justified even if the recordable lengths are not equal, but only roughly proportional to the latent track lengths. The last assumption (6) is clearly an unrealistic one; the limitations of it will be discussed in detail in the experimental section.

In this way, we get a porcupine-like hemispheric representation of all tracks from d e p t h - Z , not only of those which could be subsequently etched. The central cross-sectional area of this porcupine in the (/, Z) plane is shown in Fig. 1, where other variables are also defined. Let us now consider the density n' of the upper ends of these tracks on the surface of the hemisphere with radius R. It is equivalent to the track density no per solid angle d~. Using the isotropy assumption, one gets for the track density: dn' no ~ = const. = ~ - ,

(1)

where no is the total track density. The value of this constant was chosen arbitrarily. As the track length distribution does not depend on azimuthal angle qS, because of the axial symmetry of the problem, one obtains, using relation dff2= sin0d0d05 and integrating equation (1) over ~b: d n ( O ) = ~ d n ' =no sinO dO.

(2)

LENGTH DISTRIBUTIONS O F FISSION TRACKS IN THICK CRYSTALS This relation is also used for the different distributions of etched tracks, which are defined as the lower part of the tracks, situated between the elementary layer dZ and the surface of the dielectric (Z=0), as shown in Fig. 1.

183

from equation (3), dN dr

Z n°r2'

Integrating over Z between limits of (0, r), 2.3 Results (a) Projected-length distribution N(I). Let us consider all etched tracks with projected length equal to I. Such tracks are physically possible for

fo;dZ rZ

N(r)=

no

no Z 2 ~

=5- f r o

Finally, the rectangular distribution 0>Z~

- x//~-

12

N, , fno/2 (see Fig. 1); the sign of Z is negative for all Z = const, surfaces inside the dielectric, For Z=const., the function dn(0) (relation (2)) can be expressed in terms of length l. Since d n = n o d(cos0)

and

dN dl

?n d(cos0) - , ~(cos0) dl

dN

trJ=~0

for0R

(5)

is obtained. (c) The inclination-angle distribution N(0). This distribution is obtained by integrating equation (2) over Z between limits (0, R cos0):

0~oosg N(O)=no Jo

ZI

dl = n0-(Z 2 + 12)3/2.

(3)

To obtain the projected length distribution, one should integrate equation (3) over Z between the limits 0 > Z > _ x ~ R -2-12"

sinOdZ=noRsinOcosO no Z

N (O) = ~- R sin20.

(6)

(d) Depth-length distribution N(Z). Noticing that equation (3) is symmetric in the l and Z parameters, integrating over l, one obtains for the depth distribution a triangle similar to (4):

The assumption of homogeneity in the Z-direction is here used,

Zl N(l)=noffCte-t2(ZZ+12)3/2dZ no l

o ~//R2-12

(Z 2 + 12) 1/2 Finally one gets the triangle-shaped distribution

The final distributions (4)-(7) are obtained on the assumption that the velocity of surface etching vB =0. For the more realistic case when the bulk etching velocity is non-zero, but the velocity of track etching is much higher (O
3. EXPERIMENTAL SET-UP AND RESULTS

(b) Full-etched-length distribution N(r). As

3.1 Experimental conditions

dN -dr -=

The experimental part of this work was done with the commonly used big crystals of Durango apatite. The samples were cut along the (0001)

c~N dl

Ol dr

and

dl r l = ( r Z - Z Z ) 1/2, dr - (r 2 - - Z 2 ) 1/2'

184

M. DAKOWSKI

plane, where the tracks have large, nearly circular openings (tack-shaped tracks) (see Fig. 2). Such a shape implies at least two different etching velocities. Only one crystallographic plane was chosen for this study to diminish the influence of considerable anisotropy on the properties of track revealing and annealing (Green and Durrani, 1977).

and was used in the calculations of full projected lengths l and of full length r. F r o m the integer number of divisions lp, counted by the scanner, the projected length /c(gm) was calculated using a constant scaling-factor. Afterwards the following formulae and values were used (cf. Fig. 2): Z d = 0.7 #m,

"~//4//////~...,.

>17/////////////////'//

1/°i 12\

t

>/

- '_/

0 = arc tan(lc/Z), O' = ~/2 - O,

(8)

I = Ic + Z d • tan0, r=[(Zq_Z,l)2 + 12]1/2.

The measurements were carried out for four different annealing times (and annealing ratios), for an annealing temperature of 315"C. Below, the annealing ratios are expressed in terms of relative densities P/Po.

Flo. 2. Schematic diagram of the track with the explanation of different values used or measured in the paper.

Sample G - annealing ratio p/Po = 1.00, (non-annealed crystals) Sample F

At the start of this experiment all the samples had the fossil tracks completely annealed. The samples had also the same initial density of neutroninduced tracks. The etching conditions were chosen to be 4.5 min in 1 ~,~; H N O 3 at 23 °C, to minimize the ratio of bulk-etching velocity vB to the track-etching velocity yr. The estimated half-angle of the narrow cones of the tracks has a value 0 c < 5 . As 0 c = a r c sin(vr/Vc), one estimates the vr/v c ratio <0.1 Such a low value of vB/v ~ justifies the comparisons of the experimental results with the model calculations which neglect the influence of bulk-etching velocity. Measurements were made with a standard microscope using an objective of 100 x magnification. For length measurements, immersion oil was used. It did not change the projected lengths, as the lower side of the objective was flat. During the length measurements for each track, the following quantities were measured (see Fig. 2): (i) track length projection lp; (ii) track depth Z. The broad "entrance" of the tack-shaped track was estimated to have the mean depth of Z d = 0 . 7 # m . This value was later assumed constant for all tracks

annealing ratio P/po=0.85,

Sample H---annealing ratio P/Po =0.70, Sample K

annealing ratio P/Po =0.58.

Each of these four samples contained about 6-8 crystals. Runs of 60 to 150 tracks were repeated in a random fashion for different crystals and different samples unknown to the scanner. This was done in order to avoid the systematic errors caused by fatigue and boredom of the scanner. After rejection of biased results (three out of 37 runs) these data were summed up into the final results.

3.2 Results and their uncertainties The results of the measurements of I, Z, r and 0' ( = ~ / 2 - 0 ) distributions are presented, respectively, in Figs 3 6 for the above-mentioned different annealing ratios. For comparison, the distributions are shown resulting from the model presented in Section 2. In presentation butions (Fig. 6), which is defined direction and the

of experimental dip-angle distrione uses the angle 0 ' = 9 0 ' - 0 , as the angle between the track surface of the crystal. Values of 0'

LENGTH DISTRIBUTIONS O F FISSION TRACKS IN THICK CRYSTALS are calculated from the relation tan O ' = Z / l c (see Fig. 2). The threshold of counting is about 1-2#m in the projected-length distributions. In Fig. 3, the threshold is shifted by the transformations mentioned above ( l p ~ l c ~ l ) . In the depth (Z) distribution, the tendency to overestimate the shortest lengths is stronger than in the projected-length l distributions (cf. Fig. 4, especially H and K). It is caused by a different, more subjective way of measurement (depth screw of a microscope instead of simple counting as in the case of l). The threshold is then higher in the depth (Z) and hence also in the full length (r) distributions. For all these distributions the threshold does not depend perceptibly on the annealing ratio in the studied range of P/Po. These effects not only change the length distributions, but shift the dip angle distribution in the direction of more inclined angles. Compare, in Fig. 6G, the simple prediction with the experimental distribution obtained for fresh tracks. For the annealed samples, as the vn/v r ratio seems to rise (see, e.g., Fleischer et al., 1975), the differences go in the same direction, but are more pronounced. It is consistent with the annealing results of Green and Durrani (1977) for apatite. They showed that tracks are preferentially shortened in directions parallel to the (0001) plane compared with those at 90" to (0001). For bad etching conditions, the higher vB/v T ratio and bigger cones of tracks increase the difficulties of definition of track. Our experience shows that the effective /mi, value is then higher, and the slope of the distribution at the low lengths is less steep. These factors depend more strongly on the subjective factors. The results of correlations between the relative densities and different parameters of distributions are shown in Fig. 7. F]O. 3. Projected length (1) distributions for the samples with different annealing ratios (see text for values applicable to samples G-K): histograms: experimental results, straight lines: shapes of distributions predicted by "'vB/v 7 = 0 " model, adjusted to guide the eye through the histograms. The extrapolated maximal values for the distributions are marked by arrows. They were estimated from the comparison between histograms and model predictions. Length distributions (Figs 3 5) have a channel width equal to 1 #m.

---1

185

Sample K T.= 6 3 0

Joo

5o

844

Sample ~-

H

\ too

\\

50

o fO0 Sample

F

~- 549

__

Sample

G

_

tO0

50

J

I

5

I0

L,

~&M

~-

r5

~'m~x

186

M, DAKOWSKI \

\ I00

- -

K I00

lz

6O

L

L

50 I

0 I00 \

I

I

50 t00 N

0

F-

V

5O

I

.

,

I

.

.

,

.

I I

]

F

50 N

I 0 0 --

\

0 6

I00 50

I I

f

50

I I

0 150

\

0

r-

I

,

5

r, FIG.

I00

1

10

15 ~m

5. T o t a l l e n g t h (r) d i s t r i b u t i o n s , F o r e x p l a n a t i o n s , see c a p t i o n to Fig. 3.

additional

50

5

I0 Z,/~m

~ 15 Z~

FIG. 4, D e p t h

(Z) distributions, For additional n a t i o n s , see c a p t i o n to Fig. 3.

expla-

LENGTH DISTRIBUTIONS O F FISSION TRACKS IN THICK CRYSTALS The depth parameters Z and Zm. x show weaker correlation with relative density P/Po than do the projected length parameters T and lmaX. The maximal values lr.ax and Zmax show more linear correlations with P/Po than do the corresponding mean values.

187

The first effect may be caused by the depthmeasurement errors and annealing ratio differences discussed above; the second shows the difficulties in the interpretation of the commonly used mean values T, even for samples with the track populations having only one annealing ratio.

I

~oo

-

I

-

0,9 K

5 0 -

J,'

0,8

I

)
r

I

/j// "

0.7

x

I 0.6

o.= 5

0

/

I 0.6

II

I 07

I 0.8

o

L mox

El

Zmax

I 0,9

P/Po

N

o

I

I

I

FIG. 7. Correlation between relative density P/Po and different length variables (marked X/Xo) /x and thick broken line: the mean depth Z; [] and thin broken line: the maximal depth Zm,x; • and thick solid line: the mean projected length T; © and thin solid line: the maximal projected length lmax.

t --I~

o

I

i

i

i i Joo

--

I

G

50

O 0:5

I0 "m/4

~.5 x"/2

0', radians

FIG. 6. Dip angle (0'=~/2-0) distributions. The thin line on histogram G denotes prediction of the model.

4. DISCUSSION Our results can be compared only with the few published distributions. Bhaladari et al. (1971) showed in their Fig. 4 the experimental full-length distribution for apatite and compared it with the theoretical estimation (rectangular spectrum, as in equation (5)). Fleischer et al. [1967) mention "uniform distribution.., of observed track lengths", without going into details. Somogyi and Nagy (1972) mention the model-dependence of dip-angle distribution identical to equation (6). Nishida and Takashima (1975) measured the "pit length" for zircons without specifying if they measured the projected or the full length. They claimed that "the distributions of pit length...showed the normal distribution with the standard deviation (a) of 1.4~2

188

M. DAKOWSKI

microns". Both our Figs 3 and 5 are completely different from this picture. However comparison of projected length distributions from different papers is more instructive. Wagner and Storzer often showed projected length distributions (see, e.g., Wagner (1972), Wagner and Storzer (1975) and also Fig. 8, reproduced from Wagner and Storzer (1972)). Their distributions for induced tracks have symmetric, nearly triangular form. They claim that for the natural tracks the lower peak of the distribution is due to the thermally annealed part of the tracks. If so, the clear distinction between annealed and fresh tracks would be possible on the basis of projected length distributions. Our distributions are evidently different from the distributions reported by the Wagner Storzer group. Let us analyse the shapes of the projected length distribution shown in Figs 3 and 8 and its implications on the density dependence of the average or maximum lengths. Without going into details of the annealing mechanism, let us only assume that the annealing process changes monotonically the length of every track and that it is not a macroscopically discontinuous process, e.g. disappearing of one long track without changing the length of the other. In the simple model presented above, the integration of equation (4) and also of equation (9), for the limits of integration equal to 0 and R = I,,~,, I

'

160 --

I

F

r,

~

I

'

[

Apotite T5a 15 sec, 6 5 % HNOTroom t e m p e r a t u r e

r--' I ~-~

--5

,

I

0

I

I.

-

[.

_J

120

I

r--,,

r--, I I

.

.

.

[

.

~

Spontaneous Induced --

--

F-"

ao

;

I

:~ 4 0

j

r-J 0

yields the linear correlation between l . m / I o and P/Po- As p ~ n o 1..... therefore any decrease in the length must result in a decrease in track density. This is true for the uniJbrm distribution of full length r. Let us suppose that the nearly symmetric shape of the distribution of unannealed tracks in Fig. 8 reflects the physical reality, i.e. that there are very few short tracks. In contrast to the above situation, during the first of successive annealings this spectrum should shift in the direction of shorter tracks without appreciable loss of events. The dramatic change of density is expected afterwards, especially when tile maximum of distribution will enter into the non-measurable zone of shortest tracks: it takes place when the effective thickness of the layer containing etchable tracks diminishes abruptly. This implies the concave behaviour of the l / l o = f l P / P o ) correlation, which is ill contrast with the experiment. The experimental shape of this correlation was reported to be linear for muscovite by Bigazzi (1967), and for apatite by M~.rk et al. (1973). It was found to be convex for apatites in the abovementioned papers of Wagner and co-workers. Figure 7 shows nearly linear behaviour of l and I..... but much more convex behaviour of Z and Zm~~ in the analysed range of annealings. The convex shape is consistent with different experimental errors discussed by Storzer (1972). This analysis suggests that on the distribution of Fig. 8, the tracks normal or nearly normal to the surface (i.e. with very short projections) are counted only partially and, perhaps, not at all. This bias seems to be different for an annealed and an unannealed sample. It should be stressed that for glasses, the convex curve for the dependence of pit diameters on the relative density was explained by Somogyi and Nagy (1972) for a model calculation with high value of c./~-r. 5. C O N C L U S I O N S

I 4

8

12

16

20

P r o j e c t e d l e n g t h of e t c h c h o n n e l s on ( 0 0 0 1 ) I unit = 052/*

FIG. 8. Typical projected length distribution of fossil and induced tracks in apatites as reported by Wagner and Storzer (1972).

The results of this paper could be summarized as follows : The presented length and angular distributions of surface tracks for unannealed apatite crystals are compatible with the model calculations which assume vs/t~r=O. The differences between the simple

LENGTH

DISTRIBUTIONS

OF FISSION TRACKS IN THICK CRYSTALS

model predictions a n d experimental distributions are p r o b a b l y caused mainly by higher vB/vr value a n d by the errors inherent in the optical microscopy technique. These differences increase with growing degree of annealing. Systematic errors a n d the errors caused by the n o n - h o m o g e n e i t y of u r a n i u m distributions in the crystals are c o m p a r a b l e with, or often bigger than, the s t a n d a r d statistical errors ( ~ 2 5 %) in the type of density m e a s u r e m e n t s reported. Lower parts of our length distributions (see Figs 3, 4 a n d 5) do not perceptibly depend on the annealing rates. The m a x i m u m values of distributions I. . . . Zmax are less subject to systematic errors t h a n the average values. The values of /max iS then a more precise a n d more sensitive measure of the changes of the length distributions as a function of annealing rate, for minerals with low value of vs/v r. F o r the low v~/v T ratio a n d for good scanning conditions the correlation between projected length a n d density should be near to linear. The mixture of annealed a n d unannealed tracks does not show any noticeable double peaks in the length distributions, especially in the projected length distributions (compare in Fig. 3 the distributions G with K or G with H). The suggestion that one can distinguish the annealed a n d fresh tracks on this basis is s o m e w h a t doubtful. Moreover, for our shape of the projected length distributions a n d for the sample with mixed tracks o f different annealing ratios, the average projected length 7-loses its sense as a measure of annealing ratio. The pit diameter m e a s u r e m e n t (e.g. for glasses with higher value of vB/vr) does not contain more information a b o u t the track than do the length and angle m e a s u r e m e n t for crystals. This implies that the above d o u b t s apply also to the age correction procedures based on the measurements and separations of two peaks in the pit-diameter distributions. Acknowledgements Thanks are due to Mr J. Sobolewski for help in the analytical part of this work; to Mr R.

189

Rawicki for skillful and tireless scanning of the samples; and to Dr E. Cieslak for fruitful doubts and stimulating discussions. I am also grateful to Dr J. Burchart for his valuable contribution in the experimental phases of this work and for many constructive discussions. Last, but not least, I thank Dr Beil for a critical reading of the manuscript.

REFERENCES Bhandari N., Bhat S. G., Lal D., Rajagopalan G., Tamhane A. S. and Venkatavaradan V. S. (1971) Fission fragment tracks in apatite: recordable track lengths. Earth Planet. Sci. Lett. 31, 191-199. Bigazzi (3. (1967) Length of fission tracks and age of muscovite samples. Earth Planet. Sci. Lett. 3, 434438. Fleischer R. L., Price P. B. and Walker R. M. (1975) Nuclear Tracks in Solids, University of California Press. Fleischer R. L., Price P. B., Walker R. M., Maurette M. and Morgan G. (1967) Tracks of heavy primary cosmic rays in meteorites. J. geophys. Res. 72, 355366. Green P. F. and Durrani S. A. (1977) Annealing studies of tracks in crystals. Nucl. Track Detection 1, 33-39. Lal D., Muralli A. V., Rajah R. S., Tamhane A. S., Lorin J. C. and Pellas P. (1968) Techniques for proper revelation and viewing of etch-tracks in meteoritic and terrestrial materials. Earth Planet. Sci. Lett. 5, 111 119. M~irk E., Pahl M., Purtscheller F+ and M~irk T. D. (1973) Thermische Ausheilung von Uran-Spaltspuren in Apatiten, Alterskorrekturen und Beitr~ige zur Geothermochronologie. Tschermaks miner, petrogr. Mitt. 20, 131 154. Nishida T. and Takashima Y. (1975) Annealing of fission tracks in zirkons, Earth Planet. Sci. Lett. 27, 257 264. Somogyi G. and Nagy M. (1972) Remarks on fission-track dating in dielectric solids. Radiat. Effects 16, 223231. Storzer D. (1972) Fission track etch kinetics in glasses. Trans. Am. nucl. Soc. 15, 129. Wagner G. A. (1972) Spaltspurenalter yon Mineralen und Naturlischen Glasern: eine Ubersicht+ Fortschr. Miner. 49, 114-145. Wagner G. A. and Storzer D. (1972) Fission track length reductions in minerals and the thermal history of rocks. Trans. Am. nucl. Soc. 15, 127-128. Wagner G. A. and Storzer D. (1975) Spaltspuren und ihre Bedeutung ffir die thermische Geschichte des Odenwaldes. Aufschluss, Sonderband 27, 79-85.