Nuch, ar TJ'ack Detection, Vol. 1. No. 2, pp. 107 121. Pergamon Press. 1977. Printed in Great Britain

ETCHED-TRACK KINETICS IN ISOTROPIC DETECTORS

A. ALI* and S. A. DURRANI

Department of Physics, University of Birmingham, Birmingham B l 5 2TT, England

(Received 7 February 1977)

Abstract--As the etching of a charged-particle track proceeds in a detector, three successive phases can be recognized: the "cone phase", the "transirion phase", and the "sphere phase". Equations are derived to give the values of various useful etch-pit parameters, in particular those connected with the surface openings in the three phases. These values are of use in reconstructing the kinematics of the ionizing particle, such as its charge. mass, energy, and range.

1. INTRODUCTION

THE PRODUCTION of a visible track in a solid-state nuclear track detector (SSNTD) involves two stages. The first is the production of a sufficiently intense and continuous trail of radiation damage caused by the passage of a charged particle through the medium. The second is its revelation through the action of a suitable etchant, which progressively enlarges these damage trails as the etching proceeds. In this paper we discuss the second aspect of the phenomenon. A good deal of information about the charge, energy, and range of the ionizing particle is inherent in the size, shape, and rate of enlargement of the etch pit as a function of etching time. In order to retrieve this information, and thus gain an insight into the nature of the primary particle, it is necessary to clearly establish the various phases through which the etch-pit profile passes as the etching proceeds as a function of time and depth below the initial surface.

Equations governing track kinetics in an etched medium have been derived by a number of workers (see, e.g., Henke and Benton, 1971; Somogyi and Szalay, 1973; and Fleischer et al., 1975). In the present work, greater emphasis has been laid on the differen- tiation of the successive phases in the evolution of the etch-pit profile, and the track geometry has been studied more explicitly. The resulting equations are

formulated somewhat differently than their counter- parts in the earlier papers, and have been more firmly related to the evolutionary phases involved. The main approach adopted in the paper has been to evaluate the observable parameters in track measurement: namely, the dimensions of the etch-pit "surface opening" (its major and minor axes; radius; "length" and "width") at successive stages of etching, and of the etch-pit "profile" (i.e. the part of etched track lying below the surface and within the body of the detector). The measurement of these parameters is needed in the identification of track-forming particle and its en- ergetics.

Throughout this paper, the medium of the detector has been assumed, for simplicity, to be isotropic from the point of view of track-etching properties. We assume the detector to have a plane top-surface, referred to as "horizontal", and the etchant to work into the medium from this top surface. For clarity of treatment, we assume the track to cross the top surface of the detector and be etchable from that initial surface. The general, or bulk, velocity of etching in the medium is taken to be constant, and termed V 0. The velocity, VT, of etching along the track does, however, in general vary at different points along the particle track. In our equations, we use an average value VT to take account of this fact. We define below the various symbols used in our treatment.

*On leave of absence from Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan. 107

108 A. ALl and S. A. DURRANI

2. NOTATION USED

V~: general, or bulk, velocity of etching in the medium. (If the etching conditions change with time, then V~ should be replaced by the mean value

1 V.qdt . ~"= i o

Vr: velocity of etching along the latent track at a given point.

R : the etchable range of a particle in the medium.

';o f: P'r = t V r dt for t < to, where R = V r dr.

0< = sin- 1 (Vg/~/r), the "critical angle of etching" (Fleis- cher et al., 1975).

- - l h - ~ o Vqdt, the thickness of the top etched-layers removed, or the vertical distance traversed by the etchant in the general medium.

L = ~ g r dt for t< to, the distance traversed by the etchant along the track.

h "length" (i.e. longest axis) of the surface opening of the etch pit.

/': "width" (i.e. maximum axis ± to l) of the surface opening.

z: depth of the lowest point in the etched track profile, with respect to the final etched-surface.

r: radius of the circular part of the surface opening. a: major axis of the ellipitical part of the surface

opening. b: minor axis of the ellipitical part of the surface

opening. u: radius of the spherical part of the etched track

profile (within the body of the detector). s: projected length of the etched track, on to the

detector surface. to: time of end of cone phase and start of transition

phase. t,r: time of emergence (i.e. detection) of the upper

terminal point of the T-circle (or for short, "time of the upper T-terminal"). (See text for expla- nation.)

tcr: time of emergence of the lower terminal point of the T-circle (or for short, "time of the lower T- terminal"). (See text for explanation.)

te: time of emergence of both terminal points of the P-circle (or for short, "'time of the P-terminals"). (See text for explanation,)

ho = R sin 0<,.

f t , , l

tl,,.1= V. dt. o

I r t !

h i t = V~ dt. • o

he = V. dt. o

3. T H R E E P H A S E S O F THE ETCHED-TRACK EVOLUTION

Three distinct phases can be recognized as the etchant proceeds along the length of the latent track in an isotropie detector, as shown in Fig. 1. In this figure, X Y represents the top (horizontal) surface of a detector, and PO the direction of the latent track, assumed to be etchable from the surface. PO (which makes an angle 0 with the surface X Y) is also taken to be the etchable range, R, of the charged particle concerned. The etchant travels with velocity Vr along the track, and with velocity Vy elsewhere in the medium. When the etchant reaches point O along the track, the general surface of the detector has been etched away down to •1 Y1 ; the time taken being to = PO/ f@ Throughout this time interval t = 0 to t = to, the profile of the track is conical, and the surface opening of the etch pit is elliptical. We term this phase the cone phase.

After time to, there is no preferential etching, so that the etchant proceeds henceforth in all directions with velocity V~. This is the start of the transition phase. During this phase the bottom of the track profile becomes progressively more "rounded", while the surface opening of the etch pit at first remains elliptical and then becomes partly elliptical and partly circular. Finally, the whole profile of the track becomes spherical, and the surface opening of the etch pit becomes completely circular (at level X~ Y4 in Fig. 1). This is the start of the sphere phase, which persists ever afterwards.

4. DETECTOR THICKNESS REMOVED AT DIFFERENT TIMES O F

TRACK EVOLUTION

Let us now derive the relations for the thickness of the detector removed by the etchant, and the time taken in doing so, as the track profile enters into the three successive phases.

As stated above, the time taken by the etchant to reach the end of the track is given by

to=R/V r ( I )

E T C H E D - T R A C K K I N E T I C S IN I S O T R O P I C D E T E C T O R S 109 i

t /

1 i

H P 8 Y original surface X 4 ..Jho cone phase

start of Xl . . . . . . . ~ Y, transition ~lose

{at time ~

X2 - Y2 t = ruT

t" Xa . . . . r~ Y3 t= tp

___ \ ' \ ' \ \ , stort of X~ . . . . "- K" Y4 sphere phcr=e

(a~ time tiT )

H I

FIG. 1. The three successive phases of track evolution as the etchant proceeds along the length PO of the latent track in an isotropic detector. X Y represents the top surface of the detector, with which the track (assumed to be etchable from the top) makes an angle 0. The first phase is the "cone phase", which lasts until the general etchant, travelling with velocity V 0 in the medium, has descended to level X~ YI (taking time t o ) while the etchant travelling along the track (with an average velocity Vr) reaches point O. Throughout this time, the profile of the track within the body of the detector is conical. Beyond point O there is no preferential etching, so that the etchant proceeds henceforth in all directions with velocity V0, and the "transition phase" sets in. During this phase the bottom of the track profile becomes progressively more rounded, while the surface opening of the etch pit at first continues to be elliptical (until the general etchant has reached level X 2 Y2 at time t,r) and then becomes partly elliptical and partly circular. Finally, when the general etchant has descended to level X4Y4, the "sphere phase" sets in and persists ever afterwards. In this phase the whole profile of the track becomes spherical, and the surface opening of the etch pit becomes and remains completely circular. Parameters ha, h,r, etc., represent the detector thickness removed at times to, t,r, etc. (For definitions see Section 2 (notation). Also see text for the explanation of the T- and the P-planes and the intersections of the track profile with these planes to produce the corresponding circles represented by EK'N', etc.).

In constructing all figures (except Fig. 7), the following values of the track parameters have been used unless otherwise stated: "dip angle" 0 = 70 °; "critical angle of etching" 0 c = 30 °; etchable range R = 5 cm.

The thickness of the detector etched away in this time is

ho=Voto=R(Vo/~)=R sin 0c (2)

The etching time to and the detector thickness removed ho = R sin 0c signify the end of the cone phase and the beginning of the transition phase.

In order to analyse the subsequent stages of track etching, it is necessary to introduce the following definitions which should help clarify the picture:

(i) The track plane (or the T-plane). This is the plane which contains the unetched track as well as the normal to the detector surface. (It is the "plane of the paper" in Fig. 1, containing both PO and HO.) Note that in the case of tracks at right angles to the detector surface, any plane normal to that surface can be taken as the T-plane.

(ii) The perpendicular plane (or the P-plane). This D

is the plane which contains the end-point(O) of the unetched track and is perpendicular to both the detector and the T-plane. (It is the plane L to the plane of the paper in Fig. 1, and passing through OH.)

(iii) The T-circle. This is the intersection of the spherical port ion of the etched track with the T-plane (see, e.g., the arc K"H'N", or EK'N' in Fig. 1).

(iv) The P-circle. This is the intersection of the spherical portion of the etched track with the P-plane (this circle is in a plane perpendicular to that of the paper and so cannot be seen in Fig. l).

After the onset of the transition phase, both the T- circle and the P-circle start growing in size and their "terminal points" start advancing towards the final etched-surface of the detector. These terminal points (being the points of intersection of the circular and the elliptical cross-sections of the track profile) meet the final etched-surface of the detector at different times of

110 A. ALl and S. A. D U R R A N I

etching in the case of the T-circle; but in the case of the P-circle, both terminal points reach the detector surface simultaneously. These various properties can be visualized by referring to Fig. 1.

The transition phase starts from the etched surface X I Y 1 (Fig. 1). At level X 2 Y z , the "upper terminal point", L, of the T-circle lies on the final etched-surface of the detctor. The time of etching corresponding to the emergence (at level X2 Y2) of the upper terminal point will be referred to as the "time of the upper T- terminal", and denoted by t,r. At level X3Y3, the terminal points of the P-circle lie on the detector surface. In Fig. 1, the projection of these terminal points, onto the plane of the paper (i.e. on the T-plane), is shown as M' (the two terminal points lie on the normal to the plane of paper, directly above and beneath the point M'). The etching time corresponding to level X3 Y3 is called the "time of the P-terminals", t e. Finally, at level X4Y4, the lower terminal point, N", of the T-circle reaches the final etched-surface of the detector, and the corresponding etch-time is fit ("time of the lower T-terminal").

We now derive the values of the thicknesses h,r and hrr, removed at times t , r and fir, respectively, in terms of the observables R, 0 and 0c.

In Fig. 2,

h., r = C L = CB + B L

But B L = OL = u, the radius of the sphere (or of the T- circle); this is because the general etchant travels the same distance in descending from level X~ Y1 to level X2 Y2 as it does in reaching from point O to point L (the track etchant having arrived at the end of track, O, at the same time as the general etchant arrives at level X1Y~ from levelX Y).

Also

CB = PE = R sin Oc.

Hence

h , , r=R sin Oc+u (3)

Now the time taken by the etchant to reach L along

C F IH P C'

! ,,/ 1 / 9~,:~ V o V' .~_ _ - ¢ -~,-~ ~

K" L" N" ) Yr.

N'

FIG. 2. Determination of the detector thicknesses h,r and h~r removed at the time of emergence of the upper and lower terminal points, respectively, of the T-circle (see text for explanation). The critical angle of etching 0r = sin- 1 (V9/Vr); and u is the radius of the spherical part of the track profile (and also of the T-circle). The tangent LF' is parallel to OF and perpendicular to the radius OL; NK2 is parallel to OEC'. The lower part of the figure (e.g. arcK"H'N") is to facilitate the geometric proof for the

value ofh~r (not explicitly derived in the text, but following lines similar to those for h,], ).

the path POL is given by

R sin 8 u t, =Rj~T+u/Vg=----‘f-

K v,’ (4)

whereas time taken by the etchant to reach L along the path CL is

t; = CL/& (5) Similarly, it can be proved that the thickness h,,

removed at time t,, (the general etchant having reached level X, Y,) is given by But

where

and

The

CL=CV+ VL

CV=R sin 0 (6)

VL=OL sin LOT/=---u cos (fl+tl,) (7)

last equality results from the fact that the tangent LF’ to the T-circle is i to the radius LO; and since LF’llOF (these being the two etchant fronts, a distance u apart), hence L6F = n/2. Then LOV = LOP - Vt?P

=(7q2+~,)-(~-e)

=(e+ec)-7C/2

so that sin LdV= - cos (0 + 0,).

Hence, from equations (5), (6) and (7),

t, =R sin B-u cos (e+e,) 1

K (8)

But since, by definition of tuTI the etchant reaches the upper terminal point L simultaneously via POL and CL, hence

so that from equations (4) and (8),

R sin 0 u Rsine-ncos(f3+8,) -----f_+-=

vg v, v, (9)

t, = t;

R (sin 0 -sin 0,)

...‘= I+cos(~+~,)

points meet the etched detector surface, i.e. at level X,Y,), the surface opening consists of exactly a half- circle and a half-ellipse. The radius r of the circular part of the surface opening becomes equal to the semi- minor axis b/2 of the elliptical part at this juncture.

With further etching, the circular portion of the etch-pit opening continues to grow until, at time t,, (when the lower terminal point of the T-circle meets the final etched surface, i.e. at level X4 Y,), the whole of

(10) the surface opening becomes circular. This is the start of the “sphere phase”. Continued etching increases the

Hence the thickness h,, removed at time t,, (the

general etchant having by then reached level X,Y,) may be expressed, by virtue of equations (3) and (lo),

as

h,,=R sin 8,+ sin e -sin 8,

l+c0s(e+e,) 1 (11)

ETCHED-TRACK KINETICS IN ISOTROPIC DETECTORS 111

sin e-sin 8,

i-cOs(e-e,) I (12)

It will be proved later (Section 8) that the thickness

h, removed at time t, (the general etchant having reached level X, Y3 in Fig. 1) is given by

h, = R (sin 0 + sin (3,) (13)

It may be noted that for vertical tracks, huT= h,,

= h,, as can be verified by putting 8 = 90” in equations (ll), (12)and (13).

5. CONTOURS OF THE SURFACE OPENINGS IN THE THREE

PHASES

In the “cone phase”, and through the first part of the

“transition phase”, the surface openings of the etch pits are elliptical (except in the case of vertical tracks, when they are always circular). These elliptical shapes contmue up to the time t,,, when the upper terminal point of the T-circle meets the detector surface, i.e. to the level X, Y, in Fig. 1.

Beyond the etched surface X, Y,, part of the surface opening becomes circular. The circular part keeps growing until, at time t, (when the P-circle terminal

112 A. ALI and S. A. D U R R A N I

radii of the etch pits, but they always remain circular henceforth.

F rom triangles OKB and OBK', with KB=a~ and BK' =a 2 and the various angles as shown in Fig. 3, we have :

6. D E R I V A T I O N O F T H E D I M E N S I O N S O F S U R F A C E

O P E N I N G S

(i) Major axis of the elliptical surface openings and

Consider Fig. 3, which applies to the "cone phase" as well as to the early par t of the t ransi t ion phase while the surface openings are still completely elliptical. In this figure, PBO is the track direction, and KK' is the Hence surface opening (the e tchant along the track having reached point O and the general e tchant having descended to the level X ' Y ' in the same time, t). Let K K ' = a be the major axis of the elliptical surface opening, P BK' = 0, and K O B = B O K' -- 0,.. or

Then :

o r

y = P O - P B

= PT t - VJ/sin 0

a~/sin Oc=y/sin {n-(O+Oc)} =y/ s in (O+Oc) (15)

az/sin 0,. = ),/sin (0 - 0,.) (16)

1 l a=al +a2=ysinO~Isin(O+Oi.)+sin(O~Oi.i I

2y sin O~ "(sin O.cos 0,3

a = sin 2 0 - sin E 0,. 17)

On subst i tut ing the value of y from Eqn (14) into Eqn (17),

1 1 0 VJ ( , . . . . . . . , ~= Vot(sin - s i n 0c)

\ s i n 0c sin O/ sin 0 sin 0e

2V~t ' cos Oc (14) a = (t8)

sin 0 + sin Oc

× ¥

,

P

Ii //,i

t

', :• "qa~.I , / - - / y ,

FIG. 3. Derivation of the major axis a of the surface opening. Line KK'= a is the major axis of the elliptical surface opening (with parts KB=a I and BK'=a2); "depth" along the track is BO=y; and PK1 = Pot is the detector thickness removed. Although the figure is drawn for the "cone phase", the proof remains valid during a part of the "transition phase" (up to time t,r when the general etchant reaches level X 2 Y2 shown in Fig. 1). Beyond level X1 Y1 (Fig. 1 ; onset of the "transition phase"), the bottom of the track starts becoming rounded, but the surface opening remains completely elliptical up to time tur; for the sake of proof in this region (level X 1 }I1 to X2Y2), one may extend the track linearly beyond O, assuming that the etchant is still

proceeding along the direction PO with velocity Vr. Note that Fig. 3 has been constructed with PO = 10cm.

ETCHED-TRACK KINETICS IN ISOTROPIC DETECTORS 113

By putting Vot=h, the surface layer removed, we obtain the relation:

a=2h cos Oc/(sin O+sin 0,3 (19)

As explained in the preceding section, these ex- pressions (Eqns (18) and (19)) hold for the value of the major axis of the elliptical surface opening of the etch pit until the upper T-time, tur, i.e. during the time that the thickness of the detector etched away h < h,r (that is, down to level X2 Yz in Fig. 1). Indeed, the formalism of Eqns (l 8 ) and (19) remain s valid until the end of the transition phase ( g 4 ]I4), even though beyond time tu7 the surface opening is no longer completely elliptical.

The conical geometry on which the derivation of the major axis, a, of the elliptical surface opening is based is clear-cut in the "cone phase" (i.e. until the general etchant reaches level X1 Y1 ). Although beyond time to (the general etchant having crossed level X~Yx) the bottom of the track starts becoming rounded, this does not affect the validity of Eqn (19), which holds until the general etchant has reached level X2Y 2 (Fig. 1), i.e. until time tur. For the sake of proof of Eqn (19) in the region XIY1 to X2Yz (for the level of the general etchant), we may extend the track hypothetically along the line PO beyond point O and assume that the track- etchant is still proceeding with velocity ~'r in the original direction of the track. Thus the parameters h and 0 c used in deriving Eqn (19) remain unaffected, and

the triangular relations of proportionality (cf. Eqns (15), (16)) still hold. Consequently, parameters ax,a 2 and a for the surface opening, which is still elliptical until time t,7- (i.e. until the general etchant has reached level X 2 Y2 ), continue to have the values derived above. Indeed, as stated above, the formalism remains valid down to level X4Y4. Similar remarks apply to the minor axis b derived in the following subsection.

(ii) Minor axis of the elliptical surface openings

Consider, next, Fig. 4, with PBO as the track direction. Imagine a plane perpendicular to the track and containing point Q (which is the midpoint of line KK'). The intersection of this plane (which is not the "P-plane") with the etch cone yields a circle whose diameter is GG'. Imagine also an ellipse in a plane 9arallel to the surface of the detector and with KK' - a ) as its major axis. Then the semi-minor axis (b/2)

is the line joining the centre Q of the ellipse with one of the two points of intersection of the circle and the ellipse. (Both points of intersection are out of the plane of paper in Fig. 4 and hence are not shown.)

Then, from Euclid, (the length b being a chord (of the circle) bisected at right angles by the diameter GG'),

b , = ~/GQ. G'Q (20)

P

/ /

G 1 96-ec / ' ' /

X' Y~

0

FIG. 4. Minor axis b of the surface opening: PBO is the track direction, and Q (the midpoint of line KK') is the centre of the elliptical surface opening. If one imagines a plane perpendicular to PO and containing point Q, then its intersection with the ellipse yields a circle whose diamter is GG'. The semi-major axis (b/2) is the line joining Q with one of the two points of intersection of the circle and the ellipse; these points being directly "above" and "below" Q, and hence outside the plane of paper, cannot be shown in the figure. As in the case of Fig. 3, the proof is also valid beyond the "cone phase"; at time tp (see Section 2 for definition), when the general etchant has reached levelX 3113 (Fig. 1), the surface opening consists of an exact half- circle and a half-ellipse. Beyond level X3 Y3, the elliptical part of the etch-pit opening has a "width" less than b. Note that Fig. 4

has been constructed with P O = 10 cm.

114 A. ALl and S. A. D U R R A N I

Also, from the triangles GQK and G'QK',

GQ/sin (O + Oc)=K Q/sin(2-Oc) =K Q/cos 0c N ~

and

Q/sin (0 -0c , = K' Q/sin(2+Oc)=K 1Q/cosOc G'

But since Q is the centre of the ellipse, so that KQ =K'Q =a/2 , we have from Eqn (18),

a _ V0t cos 0 c (21) KQ=K'Q 2 s in0+sin0~

By substituting the values of GQ and G'Q in terms of Eqn (21) into the expression for b/2 above (Eqn (20)), we obtain

b/2 = g Q {sin (0 + 0c)" sin (0 - 0c)} 1/2 COS 0 c

{sin 2 0 - sin 2 0~} 1/2

= V°t (sin 0 + sin Oc)

/sin 0 - sin 0c =V~t / - -

~/sin 0 + sin 0c

Hence

/sin 0 - sin 0c b=2Vgt / - - - - (22)

~/sin 0 + sin 0c

And, by putting V0t = h, the surface layer removed, v~e obtain the relation:

b - 2h /sinO-sinO,, (23) - Ns~nn0+sin0c

Again, as indicated in Section 4, these relations for the value of the minor axis of the elliptical surface opening of the etch pit hold until time te, i.e. during the time that the thickness of the detector removed h__< hp (that is, down to levelX 3 Y3 in Fig. 1), at which time the surface opening consists of an exact half-circle and a half-ellipse. Beyond the level Xa 113, the elliptical part of the etch-pit opening has a "width" less than b; however, as pointed out in Subsection (i) above, the formalism of Eqn (23) remains valid down to level

X 4 Y 4.

(iii) Radius of the circular surface openings Consider Fig. 5, where the track PO, of etchable

length R, makes an angle 0 with the detector surface.

Xl

/ X,

X~

1 / HI P _ _ / _

\ \

oi', \ \

\ \ \ -V---

' r "

\ x / \ x /

\ N / / \ \ ~ , \ A /

I

\ / u /

\ / /

H'

Yl

y~

Y~

FIG. 5. Radius r of the circular surface opening. The etch-pit opening becomes completely circular in the "sphere phase", which sets in when the general etchant has reached level X4 Y4. The distance lead taken by the track-etchant (travelling along PO) over the general etchant (travelling along, say, HBO) remains constant once the track-etchant has reached the end-point O of the etchable range. This lead is shown by z = GH' = BO = R (sin 0 - s in 0¢). Lines OA and OH' ( = u) are radii of the spherical

track-profile.

ETCHED-TRACK KINETICS IN ISOTROPIC DETECTORS 115

The time taken by the etchant to reach point 0 along the track is given by

to= R/[/T, (1)

= 2 VotR (sin 0 - sin 0c) - R 2 (sin 20 - sin 20c)

. . r2-~2hR(sinO-sinOc)-R2(sin20-s in20c) (28)

whereas the time taken by the general etchant to reach level X'I O Y'I along path HO is

t~ = R sin 0/Vg.

The time lead for the etchant travelling along the track is

by putting Vgt = h. Eqn (28) is valid at all times beyond t ,r , as long as

the appropriate value of the detector thickness removed, h, is used.

An alternative expression for r 2, obtained by .substituting the value of he from Eqn (13) (to be proved later) into Eqn (27), is:

R sin O R R sin O - R sin O¢ t~ - to Vg ~r - V~ (24)

which corresponds to a distance lead, z, such that

z = R (sin 0 - sin 0c) (25)

This distance lead remains constant once the track- etchant has reached the end of the etchable range (i.e. point O).

Let us now determine the radius r of the circular surface opening. The proof we give below relates, for simplicity, to the case when the surface opening is the entirely circular, i.e, from the time that the general etchant has reached level X4 Y4, and thereafter. Beyond the level X4 Y4 (at time hr), the "sphere phase" sets in: the track profile henceforth remains spherical and the surface opening always circular. The formulas derived, however, are also valid for partially circular surface openings, i.e. from time ruT onwards (the general etchant having reached level X2 Yz of Fig. 1) at which time the surface opening first starts to become partly circular, as stated in Section 5.

If the total time taken by the etchant to travel along POH' or POA (Fig. 5) is t, where OH' = OA = u is the radius of the sphere (or of the spherical part of the track profile), then

u = O A = ( t - R / ~ ' T ) V o = V o t - R s i n O c (26)

The radius r of the surface opening is then given by Euclid as:

r 2 = G H " ( 2 O H ' - G H ' ) = z ' ( 2 u - z ) (27)

or, using Eqns (25) and (26),

r 2 = R (sin 0 - sin 0c)" {2 V0t - R (sin 0 + sin Oc)}

r 2 = R (sin 0 - sin 0c){2h - he} (28a)

As stated, the surface opening becomes fully circular at the start of the "sphere phase", which sets in at time ttT (the general etchant having reached level X4Y4). This is also the moment of the end of the "transition phase". However, during the transition phase, the circular part of the surface opening continues to grow relative to its elliptical part, and the "width" l' (i.e. the maximum axis perpendicular to the longest axis) of the surface opening progresses through the following sequence: Prior to time tp (i.e. for h<hv), the maximum width of the surface opening is its minor axis b, since the longest chord of the circular part of the etch pit is less than b. At time te (when the etchant has reached level X3 Y3 in Fig. 1, and h =hp), the circular part of the surface opening becomes an exact half- circle: at this juncture 2r = b. Beyond levelX3 Y3, i.e. for values of the detector thickness removed h >hp, the maximum "width" of the part-circular, part-elliptical etch pits is the diameter 2r of the circular part of the surface opening (the circular part now being always larger than the elliptical part). The etch pit becomes completely circular at level X4 Y4, and henceforth (i.e. for h>htr) the "width" of the surface opening, of course, continues to be its diameter 2r.

(iv) Longest axis of the surface openings when partly elliptical and partly circular

Let us, finally, consider the value of the longest axis of symmetry of the surface openings while they are partly elliptical and partly circular (i.e. between levels X 2 Y 2 and X4Y, in Fig. 1, or in other words for tur<t ~-~ tlT or for huT < h <= htT ).

In Fig. 6, if the track PO is extended to B, which is the point of intersection of the track direction with any final etched-surface X' Y'. then BK~ corresponds to a~ (part of the major axis of the elliptical surface opening)

116 A. ALl and S. A. D U R R A N I

X

X,

,

H P y

' [ ° l

/ i ,' I

,' I •

',. r '/ "1 ij . , ' " K' BI~O ,G [ K2 K1 4"

i i [ ~ i

i H t I

L4 Q 2 ~1

y,

F1G. 6. Longest axis I of a partly-elliptical and partly-circular surface opening. The track P O has been extended linearly to point B lying on an arbitrary final etched-surface X' Y', when the detector thickness removed is h. Beyond time t,.r (the general etchant having reached level X 2112, Fig. l), the surface opening starts becoming partly circular (the bottom of the etch pit starts becoming rounded already beyond the general-etchant level X~ Y~ ). In this figure, the vestigial conical part of the etch pit is represented by line N K x , which is parallel to the tangent OE, while the spherical part is represented by the arc K H ' N . Here K G = r is the radius of the circular part of the surface opening, and B K 1 = a2 is a part of the major axis of the elliptical portion of the surface opening. Then the longest axis of symmetry of the surface opening is given by K K ~ = I = K G + B K 1 - BG = r + a2 - B G.

where B G = O G / t a n O. Note that O H ' = OK = u is the radius of the spherical part of the track profile.

in Fig. 3. Hence, from Subsect ion 6(i) above, we have:

az = y sin 0ffsin (0 - 0~) (16)

where

y = h (sin 0 - sin 0c)/sin 0- sin 0 c (14)

with V d = h.

Hence

a2 = h (sin 0 - sin 0c)/sin 0. sin(0 - 0c)

or, by mult iplying the numera to r and the denomina to r by sin(0 + 0c) ,

h ~'sin(O + Oc) • (sin O - sin Oc))

h sin(O+Oc) (29)

• . a 2 - sin 0 sin 0 + sin Oc

Also, the radius r of the circular part of the surface opening is given by the relat ion derived above:

r 2 = 2 h R (sin 0 - s i n 0~) - R 2 (sin 2 0 - s i n 2 0,.) (28)

Now, from Fig. 6, the longest axis of the surface opening is given by

But

and

I = K K 1 =a2 + r - B G

B G = O G / t a n 0

O G = H G - H O = h - R s i n O

Hence

h - R sin 0 / = a2 + r 130)

tan 0

ETCHED-TRACK KINET1CS IN ISOTROPIC DETECTORS 117

And, using the value of a 2 from Eqn (29),

h - R s i n O tl sin(0+0~)

a2 tan 0 sin 0 sin 0 + sin 0~

( h - R sin 0)cos 0

sin 0

h { sin,0 +0~) cosO} = sinO sin 0 + sin 0~

+RcosO

(cos 0c - c o s 0 / - h ~ s i n O + s i n O ? ~ + R c ° s O

So that Eqn (30) may be re-expressed either as:

fcos 0c -cos 0] l=r + R c o s O + h ~ - - - - ~

{ sin 0 + s i n 0~ J

or since

(31)

as:

a = 2h cos O,/(sin 0 + sin 0~), (19)

I = r + a/2 + R cos 0 - h cos &(sin 0 + sin 0,3

i .e.

= r + a/2 + ( R - ~' ) c o s l 0 (32) sin 0 + sin 0cJ

These equations ((31) and (32)) for the value of the longest axis of symmetry of the partly circular and partly elliptical surface openings are valid for h,r < h < hvr (i.e. between levels X2 I"2 and X 4 Y4).

We may summarize the conclusions of Subsections (i) (iv) above as follows:

From level X Y to X2 Y2, the "length" of the surface opening is a, and its "width" is b.

From level X2Y 2 to X3Y3, the "length" is (a/2 +diametral part of the circular cross-section): and the "width" is b.

From level X~Y 3 to X4Y, , the "length" is (r +axial (major) part of the elliptical cross-section); and the "'width" is 2r.

Beyond level X4Y4, the "length" is 2r, and the "width" is also 2r. F

7. P R O J E C T E D L E N G T H OF THE ETCHED TRACK

If etched tracks are "undercut" (in the sense of Henke and Benton, 1971), i.e. "if the detector surface overhangs some portion of the track", then the projected length of the track on to the detector surface becomes a measurable parameter. Thus, in Fig. 7, if P' (~'O = q5 > rt/2, then (0 + 0c) < rt/2, so that in the cone phase the condition

0 < (re/2 - 0,3

defines an "undercut" track. Such a track continues to be undercut in the transition phase so long as the detector thickness removed h < R sin0. In Fig. 7, the track ceases to be undercut when the general etchant has reached level X"Y" (with h=RsinO); the most "projecting" point S" then lies on the final etched- surface X" Y'.

In the transition phase (for h < R sin 0), the projected length s on the final etched-surface X'Y ' is given (see Fig. 7) by:

s = u + R cos 0 + a2 - h/tan 0 (33)

i.e.

h - R s i n O s = u + a 2 t an0 (33a)

which is identical in form to Eqn (30), except that the radius r of the circular surface opening has been replaced by radius u of the spherical part of the etched- track profile, and I has been replaced by s. Then, using procedures similar to those followed in Subsection 6(iv), Eqns (30) to (32), it can easily be shown that

s = u + a / 2 + ( R - - h ) co s0 sin 0 + sin OcJ

(34)

which is cognate with Eqn (32). In Eqns (33) and (34), the radius of the spherical

portion of the track profile is given by:

. = I t -- R/~/T )V o

where t is the time taken by the general etchant to reach level X'Y ' and the track etchant to reach point O; i.e.

u = h - R sin 0c (35)

c,

"5

3

G

&

;:>

D',

J ~

._E

7 co

m e-- =

' ~ = E

. - = 7 ~ ._ ca m

E -

vii E

..Ca

" *~ i m + ~ e ~ lg ~ ! I.~

A

II

_,=~

~ " d

= o ~ E .E E ~ - ' F - .

' - -~£ '= v . - - = ~ ~ . o - Y ~ ~ , ~ = = ~ ~ - o

,..¢=

E

=1 + +

Iq

2

e- k,

c a ; Z '

VII ~ >

t " ,

e~

II

E ¢ -

.~ ..~

©~i~ ~

" - ~

-~ ~ ~ ~ - ~ ~ ~.._~

~ ~ ~ "~, ~ ~ ~ .

~- ~ . = . _ ~.~ ~

~ =

v ~ - ~ ~ , ~

~ ~ VII Al l II ~ >

4-

II

. - ~ - ~

~. o ~:~

÷ 4- ~ ~ II

II II

. ~ . = ~ ' - ~ ~-~ ._===~ oo==- ~j ~ , . ~ ~.--. ;-, ~ = ~ ,..~ .~ ~-.

0 O . i ~ ~ ~

~-~- ~ ~ ~ ;~- - '~

o ~ .~

~ l ~ e ~l o

I

~ -~ +

I 0 "~

II o .~

I

li

~ ~ u =

~.~ ~ o ~- ~ oo ~ = i

If9

4-

I

o

4-

o

0

o ©

o

©

~0

2

.o

o

©

D.

2

._=

._o

120 A. ALl and S. A. D U R R A N I

' t ~ '¥ I i I ho

Xt . . . . . . i . . . . . . . . . . . E . . . . . . . . . . .

' I '

IK i y , X' i ~---- K'

I ~ h i t I

K"

I I

"1

y"

FIG. 7. Projected length s of an " 'undercut" etched t rack on to an etch-surface X ' Y ' in the " t rans i t ion phase". The most " 'projecting" point S of the e tched- t rack profile is outs ide the surface opening K K ' when viewed from "above" (i.e. a long a line paral lel to H O ) ; hence the t rack is undercut . Here OO' = u is the radius of the spherical par t of the profile: P ' K ' = a2 is a part of the major axis of the ell iptical por t ion of the surface open ing : PK'~ = h is the detector th ickness removed. Hence the projected length K 7K' = s = K 7K + K K '~ + ( P ' K ' - P ' K '1 ) - u + R cos 0 + (a 2 - h/ tan 0). When the general e tchant reaches level X" Y" (wit h/1 = R sin 0), the most "projec t ing" point S" lies on the final etched-surface X" Y", and the t rack ceases to be undercut : the surface opening then is S 'K" and O S " - O M , i.e. r = u = z {see text). Note that Fig. 7 has been cons t ruc ted with 0 = 4 0 : 0 , - 1 5 :

R = 15 cm. A t rack is tmdercut only if 0 < (re~2 0,,).

Hence from Eqns (34) and (35),

t - c o s ( ) } s = h 1 s in0+s in0 ,

a + R ( c o s O - s i n O c ) + 2 (34a)

The relations given above for s (Eqns (33) and (34)) hold for

R sin Oc<-h <_ R sin O

During the cone phase, i.e. when h < R sin 0c, u = o, and also R, the etchable length of the track, is replaced by L, the distance actually traversed by the etchant along the track :

L = ~ Vr dt = h/sin 0c (36)

(since L/[/,, r = h/Vq ),

By making these substitutions for u and R in Eqn (34), we obtain:

a / h \ s = + / L ~ / c o s O

2 \ s i n O + s i n O ~ ]

a / I I \ = + h c o s O / . . . . . /

2 \ s in O~ sin O+sin O~J

a h cos 0. sin 0 or s=-- + (37)

2 sin 0,:(sin 0 + sin 0~)

It is to be noted that the track is "'undercut" if r < u. In Fig. 7, r is actually equal to zero in the case of etch- surface X ' Y ' , since the surface opening there is completely elliptical. In a more general case, when the surface opening is partly circular and partly elliptical, r becomes finite. At the juncture when r = u, the track ceases to be undercut; this is the case with the final etched-surface X ' Y " in Fig. 7. At this point, the

E T C H E D - T R A C K KINETICS IN I S O T R O P I C D E T E C T O R S 121

distance-lead, z [cf. Eqn (25)), becomes equal to both r and u.

It has been stated above that Eqns (33) and (34) hold in the domain h < R sin 0. At h = R sin 0, we get from Eqn (28):

r z = 2R 2 sin 0(sin 0 - sin Oc)

- R e (sin 20 - sin 20c)

: R 2 (sin 20 - 2 sin 0 sin 0,. + sin 20c)

= R 2 (sin 0 - sin Oc) z

so that

we get, by equating the parameters r and b/2 at h =hp:

h e ~ s i ~ s in2/Sin0-s in~)=2hpR(s in0 - s i n 0c)

- R 2 (s in20- sin20,) (39)

On simplifying Eqn (39) we obtain:

hp = R (sin 0 + sin 0c) (13)

which is the relation that we hade promised at the end of Section 4 to derive.

r = R (sin 0 - sin 0c)

But this is identical with the expression for the distance = lead z of the etchant along the track over the general etchant (cf. Eqn (25) and Fig. 5).

Hence, the etched track ceases to be undercut when

r = u = z = R (sin 0 - sin 0c) (38)

at which juncture the detector thickness removed h = R sin 0 (i.e. at level X " Y " ) .

8. D E R I V A T I O N O F T H E R E L A T I O N F O R h e

It has been stated above (cf. Sections 5 and 6) that at level X3113 (Fig. 1), when both the terminal points of the P-circle meet the detector surface and the detector thickness removed equals hp, the surface opening of the etch pit consists of exactly a half-circle and a half- ellipse. At this juncture, the radius r of the circular part equals the semi-minor axis b/2 of the elliptic part.

Now, from the relations

9. C O N C L U S I O N

All the parameters derived in the preceding sections are collected together in Table 1, along with approp- riate comments. It will be seen that a number of useful equations have been derived in the foregoing, in particular those governing the surface openings and the detector thickness removed at various important junctures in track-etch kinetics in an isotropic de- tector. These equations should help in the easy calculation of parameters connected with the track- forming particles, e.g. their etchable range R, their average track-etch velocity VT, and by inference their charge and energy. These equations agree with those derived by other authors (e.g. Henke and Benton, 1971 ; Somogyi and Szalay, 1973), but we have related them more specifically with the successive phases in the evolution of an etched track.

Acknowledgements--Anwar Ali wishes to thank the Pakistan Atomic Energy Commission and the Colombo Plan autho- rities for leave of absence and financial support, respectively. The support of the Science Research Council for our SSNTD work is gratefully acknowledged.

and

r 2 = 2hR (sin 0 - sin 0c) - R 2 (sin 20 - sin 20c)

/ s in 0 - sin 0c "~ 1/2

R E F E R E N C E S

Fleischer R. L., Price P. B. & Walker R. M. (1975) In Nuclear Tracks in Solids, Chapter 2. University of California

(28) Press, Berkeley. Henke R. P. & Benton E. V. (1971) On geometry of tracks in

dielectric nuclear track detectors. Nucl. Instr. Meth. 97, 483489.

Somogyi G. & Szalay S. A. (1973) Track-diameter kinetics in (23) dielectric track detectors. Nucl. Instr. Meth. 109, 211

232.