The relationship between tracks in solid state nuclear track detectors (SSNTD) and the submicroscopic kinetic theory

Pergamon

Radiation Measurements, Vol. 26, No. 4, pp. 577-583, 1996 Copyright © 1996 ElsevierScience Ltd Printed in Great Britain. All rights reserved PII: S1350-4487(96)00029-7 1350-4487/96 $15.00+ 0.00

THE RELATIONSHIP BETWEEN TRACKS IN SOLID STATE N U C L E A R TRACK DETECTORS (SSNTD) A N D THE SUBMICROSCOPIC KINETIC THEORY RUBEN OMAR MAZZEI Departamento de Radiobiologia, CNEA, Av. del Libertador 8250, 1429, Buenos Aires, Argentina (Received 15 November 1992; revised 17 October 1995; accepted 5 March 1996)

Abstraet--A theoretical model for submicroscopic kinetic of track formation is described. Three theoretical methods that yield track profiles corresponding to normal incidence ions are proposed. Track profiles obtained by means of the three methods are in good agreement with each other and compare favorably with experimental results. They can be applied to describe the track development all the way from very short ("new born" tracks) to long etching times (track diameters > 1 ~tm). Copyright © 1996 Elsevier Science Ltd

1. INTRODUCTION The classical track kinetic theory (CTKT) has been described extensively (Somogyi and Slazay, 1973; Paretzke et al., 1973; Ali and Durrani, 1977; Fleischer et aL, 1975). In this, the physical damage zone is considered to be restricted to a straight line. Mazzei et al. (Mazzei et al., 1985, 1986, 1988a, b; Mazzei and Cabanillas, 1993) described a submicroscopic track kinetic theory that allows the analysis of tracks inside and outside the physical damage region. This submicroscopic evaluation is a general one, extending from the CTKT regime for large tracks to the kinetics of "new-born" tracks. The purpose of the current study is to compare results obtained using different submicroscopic evaluations and to elucidate the shape of the submicroscopic etched tracks when etch rate changes both along and perpendicular to the particle trajectory. In this work, discussion will be restricted to perpend!cular ion incidence in homogeneous and isotropic detectors. When an ion with atomic number Z and energy per nucleon E/n falls on a detector it produces a damage zone around the incidence axis X. In inorganic detectors the ion can produce atomic displacement; and in polymers it can also lead to chain scission, free radical production, excited molecule production, gas generation, etc. Moreover, post-irradiation rearrangement of damage, such as transport and diffusion processes and temperature gradients within the track, should be considered. A careful analysis of these processes should be performed considering the temperature radial profiles of the initial damage-region and their evolution in time to find the final damage-region.

So, the charged particle creates a cylindrical region easily attacked by a suitable reagent. The etching solution creates holes along and around the particle path. The way in which the damage is related to the chemical etching process is far from being understood. However, the condition that equal chemical etching corresponds to equal damage S(Z, E / n ) = S(Z', E'/n'), must be fulfilled. The CTKT considers only two quantities to find the theoretical track profiles: the etching track velocity in the damage zone, ~ and the bulk velocity outside it, ~. I~,depends on the measurement distance along the incidence ion axis. Moreover, the theoretical track profiles can be derived by applying the least-time method (Fleischer et al., 1969). On the other hand, when a radial distribution of damage from the ion incidence axis is present a radial attack rate of the chemical etching must be considered and fundamental information on the geometric distribution of damage can be found from the corresponding experimental track profiles. In view of this, a generalization of the CTKT must be made to include a radial variation of track velocity and to analyze track profile at a submicroscopic level (Mazzei et al., 1985, 1986, 1988a, b; Mazzei and Cabanillas, 1993). Therefore, it is valuable to perform a detailed analysis for the relation between radial track etching velocity, radial damage profile and the experimental track profile. The track profile evolution can also be analyzed applying the variational principle (Mazzei et al., 1988b). In this way the least time trajectories can be evaluated for several etching times. The trajectories are considered from each arbitrary point of the track profile to any point of the non-etched detector surface. 577

578

R. O. MAZZEI

Mazzei and Cabanillas (1993) described a submicroscopic evaluation that allows the analysis of tracks inside and outside the physical damage region. The CTKT equations for longer etching times are presented in Section 3.1. In Section 3.2 the submicroscopic nuclear track kinetic theory (STKT) is described (Mazzei et al., 1985, 1986). It defines different wavefronts generated at arbitrary points of the track and the points belonging to the wavefront envelope. In Section 3.3 the variation principle is applied to find--for V~(y) function--the theoretical nuclear track profile. In Section 3.4 the "law of refraction" is applied for the case V~(x,y). Therefore, by using a computational loop the etching trajectories are obtained. The equations obtained in Section 3.1 (CTKT equations) are reproduced using the formalisms described in Section 3.2. Finally, theoretical results obtained using the equations in Section 3.2 can be reproduced using equations obtained in Section 3.3. Equations in Section 3.4 allow the reproduction of track for longer etching time (Section 3.1). Moreover, an experimental result related to charge state transitions inside a solid target is reproduced using these equations.

axis; the position x = 0 is on the original detector surface. The effective time t' of the wavefront generated by each arbitrary point x0 (Fig. 1) of the track must be taken into account to analyze the kinetics of chemical attack (Fleischer et al., 1969; Somogyi and Almasi, 1980). The effective time of chemical attack in Fig. 1 is:

dx' t' = t - ~o~° V,(x')

(1)

So: t'

=

( y2 + (x - x0)-')1~-' = t -

v~

fo'° V,(x') dx'

.

(2)

A family of spheres can be found by imposing: B(xo) = -

( y'- + (x - xo)'-)".I ~° d x ' V~ + t V,(x'----3 = O.

(3) The track wall can be considered as the envelope of these wave fronts and is determined by: B(xo) = O,

dB( xo) ~Xo - O .

(4)

2. MATERIALS AND M E T H O D S A beam of 36 MeV L'C of 4 + state charge was used. Irradiation was performed at the Buenos Aires Tandem accelerator. Macrofol-E foils of 300 lam thickness were used as solid state nuclear track detectors. After irradiation the foil was chemically etched in a PeW solution (15 g KOH + 4 0 g CHsCH2OH + 45 g H,O) at 33 + 0.1°C during an etching time of 4'30". A shadowing technique was used to obtain Pt and C layers. The sample was dipped for 2 h in chloroform to dissolve the Macrofol and the replica was mounted on a copper grid. Particular care was taken to fold a portion of the replica so that some tracks clearly protrude from the borders of the replica (Mazzei et al., 1984). The track profile was analyzed in a model 300 Philips transmission electron microscope (TEM). 3. DESCRIPTION O F THE THEORETICAL MODELS 3.1. Classical track kinetic theory ( C T K T )

The CTKT considers the physical damage zone as restricted to a straight line (Mazzei et al., 1988a, b; Mazzei and Cabanillas, 1993). Figure 1 represents a theoretical track profile for a V,(x) function. The corresponding ion was considered to enter the detector along an incidence axis x at the position y = 0, y being the perpendicular to the ion incidence

3.2. Submicroscopic track kinetic theory ( S T K T ) The STKT estimates the envelope of the different wavefronts generated in each arbitrary axis (.v0) parallel to the ion incidence axis. The track profile is the envelope of the envelopes of the wavefronts generated in the different .v0 axes. The equation that describes the chemical etching kinetics--for a I4,(.v) function--is (Mazzei et al., 1985, 1986):

B(xo, .vo) = t

x0 V,( Yo)

r f' dp ( Y : Yo) J,o V,(p)

(5)

h = Vb.t - I

(o). X

/Vt(x )l

Fig. 1. Schematic view of an etch-pit profile for V,(x) increasing along the track for normal incidence.

SSNTD AND KINETIC THEORY

579

with V,(Y0) the track velocity on the y0 axis and r = - [( y - - y 0 ) 2 "~- (X -- 20)2] 1/2. The points belonging to the wavefront envelope are defined by the following conditions: B(xo, Yo) = 0,

OB(xo,Yo) = 0, OB(xo,Yo) = 0. ÙXo

(6)

Oyo

I

,_

y2 . . . . . . . L _

From this, the following equations are obtained: x = t" V,(y0) -- [FV~ - (y-),0)2] ':

(7)

t'[ V,,(Yo)] "[12"V~(Yo) -- ( Y -- yo)2]'2' 6~ Vt(Yo) 8Yo = I-" 8V'(Y°) I+ ( y -- yo).[Vt(yo)] l ~Y0

Yl

. . . . . . . . . . .

(8) D

where

xI

I( Y0, Y) = I

ff

dp V,(p) '

=

(9)

X2

X

Fig. 2. Schematic representation of a track showing the relation between (x., v ) (x_, v:) and 0, = 0(x, v,).

0

From equations (7)- (9) the theoretical track profile function x ( y , t ) can be calculated. Including the x dependence in equation (5), the following equation can be obtained:

£ v° B(xo, v ) = t -

dx ~ V~(x', )'o)

coordinates with z = 0. For short etching times, track velocity variations only along the y axis can be considered. For a V,(y) function, the time necessary to reach a point (&, y~) of the track profile following the arbitrary trajectory y(x) and starting at a point (0, Y0) of the non-etched detector surface is: [1 + ( zdyd . "Y' v ]~j'

-2 (y

f' Yo) ),

dp / " g(Xo + (Vo - p).~ x - xo \

(10) t=

(7o--;))'P'.

-

When the physical damage region is considered as a straight line, an agreement between CTKT and STKT theories should be expected even in the most general case when a dependence of x is included in the track velocity. Since V,= Vb outside the incidence axis, when Y0 = 0 the CTKT equation is obtained [equation (3)]: B ( x o , O) =

-

( y~- + ( x - x0)2) ' ~- + t v~

')(x ) ' d x

where f(x) =

v,(y(x))

(13)

Applying the variational principle to the above equation, the Euler-Lagrange equations for the function f c a n be obtained (Mazzei et al., 1988b). From these equations, Fig. 2 and dy/dx,_,, = 0, the relation between y and x is obtained: 1+

~

=

~ )

= 1 +tan-'(0)

(14)

£~" dx' g(x',0)

(11)

x =

f

'

dp

.

(15)

On the other hand for Yo ¢ 0 equation (10) is: B(x0, yo) = t

X0

V~

r

V~ "

(12)

From equation (6), x = Vb.t, equation (5). That is, a plane wavefront is obtained in agreement with the CTKT. In an unified way the chemical etching can be considered as a plane wavefront (bulk region) ore as a track profile by just imposing either Y0 4:0 or yo=0.

Assuming a certain V, function and solving equations (13)- (15) for each Y0 value the minimum time trajectories (X, yo) are obtained. For a given etching time t, the last points of the minimum time trajectories belong to the theoretical track profile. For a V,(y) function the results of variational principle and the STKT (Section 3.2) are in good agreement and agree with experimental observation in Macrofol-E (Mazzei et al., 1988b).

3.3. Variational principle applied to a V,(y)junction Figure 2 is a schematic representation of a theoretical track profile. Cylindrical symmetry around the ion incidence axis is assumed. The chemical track profile is analyzed on the x,y

3.4. Law of refraction Born and Wolf (1959) demonstrate that the "'law of refraction" can generally be derived from variational principles and that it is implicit in the

580

R . O . MAZZEI

equations resulting from it. For the etching trajectories the "law of refraction" is implicit in the Euler-Lagrange equations that result from minimizing the etching time t [equation (13)] (Mazzei and Cabanillas, 1993). For a V~(y) function and taking into account equation (14) and Fig. 2 and "law of refraction" the following equation is obtained: sinflt V,(y,) sinfl.~- V,(y,.)

(16)

Considering Fig. 2 the following equations result: Y2 = Y, + V,( y,).At.sin(O,)

(17)

x2 = x, + V,( y,).At.cos(O,)

(18)

and from equations ( 1 6 ) - (18): y~)-~.At x, = x, + (V,( V,(y,,)

y,.=y,+

Vt(y,)'At"

I1--[

~v,(,,) ..]j

(19)

l 'l

.

(20)

In this way the new point (x_,, y,) at t + A t is determined from the old point (x,, Y0 at t applying the "law of refraction" and At is as small as required. The results obtained by Mazzei et al. (1985, 1986, 1988a, b) were reproduced using equations (19) and (20) for different etching times and a V,(y) function with a lesser computational time (Mazzei and Cabanillas, 1993). Moreover, when the velocity of chemical attack varies both along and perpendicular to the heavy ion incidence axis and explicitily with the etching time t a (Vt(x,y,t) velocity must be taken into account. In order to find the position at t + At from the position at t, a computational loop is proposed: x: = 0 y_, = y0 cos(00 = 1 .

(21)

For i = 1 to N sin(00 = (1 - cos20~)'~-~

(22)

x, = x2 Yl =Y2

(23)

y~. = y, + V , ( x , , y t , ( i - l ) . A t ) . A t . s i n O ( x ~ , y ~ , ( i -

with/3,,/32 measured from the perpendicular to curves V~(x,y) = const. Then: t a n / 3 - OV,l~y

(28)

OV,/Ox "

For n >/3 > n/2; fl' = ~ - n; 0~ > fl' + n/2: cos0: = - cos//'.[l - (k.(sinfl'.cos00 -- (1 -- (COSOi)2)l/"'COSfl'))2] '~2 -

-

(sin/~'.k.(sinfl"cos0,

- (1 - (cos00-')'ncosff).

(29)

Similar equations can be obtained for the other three cases (Mazzei and Cabanillas, 1993). These equations yield the track profile from long etching times to short etching times and from diameters inside the damage zone (10nm) to diameters > 1 ~tm. By using these equations, experimental tracks for all etching times can be reproduced.

4. RESULTS AND DISCUSSION Assuming that: Vt(x, y) = Veil + g ( x ) ' ( a / h ) . e x p ( - b'y')]

(30)

the theoretical track profile can be found. Here g ( x ) = b s + a s.x. Each incident ion on the foil and with a given etching condition is characterized by a specific set of parameters b s, a t, b, a/h and c. Where a/h, b and c must be determined experimentally from newborn tracks (Mazzei et al., 1986) and g ( x ) allows the simulation of the velocity of chemical attack when the damage varies along the incidence axis (Mazzei and Cabanillas, 1993). a s > 0 simulate increased damage and a, < 0 decreased damage. For a~ = 0 and b~ = 1 a V,(y) function is obtained.

l)'At) (24)

x 2 = xt + V,(x~,y~,(i- 1 ) . A t ) . A t . c o s O ( x j , y ~ , ( i -

1)'At).

(25) Next: with N = t / A t , A t is as small as required and O, = O ( x , y , ( i - l).At) can be calculated as indicated in the following for a V~(x,y) function. From Fig. 3 a relation similar to "Shell's law" is obtained: sin/?,. = k'sinflt Vt(x2, Y2) k = Vt(x~,yO-

(26) Vn V.

(27)

IlL X

Fig. 3. The relationship between O(x, YO and O(x:, y.) for n > / / > n/2 and 0, > / ~ - n / 2 . The line d V , ( x , y ) / d x = O divides two regions of different V, values.

SSNTD AND KINETIC THEORY

581

Fig. 4. STKT track profile and variational principle trajectories for 35 s of etching time for a/h = 1.95, b=4.56x 10-Tandc=2.82.

4.1. Comparison between variational principle and

STKT results Figure 4 shows an overlapped comparison between the calculated variational principle trajectories (Section 3.3) and S T K T track profiles (Section 3.2) for a/h = 1.95, b = 4.56 x 10 -7, c = 2.82 and t = 35 s. A slight difference is observed between the

two theories. This theoretical track profile allows the adjustment of microphotographs of observed track profiles (Mazzei et al., 1988b). 4.2. Large etching times When the physical damage region is considered as a straight line or for large etching times an agreement

709

532

354

177

250

500

750

(0.1 nm) Fig. 5. Variational principle trajectories for 50 s of etching time for a/h = 1.95, b = 6.16 × 10 -7 and c=5.

582

R. O. M A Z Z E I 24,106

18,079

12,053

8500

17,000

25,500

34,000

x(nm)

Fig. 6. Theoretical track obtained for long etching times. Decreasing damage along the x axis with t = 2000 s, a/h -- 1.95 b = 4.56 x 10- 7, c = 2.82, a 8 = - 0.0001 and be = 4. among the C T K T the S T K T and variational consideration should be expected. Figure 5 shows that using large c values a small damage zone can be simulated. Here a/h = 1.95, b = 6.16 x 10 -7, c = 5 and t = 50 s (short etching time). It can be seen that the track profile is not conical near the ion incidence axis. O n the other hand, for points that lie far from the axis (in the bulk region) the track profile is conical and the three theories are in agreement. F o r large etching times the damage varies along the x axis and a s =# 0 must be used in equation (30). In Fig. 6 it was assumed that t = 2000 s, a/h = 1.95, b = 4.56 x 10 -7 , c = 2.82, a s = - 0.0001, bg = 4 in the equations from Section 3.4. The dotted line in Fig. 6 indicates that V,/Vb= 1/ sinct decreases with x. This behaviour was expected for decreased V~(x) in microscopy tracks (Somogyi and Almasi, 1980). 4.3. Discontinuities in the track profiles 4.3.1. Damage fluctuations. As shown in Fig. 5 of H a m m et al. (1985) the statistical fluctuations in

energy loss depend on the dimensions of the track segments considered. Thus, it is necessary to estimate the fluctuations in damage for each particular ion and the dimensions of these regions. Moreover, the changes resulting from statistical fluctuations would produce asymmetrical changes in the track profiles. Those can be simulated considering equation (30) and supposing abrupt changes in g(x) (Mazzei and Cabanillas, 1993). Figure 7 shows a theoretical track profile obtained considering a n o n u n i f o r m region of damage in 150 _< x250 and 50 _< y _< 150. The law of refraction (Section 3.4) was used to follow the etching trajectory throughout the n o n u n i f o r m region (Mazzei and Cabanillas, 1993). In this case it was considered: [Vt/Vb - l]~,a,~,on/[Vt/Vb - 1]o,,,d,~,~o~= 0.6, t = 40s,

a/h = 1.95, b = 6.16 x 10 -7 and c = 2.82. 4.3.2. Charge changing events. The target damage produced by an energetic heavy ion is directly connected to its charge state inside a solid target (Schramm and Betz, 1992; Cowern et al., 1984). U p until now, two studies of ion beam charge state transitions inside a solid target have been reported

425

319

¢a 213 O

106

150

300

450

0.1 nm Fig. 7. Inhomogeneities in the physical damage region.

600

SSNTD A N D KINETIC THEORY 1222

583

The results of variational principle, STKT and the "law of refraction" are in good agreement and agree with experimental observations in Macrofol-E. Using the preceding equations we can obtain V~(x,y) for the best agreement between the theoretical and experimental track profile. So, different V~(x,y) functions must be compared to obtain information about the damage zone, using different ions with different energies, charge states and using different etching conditions. These theoretical methods together with the replica method, yield information about the charge changing process in the material, the local fluctuations in the undamaged material (V0 and the local fluctuation in the damaged material (I0. With the incorporation of a time dependence in V~ and using the equations shown above, it is possible to analyze processes such as induction time, chemical attack transients and time-dependent concentrations and temperature in region as small as a few square nanometers.

y(O, 1 nm)

917i

Fig. 8. Theoretical result with microphotographs of track replicas in a half by a half fashion for t2C for 4 + incidence charge state and 4'30" etching time. A possible electron capture can be observed at x = 110 nm.

Acknowledgements--The author wishes to thank Dr D.

Schinca for many helpful suggestions.

REFERENCES (Mazzei and Cabanillas, 1993; Mazzei et al., 1994). In those investigations they tried to find distinctive signs of ionic charge changing events by analyzing track profiles in bombarded SSNTD. The difference found among energy loss values for an ion that penetrates a solid target in different charge states should be shown in a target damage variation disclosed by a change in the detector etching velocity, generating a discontinuity (break) in the track profile. Figure 8 shows theoretical results with microphotographs of track replicas in a half by half fashion for 12C for 4 + incidence charge state and for 4'30" etching time. It was considered a/h = 1.87, c = 2.1 and b = 0.00001 in the equations from Section 3.4. One electron capture was assumed at x = 110 nm.

5. CONCLUSIONS It has been shown that the "law of refraction" can be used to calculate the new etching directions on the trajectories and to give insight into the etch track profiles from short etching times--new-born tracks-to large etching time, track sizes > 1 lam. When the physical damage region is considered as a straight line or for long etching times, the equations yield the ones of the CTKT even when a dependence with ion incidence axis coordinate in included in the track velocity.

Ali A. and Durrani S. A. (1977). Nucl. Tracks Det. I, 107. Born M. and Wolf E. (1959) Principles of Optics, pp. 130, 724. Pergamon Press, Oxford. Cowern N. E. B., Read P. M., Sofield C. J., Bridwell L. B., Huxtable G. and Miller M. (1984). Nucl. Instr. Meth. B2, 112. Fleischer R. L., Price P. B. and Woods R. T. (1969). Phys. Rev. 188, 563. Fleischer R. L., Price P. B. and Walker R. M., pp. 50-57. University of California Press, Berkeley (1975). Hamm R. N., Turner J. E. and Wright H. A. (1985). Radiation Protection Dosimetry 13, 1-4, 83. Mazzei R. and Cabanillas E. (1993). Nucl. Instr. Meth. B 74, 405. Mazzei R., Bernaola O. A., Molinari de Rey B. and Cabrini R. (1984). Nucl. Tracks 9, 3-4, 219. Mazzei R., Bernaola O. A., Saint Martin G. and Molinary de Rey B. (1985). NucL Instr. Meth. B. 9, 163. Mazzei R., Bernaola O. A., Saint Martin G., Bourdin J. C. and Grasso J. C. (1986). Nucl. Instr. Meth. B 17, 275. Mazzei R., Saint Martin G., Bernaola O. A., Bourdin J. C. and Grasso J. C. (1988). Nucl. Instr. Meth. B 34, 237. Mazzei R., Grasso J. C., Bernaola O. A., Bourdin J. C. and Saint Martin G. (1988). Nucl. Instr. Meth. B 34, 74. Mazzei R., Nemirovsky I. and Cabanillas E. (1994). Nucl. Instr. Meth. B 93, 288. Paretzke H. G., Benton E. V. and Henke R. P. (1973). Nucl. Instr. Meth. 108, 73. Schramm R. and Betz H. D. (1992). Nucl. Instr. Meth. B 69, 123. Somogyi G. and Almasi G. Y. (1980). Nucl. Instr. Meth. 173, 21. Somogyi G. and Slazay S. A. (1973). Nucl. Instr. Meth. 109, 211.