Nuclear Instruments and Methods in Physics Research B34 (1988) 74-80 North-Holland, Amsterdam

TlXE S~~~ROSCOPIC TRACK KINETIC THEORY AND THE VA~~ONAL P~NCIPLE

R. MAZZEI, J.C. GRASSO, O.A. BERNAOLA, J.C. BOURDIN and G. SAINT MARTIN

Departamento de Radiobiofogia, Comision lvacional de Energia Atomica, Buenos Aires, Argentina

Received 8 June 1987 and in revised form 11 March 1988

The variational principle was used to obtain the general equations for chemical etching track profiles inside and outside the physical damage region produced by energetic ions in Makrofol E. The track profiles developed at very short etching times were used to obtain the velocity profiles in order to perform computer determinations of track profiles for longer etching times. Track profiles obtained by means of the variational principle and the submicroscopic track kinetic theory are in good agreement, and compare favorably with experimental results.

1. Lntroduction

The energy transferred by ions passing through matter is carried outside the ion incidence axis mainly by secondary electrons. Therefore, the microscopic dose decreases radially from the incidence axis in generating the physical damage region. When the physical damage region is considered in describing the etching of ion tracks in solid state nuclear track detectors, it is neces- sary to evaluate the radial dependence of the etch velocity in order to analyze the theoretical track profile.

The submicroscopic track kinetic theory (STKT) re- ported by Mazzei et al. [l-4], includes the radial depen- dence of the track velocity and is in agreement with experimental data. This theory estimates the envelope of the different wave fronts generated in each arbitrary axis (ys) parallel to the ion incidence axis. The track profile is the envelope of the envelopes of the wave fronts generated in the different y0 axes. The theory is also in agreement with the classical track kinetic theory (CTKT) for long etching times (bulk region) IS].

For very short etching times, the chemical etching track profile is defined by a function similar to the one describing the track velocity profile [4]. Thus, in order to estimate the track profile for every etching time, we obtain the track velocity radial function from experi- mental data for “new born” track profiles.

The track profile evolution can also be analyzed applying the variational principle [6]. In this way the least time trajectories can be evaluated for several etch- ing times. The trajectories are considered from each arbitrary point of the track profile to any point of the non-etched detector surface. In section 1 of the results in the present work, the variational principle is applied to obtain chemical etching theoretical track profiles

0168-583X/88/$03.50 Q Blsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

using a specific radial dependence of the material etch rate (I$( y)). In section 2 this theory is used to obtain CTKT and STKT results. In section 3 we compare the theoretical track profiles from STKT and variational principle, both obtained using the same Vr( y); experi- mental data is shown to be fitted with both theories.

2. Materials and methods

Beams of 2 MeV ‘Li (charge state +l), 2 MeV 160 (charge state + 1) and 6 MeV “C (charge state +2) were used. The irradiations were performed at the Brookhaven National Laboratory tandem Van de Graaff accelerator. Makrofol E (Bayer AG) foils of 300 pm thickness were used as solid state nuclear track detec- tors. After the irradiation the foils were chemically etched in a PEW solution [7] (15 g KOH + 40 g CH,CH,OH + 45 g H,O) at 33.0 (+ 0.5) o C during an etching time of a few seconds. In order to obtain the track replicas the technique described in [1,3,8] was applied and the track profiles were analyzed with a Philips 300 electron microscope.

3. Results and discussion

3.1. The variational principle track profiles

We consider a track which enters the detector material with an incidence axis x at position y = 0, where y is an axis perpendicular to the ion incidence and x is the distance below the original surface. The time necessary to reach a point (xi, yi) of the track profile following the arbitrary trajectory y(x) and start-

R Mazrei et al. / Submicroscopic track kinetic theory 15

Fig. 1. Characteristic miaimum-time trajectory starting at (0, Yd.

ing at a point (0, yO) of the non-etched detector surface is

t = gj-(x) dx,

where

f( X

) = { 1+ b'(~flzy2

VT/T[Y(X>I * V,( y ) is the track velocity as a function of the coordi- nate y and y’(x)= dy/dx. Cylindrical symmetry around the ion incidence axis (x) is assumed. The chemical attack track profile is analyzed on the x, y coordinates with z = 0 [l]. Sufficiently short etching times were considered in order to be able to take into account the track velocity variations onIy along the y axis 143. In fig. 1 we show the boundary conditions as: a) y(x,) ==yl (the point (x,, yi) belongs to the track profile) b) The left hand end-point can move freely along the y axis (x = 0).

Applying the variational principle to the etching time, we can obtain the Euler-Lagrange equation for the function ( f) IS]

and

dy dx X=0=

0.

These results show that at the beginning the y(x) curves are perpendicular to the detector surface.

From eq. (3)

$ + $(ln VT){1 +y’2) = 0

or

(5)

(6)

From eqs. (4) and (6) we have

p is the distance to the ion incidence axis. For each ya eq. (7) corresponds to the burns time trajectory starting at the point (0, yO) (fig. I). Eq. (1) establishes the time necessary to reach the point (xi, y,) along the ~nimum trajectory.

Assuming a certain V, function, eq. (7) can be solved numericaIly for each ye value. Then, for each y value, the point (x, y> of the trajectory is obtained and f(x) in eq. (1) can be evaluated. The process concludes when a value of x that satisfies eq. (1) is reached for the etching time considered. This way, for a given etching time, the last point of the minimum time trajectory can be compared with the corresponding point of the experimental track profile.

3.2. Variational principle applied to rely short or to large etching times

In this section we show relation of the variational principle results to the short time behavior (the STKT description) and to long time behavior (the CTKT description) of the track etching. From eq. (1) we have

dx vT ---““.=

dt (1 +y’2)1/z.

For short etching times (t = 0) x = 0 and using eq. (4)

dx

dt *=* I = vThl>-

In other words, when f = 0 and for each ys, the trajec- tories are pe~endicul~ to the detector surface and the height-variations of the track profile coincide with the track velocity profile. This result is in agreement with that from the STKT [4].

On the other hand, far enough from the ion inci- dence axis and taking into account that the track veloc- ity is a radially decreasing function, we have: V,(p) = V,, where V, is the etching velocity of the bulk detector material in the absence of ion track damage. From eq. (7) and the geometrical relations shown in fig. 1

or

Therefore

vb sin B= -. vTIT(h)

76 K. Mazzei et al. / Submicroscopic track kinetic theory

Then, for large y values the minimum time trajectories are almost straight lines with the slope given by eq. (8). As ;Vr decreases as yc increases, the straight line of steepest slope is obtained for ye tending to 0 in this equation. Therefore the contribution to the track profile fur large enougb y values is given by the trajectories correspond@ to y,, values near yO = 0. In this limit sin 8 = f/V (with v= V,(O)/V,).

In order to pruve that the minimum time trajectory for ya = 0 is the straight line y = 0 we use the eq. (7) and have

(9)

For any possible trajectory between the points (0,O) and (x1, 0) the eq. (9) states that dy/dx f y_o = 0. Con- sidering that dy/dx 1 y= o = 0 and that the arbitrary point (x1, 0) can be taken as close to (0,O) as desired, the unique trajectory satisfying these conditions is y = 0. Con&dering y = 0 in eq. (1)

t=

There x1 = V,(O) t. Thus the point that belongs both to the track profile

and to the ion incidence axis is shown to correctly translate with 5/#) velocity.

The effective time ( f ‘) of the wave front generated by each arbitrary point (x,, ya) of the track must be taken into account to analyze the kinetics of chemical attack [1,4]

E is the total etching time, V,&) is the track velocity ori yO axis and r2 = [(x - x0)’ + (y - y,)‘]. The equa- tion indicates that the,time necessary for the chemical attack of the track to reach a preestablished positiun from (x0, ~~‘0) must be the same as the total time minus the time taken to reach point x0 with the maximum velocity V,( ya) on the y,, axis.

The STKT defines the different wave fronts gener- ated by each arbitrary point (xOJ yO) of the track, and the points belonging to the wave front envelope. The equation that describes the chemical etching kinetics is

/4]:

and the points belonging to the wave front envelope are defmed by the following conditions

From the last two equations the x0 and y, dependence can be removed and from the condition 3(x,, yO) = 0, JJ(X, t) is obtained. Also, by using aS(x,, yO),Gx, = 0 and B ( x0, yo) = 0 for each y. axis the corresponding envelope that starts on this axis was obtained

FX(.Y> Jo, t)l* The coordinate along the ion incidence axis is

x = WY& - (f WYa>l” - fu -YO)2)1/2~ W>

where

I = s YdP/W& YO

Therefore, the chemical etching track profile in the STKT is defined by the envelope of the wave front envelopes when y. is varied.

In contrast, the CTKT considers the physical damage region as a straight line (with no transverse dimensions). Under this condition and using the STKT equations [.5], we have

Imposing:

B(.q)=O and i3x= , W%> 0

0

the track profile JJ(X, 6) is obtained. In figs. 2(a)-(d) an example is shown of calculated

variational principle trajectories [eqs. (1) to (‘7)] and STKT track profil.es [eq. (lO>]. The results are over- lapped for comparison and correspond to the case of 2 MeV J6Q for etching times of 20, 35, SO and 120 s respectively.

For both theories the equation used for the track velocity as a function of y is [4]:

V,(y) = v&-t- (A,/%) e-aye]

and the data from Mazzei et al. ]4] was used for the A/h, I3 and C vahtes as given in caption for fig. 2.

In fig. 2 the trajectories start perpendicular to the detector surface (y axis) and curve towards the corre- sponding STKT track profile, reaching it in an ap- proximately perpendicular fashion. These trajectories correspond to the minimum principle theory. The track profile corresponding to the minimum principle calcula- tion is determined by the last point reached by each trajectory for any given etching time. A slight difference

R Mazzei et al. / Submicroscopic track kinetic theory

988

zm

I/l/ / -.e-- / I,

r-‘,Y /’ I

Fig. 2. V~a~o~al principle trajectories and STAT track profiles ~lc~ated for 2 MeV I60 for (a) 20 s; (b) 35 s; (c) 50 s; and (d) 120 s of etching time. (A/h = 1.95, B = 4.56327 X Be7 and C = 2.82).

is observed between the two theories. This ~fference increases with etching time. Nevertheless, the greatest difference is less than 10 A at an etching time of 120 s; this value is the resolution of the experimental data [4].

The smallest difference is observed for those trajec- tories that start near the ion incidence axis and for those close to the bulk region (large y values). For each etching time the greatest difference corresponds to minimum time trajectories with y values varying be- tween 100 and 200 A. Therefore the greatest difference is in the region where the track velocity has the greatest radial variations. For the shortfst etching time data (20 s) the difference is less than 1 A. This time corresponds to %ew born” tracks and the results are in agreement with the analysis of section 3.2.

Fig. Z(d) shows that the bulk region has been re- ached at this point and the track profile is approxi- mately conic, as described by the CTKT_ Even though

Table 1 Ion track y values corresponding to the point of rn~~~u~ difference between v~atio~~ theory and STKT for various track velocity parameters: A/h, C, In B. These parameters correspond, in the order given, to the physical cases of 2 MeV 7Li, 6 MeV 12C and 2 MeV Ifi0 in Makrofof E.

A/h C InB Y IAl

1.19 2.51 - 13.0 170 1.43 2.45 - 12.3 147 1.95 2.82 - 14.6 182

R. Mazzei et al. / Submicroscopic track kinetic theory

,b 1mzmmn4@l5mmn.7m

X(A)

Fig. 3. Variational principle trajectories and STKT track profile for 50 s of etching time. Calculations are for the case (a) A/h =X95, B = 6.158692 x lo-‘, C = 2.82; (b) A/h = 1.95, B = 6.158692 x 10W9, C = 2.82; and (c) A/h = 1.95, B = 6.158692 X 10y7 and C = 4.

the greatest difference is observed for the longest etch- ing time, it is important to note that for large enough values of etching time the prevailing Ys’s correspond to Y, = 0, as reported in ref. [l] and analyzed in section 3.2.

Now we will discuss the differences observed in fig. 2

and analyze, using eq. (S), under what conditions the STKT and the vacations principle theory coincide. In the STKT the wave fronts were generated considering straight line trajectories from each wave front gener- ating point (x0, Ys). However, eq. (5) shows that in general these are not minimum time trajectories because

is not equal to zero.

Using

v,(Y)=++

we have

vb +=BCyC-’ ---I .

[ 1 VT (11)

As Y tends to 0 (with C: > l), VT tends to VT(O) and therefore + tends to 0. On the other hand, when Y tends to co, V, tends to Vt, and again Cp tends to 0. Thus, good agreement is obtained for both small and large y values. Furthermore, in both theories when y is large enough to be outside the physical damage region, the chemical etching track profile can be considered as a plane wave front, as can be observed in figs. 2(a)-(c).

In order to analyze the region of maximum dif- ference we impose d+/d y = 0 and have

ByC=ln [

&z)(C-1)

I (ByC)C- (C- 1)

By iterative calculation of this equation we obtained

R Mazzei et al. / Submicroscopic track kinetic theory 19

table 1. This table shows that the main difference be- tween the STKT and variational theory is found in the 150 to 200 A region of y values, in agreement with figs. 2(a)-(d). For very short etching times, a slight dif- ference is observed because the trajectories are almost straight lines [see fig. 2(a)]. For large etching times the

prevailing y, values are those near 0 and a slight difference is obtained again [see fig. 2(d) and eq. (ll)].

In order to analyze further the difference in the track profiles between both theories we studied the variation of Cp as a function of B in eq. (ll), assuming A/h and C as constants (A/h = 1.95; C = 2.82). The maximum

Fig. for

.4. Overlapping of STKT and vtiational principle half track profiles are given on a microphotograph of an observed track 35 s of etching times for: (a) 2 MeV 160 using A/h =1.95, B = 4.563527~10~’ and C = 2.82 and (b) 6 MeV 12(

A/h = 1.43, B = 4.551744~ 1O-6 and C = 2.45.

: profile 3 using

80 R. Mazzei et ul. / Submicroscopic track kinetic theory

difference (d+/dB = 0) is obtained when B = 6.16 x

10e7 (fig. 3a). Fig. 3b is obtained with B = 6.16 x 10-g. Fig. 3(a) shows that using the B value that maximizes

the difference results in a difference of less than 6 A. Upon decreasing B by two orders of magnitude [fig. 3(b)] the difference tends to 0 (is less than 1 A), as would be expected for B tending to zero in eq. (11).

In fig. 3(c), A/h = 1.95; B = 6.16 X 1O-7 and C = 4. The variation of C from 2.82 [fig. 3(a)] to 4 [fig. 3(c)] induces a variation of the track radius from 300 to 50 A and the difference between both theories in this case is less than 1 A.

Figs. 2(a)-(c) (20, 35 and 50 s) indicate that the track radius is approximately constant with etch time, in agreement with Mazzei et al. [5]. F~he~ore, the figures show that when the etching time decrease, the x values tend asymptotically to Vt, t for large y values. Therefore, these results may be used to decide whether or not we are in the presence of a “new born” track.

Fig. 4 shows an overlapping of both theoretical results with ~crophotographs of track replicas taken in a transmission electron microscope for an etching time of 35 s. Fig. 4(a) is for the case of 2 MeV I60 and fig. 4(b) is for 6 MeV ‘*C in Makrofol E. In order to achieve an easy visualization, only the theoretical curves of half the track are shown. Since the experimental data show some dispersion in the heights of the tracks, a coincidence of theoretical and experimental heights was imposed at y = 0 [4]. These figures show good agree- ment between both theories and the experimental data. The differences here were found to be less than the resolution (10 A) of the experimental data.

It is worth mentioning that each track profile is easily visualized using only a few ye values when the STKT approach is applied. On the contrary, many yO’s are needed when the minimum principle theory is used. Furthermore, as the STKT uses simpler algorithms than the minimum principle theory and as only a slight difference between both theories was observed, the STKT is suggested as the more useful theory to outline track profiles of various ions at different etching times.

It is important to note that in all the present analysis we did not consider the chemical concentration gradi- ents that are necessary to drive the diffusion of the

etchant into the track and the etch products out of the track. Since the experimental data are in agreement with both theories, this effect is probably not significant for tracks which are as open as these are.

In conclusion, we have shown that the variational principle can be used to theoretically describe and give insight into the etch track profiles in solid state nuclear track detectors. The results of variational principle and sub~cros~pic track kinetic theories are in good agree- ment and agree with experimental observations in Makrofol E.

The authors would like to thank Dr. P. Thieberger of the BNL for the foil exposures, Dr. I. Nemirovsky for many helpful suggestions and D. Prahic and A. Petra- galli for technical assistance. G.S.M., J.C.B. and J.C.G. are holders of a fellowship of the Comision de Investi- gaciones de la Provincia de Buenos Aires, Bunge y Born Foundation and Consejo National de Investigaciones Cientificas y Tecnicas respectively. This research was partially supported by the Scien~ic-T~~olo~c~ Co- operation Program (126/X5) undertaken by National Science Foundation-Consejo National de Investiaciones Cientificas y Tecnicas.

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