A spatial track formation model and its use for calculating etch-pit parameters of light nuclei

NUCLEAR INSTRUMENTS AND METHODS 147 (1977) 11-18 ; (~) NORTH-HOLLAND PUBLISHING CO. Session 1. Track theory

A SPATIAL TRACK FORMATION MODEL AND ITS USE FOR CALCULATING ETCH-PIT PARAMETERS OF LIGHT NUCLEI* G. SOMOGYI

Institute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen, Hungary R. SCHERZER, K. GRABISCH and W. ENGE

lnstitut./~r Reine und Angewandte Kernphysik, Universitiit Kiel, Kiel, W. Germany

A generalized geometrical model of etch-pit formation in three dimensions is presented for nuclear particles entering isotropic solids at arbitrary angles of incidence. With this model one can calculate the relations between any particle parameter (Z = charge, M = mass, R = range, 0 = angle of incidence) and etching or track parameter (h = removed detector layer, L = track length, d = track diameter, etch-pit profile and contour) for track etching rates varying monotonically along the trajectory of particles. Using a computer algorithm, calculations have been performed to study identification problems of nuclei of Z = 1-8 registered in a stack of polycarbonate sheets. For these calculations the etching rate ratio vs residual range curves were parametrized with a form of V -1 (R)= 1-~agexp(-biR) which does not involve the existence of a threshold for track registration. Particular attention was paid to the study of the evolution of etch-pit sizes for relatively high values of h. For this case, data are presented for the charge and isotope resolving power of the identification methods based on the relations L(R) or d(R). Calculations were also made to show the effect of the relative (parallel and opposite) orientations between the directions of track etching and particle speed on etch-pit evolution. These studies offered new identification methods based on the determination of the curves L(parallel) vs L (opposite) and d(parallel) vs d(opposite), respectively.

1. Introduction Our aim with this work is twofold. We should like to show, on the one hand, that track etching kinetics can be expressed in such a generalized form that it allows the calculation of any etch-pit parameter of the particles entering isotropic track detectors at an arbitrary angle, supposing a track etching rate changing monotonically along the trajectory. Thus, the present work can be regarded as an addition to and improvement on a previous work of ours~). The application of the above theoretical model requires the knowledge of the function describing the changes in the track etching rate along the trajectory. The other aim of our work is to investigate, by producing such functions and applying the developed spatial track-formation model, the tendencies of the changes in some etch-pit parameters in the case of the application of extended etching times for light ions entering polycarbonate detectors. In the calculations related to the not sufficiently studied extended etchings we considered all results obtained in the course of our * This paper was read at the 9th International Conference on Solid State Nuclear Track Detectors, Neuherberg/Mi.inchen, September 30 - October 6, 1976. The complete Proceedings will be published by Pergamon Press, Oxford and New York.

studies on the revision of the concept of registration threshold2).

2. Description of spatial etch-pit formation First of all we shall outline the problem to be solved. After an etching period measured with a given layer removal h, the different etch-pit parameters are to be calculated (such as L = theoretical track length, Lr=registered track length, D = big axis of the pit, pit profile and contour, etc.) if we know the particle parameters [ 0 = angle of incidence, E0, R 0 = energy of incidence and 'starting' range, respectively, and the V(R) = VT(R)/V B function, i.e. the changes in the etch rate ratio as a function of the residual range R, where R is the damage path length measured from the end of the particle trajectory]. Since the solution of such problems is significant especially in charge identification studies performed with the stack technique, the investigation of the problem will be extended to the etched track cones developing parallel with and opposite to the incidence of the particle entrance (henceforth forward and backward pits). The coordinate system applied in writing up the problem and the denotations can he seen in fig. 1. The figure shows the middle section of the forI. TRACK THEORY

12

G. SOMOGYI et al.

the etching times determined by the integral

" BACKWARD PIT ~ x ~ , ~ ,

~,

} /

o dX/VT(X),

s

",z- ?

3."%.,2, '-

ETCHED / "

f.f

I .......

\

A.

o) L~ ~/~'x#"/ " - - (% '1'

h

\1

I

/

ORIGINAL

,

_

and grow in radius at a rate of VB. With the parameter x0, and with xs characteristic of the starting point of the etch-pit formation in the so-called parallel (M') and antiparallel (i"~) etching situations, the equation of the bulk etching spheres in the coordinate system given in fig. 1 is

B(xs, Xo) = (X-Xo)2+y2+z2-r2(xs, Xo) = O, oRwARo Pfr N¢ ¢,

k

~J V

",

LI"

,,

", }

(X=Ro;O)--'..L \-,-

Fig. 1. Schematic view of forward and backward etch-pits evolving in a stack of detectors.

ward and backward pit of a particle passing through at the interface of two sheets of a stack after layer removal h. For the simplest way of discussing etch-pit formation it is expedient to fix the origin of the coordinate system at the intersection of the original detector surface and the particle trajectory so that the x axis should coincide with the particle trajectory. The y axis lies in the plane determined by the particle trajectory and the big axis of the pit. In this way any etch-pit parameter can be derived from the solution of the following basic problem: One must write up the equations of the planes parallel with the original detector surface as well as that of the wall of the etch-pit (envelope) and then one must find the common solution of the two equations at a parameter value C chosen depending on the quantity to be calculated, where C denotes the distance of the plane from the etched detector surface. Thus, for the original unetched detector surface C = - h , for the etched surface C = 0 and for the plane intersecting the end of the trajectory C=__Cm~_Lrsin O=L sin O-h. The equation of the planes parallel with the original detector surface, after the introduction of the changeable parameter C, can be given in the form x - y ctg 0

h+C

sin 0

0.

and the equation of the envelopes of these spheres derived from the condition 3B/Oxo = 0 is c~r Xo--X+r(Xs, Xo) 8x ° = 0. (3) The sets of parametric eqs. (2) and (3) are obviously the equations of the surface forming the walls of the etch-pits. Until the preferential etching reaches the end of the etchable trajectory, i.e. x0
B(x~, Ro) = ( x - R o ) 2 W y 2 + z 2 - r Z ( X s , Ro) = 0.

(4)

This represents spheres starting from the end of trajectory as a centre and increasing in radius at a speed VB. Thus, it will be referred to in short as the "standing balls" equation. Our next task is to express the radius of the bulk etching sphere in the above equations with etching parameters. If the total etching time resulting in a layer removal h is t, and the time of etching along the trajectory at a speed changing according to the function VT(x) between the points x~ and xo, is t(xs,xo), then it is obvious that

.(x~, Xo) = VB[t-t(xs, Xo)]

yo = VB[t-

VTI(X)dx] = h-H(x~,xo) , s

(5)

(1)

To produce the equation of the etch-pit wall, we must write the equation of the bulk etching spheres that start gradually delayed from the successive points x0 of the track axis as centres, after

(2)

where H(x~, xo) is a notation introduced for the integral H(x~, Xo) =--

f: o V - l ( x ) d x .

(5')

SPATIAL

Hence c~r OH

OXo

-

--

OXo

=

v-l(xo).

TRACK

(6)

It should be noted here that in the coordinate system used, and fixed to the point of incidence, V(x) can be obtained from the V(R) function through the transformation R--,Ro-Qx, where Q(Ti") = + 1, Q(I"+)= - 1 and R0 is the range of the particle entering the detector. Substituting into eqs. (2), (3) and (4) the values r and Or/Oxo given by the formulas (5) and (6), we can get the practically applicable formalism describing the etch-pit formation. One more detail left is the clearing up of the meaning of the Xs parameter. In our formalism Xs is the coordinate of the point where the formation of the etch-pit begins. Supposing a function V(xo) changing monotonically along the particle trajectory, the V(xo)sin 0> 1 condition allowing the appearance of the etch-pit can obviously be satisfied only for given track segments with certain )Co coordinates. If the condition V(xo) sin 0 = 1 (7) is fulfilled at the point x0 -= - x c on the axis of the backward track, then xs(tt)=xs(l"$)=0, and the origin of the coordinate system is the same as in fig. 1. If, however, the condition (7) is fulfilled at the point )c0-=xc lying on the axis of the forward track, then in the (1"~) situation no etch-pit can be formed at all, whereas in the (tt) situation it is possible only after a layer removal reaching the xs(l"t) = xc coordinate. In the latter case the coordinate system used for the actual calculations is to be transformed into point xc, where the etch-pit formation starts. To attain this, in eqs. (1), (2) and (3) the substitutions Xo-OXo-Xc and h--,h-hc must be performed, where the value of x~ can be calculated, if we know the concrete form of the function V(xo), from the solution of the equality (7) for )c0=x~, and the value of the hc, the socalled critical layer removal hc =Xc sin 0. 3.

Results

In the following we give the formulas derivable from the basic correlations (1), (2) and (3) for the calculation of the particular etch-pit parameters, then their application is illustrated on some typical examples. 3.1. TRACKLENGTH In charge identification problems most frequent-

FORMATION

13

MODEL

ly the knowledge of the so-called cone lengthrange L(Ro)h.z correlation is necessary. To produce this we may use the facts that z = y = 0 , and x = x0 = L, thus from eq. (2) r = 0. Thus on the basis of the formula (5) we get the relationship

h=

V-I(Ro-Qx)dx,

if xs = 0,

(8)

o

and

-x

h - hc --

IL

~V- 1 (Ro - x) dx,

if Xs = x¢.

(9)

do

In the case Xs=Xc we took into account that Q = Q(tl")= + 1. In the above formulas L means the theoretical track length. If we want to calculate the actually observed Lr, so-called registration track length, then the relationship L, = L - (h/sin 0)

(10)

must be used. Let us call attention to an interesting case, which can be frequently realized with the tracks of light nuclei. With backward etch-pits, if the formation of the etch pit is started, the inequality V(x0= 0)sin 0>1 must obviously be satisfied as early as on the detector surface. However, as etchpit evolution proceeds, the product V(x0)sin 0 tends to 1. In consequence, if 0 = 9 0 ° and the function V(xo) changes rather steeply, the case can occur even after a short period of etching that the registration track length disappears, nevertheless there exists an etch-pit. The h = h0 layer removal performed at the moment of the fulfilment of the condition Lr--0, can be calculated from the relationship

lho/sin 0 ho =

V-

(11)

1 ( R 0 ...[_x ) d x ,

dO obtained from eqs. (8) and (10). 3.2. TRACK PROFILE, BIG AXIS Let us consider now the determination of the track profile in the middle section of the etch pit, i.e. in the plane determined by the x-y axes and, as a special case of the problem, the calculation of the big axis of the track. The solution for the z = 0 case is being sought for from the "travelling balls" equation in the parametric form y(x0, C), h(xo, C). Expressing x from the relationship (3) and substituting it into the formulas (1) and (2), then solving these for y and h we can obtain the I. T R A C K T H E O R Y

14

G. S O M O G Y I

basic equations of the track profile: h = H ( x s , Xo) +

x o - H ( xs , Xo) - C 1-

V- 1(Xo)(sin 0 + 14I/cos 0)

Xo sin 0 - H ( x s , Xo) - C y, = w, F Uo) - -

, (12a)

(12b)

T .w,. c. o.s o '

where Wi = ( - 1 ) i \ / ' [ V 2 ( x o ) - I

i = 1,2.

],

(13)

The course of the practical calculations is as follows: at fixed h and at C values fixed in the 0 ~< C ~< Cm interval for i = 1, 2 we seek for the x0; solutions of the formula (12a). Substituting these into the formula (12b), we get the y; values. The right and left hand side points of the track profile are formed by the points at a distance y / s i n 0 from the track axis, lying in a plane parallel with the etched detector surface and lying below that at different C depths. The big axis of the etched pit, as is seen obviously from fig. 1, we can obtain by using the yl and Y2 values for the case C = 0, from the relationship O = y l ( C : O) d- y 2 ( C : 0)

(14)

sin 0 3.3. TRACK CONTOUR, SURFACE OPENING The most general problem of track etching kinetics is the determination of the track contour (pit wall). The description of the form of the surface opening of the pit is obviously only a special case of the problem. Let us produce the solution from the "travelling balls" equation in the parametric forms Z(xo,y, C) and h ( x o , y , C). Expressing x from the relationship (1) and substituting it into the formulas (2) and (3), then solving them for z and h we get the basic equations of the track contour: Z

_+

F Xo))

_

_

et al.

0 ~< C~< Cm the value of C. Then we try to find the solutions for x0 in eq. (15a) for the various y values changed in the interval y~(C)<~y<~y2(C), where yl(C) and y2(C) are the coordinates of the big axis of the given track contour, thus their calculation takes place by using the formulas (12a) and (12b). Substituting the so-obtained x0 values into the formula (15a) we can calculate the quantities z(xo,y, C). Then we can produce the track contour from the (z, y/sin 0) pairs of points. The shape of the surface opening of the pit can obviously be got in the above way in the case of C=0. Here one should mention a question of calculating technique. The formalism used so far separates the calculations according to the sign of h for the forward and backward pits, i.e. in the equations describing pit formation in the (M) situation one must perform the h--, +lhl substitution, and in the (H) situation the h--,-Ihl substitution. 4.

V(R)

functions

for light nuclei

Applying the above formalism describing etchpit formation, calculations were made with nuclei with charges Z = 1-8 in connection with track length, diameter, contour and profile formation, in the case of longer etching periods. In the following some characteristic results of our investigations will be presented. Our calculations refer to the case mentioned in ref. 2 and denoted by PC3, which represents Makrofol-E foil etched in 70°C PEW solution ( = 15 g KOH + 40 g C2HsOH + 45 g H20). It should be noted that with this etchant higher track registration sensitivity can be obtained than with the usual solutions of NaOH or KOH in water2). First we had to produce the V(R) and V(x) functions, valid for the tracks of light nuclei in PC, in a form that the integral H ( x s , Xo) in the formulas describing etch-pit formation could be explicity calculated. For this aim the series consisting of the exponential terms of form 2

-

y c t g O + sinO

xo

,

(15a)

V-I(R)

= I -

~

ai e-~'R,

(16a)

ai e-b'~R°-Qx}

(16b)

i-I 2

h = H ( x ~ , Xo) +

Xo sin 0 - y c o s 0 - H ( x , , Xo) - C

V-I(X)

= 1 -- ~ i=1

1 - V - ~ (Xo) sin 0

(15b) The course of practical calculations is as follows: we fix both the value of h and in the interval

were chosen, where the changes of the etching rate ratio along the trajectory are described in coordinate systems fixed to the end of the trajectory (16a) and to the point of the incidence of the particle (16b). Using the values a; and b; given in

S P A T I A L TRACK FORMATION 1.

Table

NUCLEI

Parameters

6.564 E-2

aHe

e q . V (~)=I--

a 2

a1

IH

of

"02 I

( pb ~ - l )

7.740 E-5 3.025 E-2

6.435 E-1

for varlous

4.696

(pro-) 8.185 E-2

E-1

1.280 E-1

1.46

E-~

MODEL

15

3 MeV

nuclei

90 85 8-0-75-70 -

i ~ .1

t,MeV 90-~_

%~_

6&

2.5

,

"-, ,, --.

",

,

5~. 7L±

8,500 E-1

6.000 E-2

4.000 E-2

1 .00

E-3

9Be

9,524 F-1

8.435 E-2

2.360 E-2

b.10

E-d

11R

9,973 E-1

9.249 E-2

1.352 E-2

"~.I0

R-4

20

12C

1.018 E-O

9.447 E-2

9.O50 E-~

1.90

E-4

30

14N

1.000 E-0

9.450 E-2

5.500 E-3

1.50

E-4

45

160

9.853 E-1

9.400 E-2

5.690 E-3

1,00

E-4

3.5

1

?

:i tL4351

\t

Remark:

the a b b r e v i a t l o n

E-n equlvalent

wlth.10 -n .

L

5. Results of calculation For the theoretical investigation of the properties of etch-pit formation a computer programme was elaborated and the evolution of the forward . . . . . . . .

t

,

.......

i

. . . . . . . .

10 ~

x

. . . . . . . .

PC

' I

01 ~

,

JH

OOt

1c~

.

.

.

.

.

.

.

ib'

3He'He

'L'LJ

Ib~

R[/um]

i,0 45

55

table 1, this series fitted the V(R) curves derived from the V(REL) response curve denoted in fig. 2, with a 2% accuracy up to the residual range denoted by Rmin. It should be mentioned that if the calculations are performed instead of an isotop of mass M1 given in table 1, with an isotope of mass M2, only the transformation b~2=b~lM~/M2 and ai2 = a n is necessary.

100

, k, "~

;/ J,/ ,/

ib'

.

.

.

.

.

.

.

io'

Fig. 2. Calculated curves showing the excess etching V - 1 - - - ( V T - V ~ ) / V B as a function of the residual light nuclei registered in MakrofoI-E foil etched in lution at 70°C. The calculation was based on the curve V = l + 0 . 0 9 6 R E L 2.82 obtained for PCI in ref. REL is expressed in MeV cm2/mg.

rate ratio range of PEW soresponse 2, where

1Mm

=

~v~?

Fig. 3. Calculated track contours for 3 M e V (R0= 13.6/1m) and 4 MeV (R 0 = 20.5 ,urn) alpha-particles as a function of the angle of incidence, at 6 ~ m layer removal in polycarbonate sheet using the V ( x ) = 1 + e x p [ - 0 . 1 7 (R 0 - x ) + 0 . 4 ] function in the calculation.

and backward etch-pits for light nuclei with charges between 2 and 8 were studied in polycarbonate sheets. For these calculations at Z = 3-8 the V(R) functions given in table 1, at Z = 2 the more exact V(R)= 1 + e x p ( + 0 . 1 7 R [ ~ m ] + 0 . 4 ) function (see ref. 2) was applied, which includes an asymptotic change of the etching rate ratio along the trajectory in the so-called registration threshold range. The capacity of our computer programme is well illustrated by fig. 3, which shows the forward etch-pit contours of 3 and 4 MeV alpha particles at different angles of incidence on the etched detector surface, at a layer removal of 6 ~m. The calculated curves display well a phenomenon which is well-known from practice, i.e. the formation of track profiles gradually narrowing towards the small angles of incidence, and with drop-like cross section, the big axes of which increase in the beginning, then gradually decrease. Further, we can clearly see as another consequence, very remarkable from the practical point of view, the appearance of the energy dependence of the critical angle for track revealing (track disappearance occurs at 24 ° for 3 MeV, whereas at round 60 ° for 4 M e V alpha-particles). It is clear that such calculations can give very useful information about the character of the expectable track images under complex irradiation circumstances. I. TRACK THEORY

G. S O M O G Y I et al.

16

h

'°B(Ro=80 IJm) 70.

5

--

~.

60-

\

10

~\~

\ 15

- - - -

_

5O

\~

E

20

----

25 30

-

~_

.\,

\ .....

\

-- - - " \ " "

3O .

\.

',,,

-%, \ I0

Fig. 4. Calculated track profiles for l°B ions of 200 p m "starting" range as a function of layer removal in PC foil, where the " s t a r t i n g " range is the one measured from the unetched detector surface to the end of the trajectory.

0

50

0

tO0

i

Our calculation programme allows the tracing of the evolution of the etch-pit profile which decisively determines the optical contrast and so the observability of the tracks. The result of such a calculation is shown in fig. 4, where we can see the forward pit profiles for a l°B nucleus entering polycarbonate foil at 45 °. The evolution of the big axis of the track takes place in the range bordered by the broken lines. One can clearly see the sharp difference between the character of the left and the right hand side track profile, which results in the comet-tail like appearance of the oblique tracks. Another objective of our calculations was the study of the charge and isotope resolving power attainable in identification investigations performed with the stack technique, at longer etching periods. Parallel with this it was also studied whether analogously with the usual identification technique based on the track length-range curves, the track diameter-range relations can be applied for identification purposes. The latter relations can be reckoned with primarily in the case of light nuclei incident on the detector at right angles, as a supplementary identification method, or one that can be conveniently applied in itself. As a model example of these calculations the case of the ~°B and l~B isotopes incident normally to the detector surface was chosen. The results calculated for the backward pits are shown in figs. 5a and 5b. It is obvious that in the given case the isotope resolving power of the track diameter-range method exceeds that of the track length-

200

150

250

R,E,~m J i

L

J

i

2bo

2;0

7O

50 ,

', ,

.. •

roB u

50-

.40-

% 30-

2O

0

o

s'o

,oo

1;o R.g,,oro 2

Fig. 5. Calculated track length (a) and track diameter (b) o f the backward etch-pit of B isotopes at right angles of incidence as a function of the " s t a r t i n g ' ' range in PC foil, in the case of extended etchings.

range method (resolving power was defined as the difference of the track diameters and track lengths of l~B and ~°B nuclei at a given range). The changes in resolving power related to the two methods are shown as a function of layer removal in fig. 6, as it is obtained from the calculation formalism together with the sign of the calculated quantities. According to the sign the data for forward and backward tracks are clearly separated.

SPATIAL

- - * ' L = L( B)-LICB) ---a¢ =d( B) -d('B)

TRACK

FORMATION

-~Oh

-30,

60 !

fU-ZTAT~ L,. ~,, I

5o~

~o (~,~1

17

uC / ~ _ _

~2 t

-50,

MODEL

"20

.,oo ;7Be< / / /

•

-10

,'o

io

20

~o ^ g/Jm2

50

%

-I

9 ~ //// ~,; / / 7 / •

7L, 3o /

I t O 0 - J. - "

li,

Fig. 6. Calculated resolving power for B isotopes o f the identification m e t h o d s based on track length and track diameter measurements as a function o f layer removal. The parameter is the " s t a r t i n g " range in PC foil. ....

~

....

~o . . . .

~ ....

,~ . . . .

~ ....

;,o- - ~

d"#,,,,!

The nuclear charge resolving power of the identification methods based on track diameter and track length measurements is compared in the range of light nuclei, in fig. 7. It is seen that at long etching periods the theoretically expectable resolving power of the track diameter method seems to be very favourable, and experimental investigations towards these ends seem to be worth performing. One of the serious experimental problems of the above-mentioned two identification methods is the exact and quick determination of the range related to a given track diameter or track length, since

5o

50 ~

•

40~

',, ',

,

"

\

~"

d'A

\

~0

LY

i

,o-

0

o

'L, ~o,

tbo

'/B

2bo

.... "N

!2c

.

,

300

,

,

too

•

'~o

,

500

.

,

6oo

.1o

~

700

.LO

Ro [ / , A m ]

Fig. 7. Calculated curves showing the resolving power for light nuclei of the track diameter vs " s t a r t i n g " range and the registered track length vs " s t a r t i n g " range relations, at extended etching producing 40,um layer removal in PC foil.

Fig. 8. Calculated curves showing the resolving power of a new identification method based on the m e a s u r e m e n t s of etchpit diameters of connected forward and backward pits in a stack of PC foils.

finding the end of the particle trajectory requires successive time-consuming etching procedures. The question emerges whether one can find a track parameter that may take the role of range in the course of identification. Our calculations to this effect showed that at longer etching periods the forward and backward etch-pit diameters or lengths relating to the same range may serve as suitable pairs of points for identification. The relation between the diameters of the forward and backward etch-pits seen on the adjoining foil surfaces in a stack of detectors falls apart into the lines characteristic of the particle charges and masses seen e.g. in fig. 8. A similar tendency can be obtained for the forward length - backward length relation. The resolving power of the method increases towards lighter nuclei and smaller ranges. This new method of identification appears to be relatively simple from the measuring technical point of view. Its experimental check-up and the determination of the limits of its application is the task of a further study. In the present paper we only wished to point to the theoretical possibilities of the method. At the same time with the calculations presented in this paper we tried to call attention to the so far unrevealed pieces of information implied in the theoretical geometrical model of etch-pit evolution. I. T R A C K T H E O R Y

18

G. SOMOGYI et al.

The authors would like to thank Prof. E. Bage and Prof. D. Ber6nyi for the facilities of the Institute. This work was in part supported by the Deutsche Forschungsgemeinschaft.

References t) G. Somogyi and S. A. Szalay, Nucl. Instr. and Meth. 109 (1973) 211. 2) G. Somogyi, K. Grabisch, R. Scherzer and W. Enge, Nucl. Instr. and Meth. 134 (1976) 129.