Radiation Measurements 40 (2005) 160–169www.elsevier.com/locate/radmeas

LightMeV-ions etching studies in a plastic track detector

M. Fromm∗Laboratoire de Microanalyses Nucléaire Alain Chambaudet, UMR CEA E4, Université de Franche-Comté 16,

route de Gray F-25030 Besançon Cedex, France

Received 27 August 2004; accepted 4 April 2005

Abstract

Following the example of the development of photography, etching not only reveals the track at an observable scale but alsomemorises information about the latent track formation in the detector material. Searching such information in etched trackswas one of our quests during the 20 past years, using notably poly allyl diglycol carbonates (PADC) from various origins andexposed to MeV ions. After a brief recall of the physical description of primary Matter–Particle interaction and the consecutiveradiochemical modifications induced in a polymer, we will discuss the “two etch-rate” etching models pertinence. Surfaceparameter measurements such as etched track radii restore information about a fraction of the particles range which couldconsequently lead to rough estimates of the ion response functions. This is particularly the case when the primary energy lossshows important variations. It is one reason why we preferred to emphasise the importance of a track-depth measurementmethod. The region of the particles stopping point is of a particular interest for checking those aspects as it lies near to theBragg-peak. Analysis of light ion tracks(1< Z <6) in the initial energy range 1–2MeV/amu are described. The measuredresponse functions appear distorted compared to those obtained for primary energy loss and the possible explanations for thisobservation will be examined and illustrated. Implications on related PTD’s applications will be discussed.© 2005 Elsevier Ltd. All rights reserved.

Keywords:CR-39; Nuclear track etching; Response function

1. General considerations

Particle track detectors (PTD) were first used under theform of a corpuscular photographic plate called nuclearemulsion (Powell and Occhialini, 1947). Lots of physicistshad studied alpha tracks from the alpha-decay of very heavynuclei in the emulsion, from very early times, in the 1890s,and even earlier (Blau, 1925; Blau and Wambacher, 1937).In 1959, the pioneering work of Silk and Barnes (Silk andBarnes, 1959), who had imaged fission fragment tracks inmica using transmission electron microscopy, opened theway to analyses of fossil nuclear tracks in minerals. In 1958,

∗ Tel.: +33381666560; fax: +33381666522.E-mail address:michel.fromm@univ-fcomte.fr.

1350-4487/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.radmeas.2005.04.028

Young proposed the etching of radiation damage in lithiumfluoride (Young, 1958). In 1975, Fleischer, Price andWalker,three American researchers, initially working at the GeneralElectric Research Laboratory, published a book in whichthe principles and applications of track etching are exposed(Fleischer et al., 1975). They wrote later in 1982 “Heav-ily ionising particles produce radiation damage tracks in awide variety of dielectric solids. Chemical etching rendersthe tracks easily visible and can be used quantitatively tomeasure the properties of the nuclear particles” (Fleischeret al., 1982). In the nineteen’s, latent tracks were even ob-served in metals (Dunlop et al., 1990) or in semi-conductormaterials (Canut et al., 1998), several additional books havebeen published dealing with particular topics of the nucleartracks and their applications (Durrani and Bull, 1987; Spohr,1990; Durrani and Ili´c, 1997; Fleischer, 1998).

M. Fromm / Radiation Measurements 40 (2005) 160–169 161

Fig. 1. The time scale of physical and chemical processes involvedin the nuclear track formation. On the top left of the figure, thenuclear track etching process in a thin foil is sketched. In this case,information about damage density variations (increase, decrease)is visualised through the etched track wall curvature (convex,concave).

Tracks produced by charged particles in external detec-tors, are nowadaysmainly registered in a polymeric isotropicmaterial composed of poly allyl diglycol carbonate (PADC).The name CR-39, often used instead of PADC, is actuallyan abbreviation for Columbia Resin #39 because it was the39th resin formula developed by the Columbia Laborato-ries. CR-39� is a registered product name of PPG industriesInc., the monomer was patented in 1947 by the ColombiaSouthern Chemical Company. Its use as a solid state nu-clear track detector (SSNTD) was suggested in the seven-ties where two papers were published in a short time range(Cartwright et al., 1978; Cassou and Benton, 1978). Duringthe past decades, this polymer was applied to the fields ofcosmic ray detection, plasma physics, nuclear physics, en-vironmental studies and dosimetry, health and biophysics.Track etching in PADC is generally achieved by using a

hot and concentrated basic (NaOH, KOH) solution, some-times with alcohol in addition (potassium–ethanol–water,PEW), which accelerates the etching process. Carbonatesreact with bases in solution. The main idea of track etch-ing is that it exists a region of the polymer, in the vicin-ity of the particles trajectory, where chemical modificationsspecifically enhance the etch rate. The volume of the PADCsurrounding the latent track is etched with the bulk mate-rial rate which acts isotropically at a lower velocity. Thus,an etch pit is created and it can be observed under a mi-croscope. Etching raises the question of time scales, on avery wide range. The etching process itself is generally per-formed during hours, but the primary interaction that givesbirth to a track does not need more than 10−18s to takingplace.Fig. 1 summarises the time scale of the physical and

chemical processes that are involved in the nuclear trackformation (Turner, 1995).The dynamics of nuclear track formation in polymers is

quite complex. In addition to the temporal aspects of radio-chemical processes, the spatial distribution of permanent,reactive or modified defects will affect the track structure.This, in turn, gives raise to the question of memorisationcapabilities of etched tracks over time, due to possible mod-ifications of the latent tracks (aging, fading, annealing, etc.).The length of a latent track is generally considered constantand equal toR, the range of the initial charged particle.The range of a high-energy ion in a given material can bedetermined using the Bethe theory (Bethe, 1930) and im-proved computation codes. Computer codes have been andare intensively developed which enable the ranges and en-ergy losses in compound materials to be calculated (Bentonand Henke, 1969; Ziegler et al., 1984; Dörschel and Hen-niger, 2000). Range is an essential value. The statistical lon-gitudinal and radial straggling calculations are incorporatedin some of the codes. The radial dimensions of a latent trackwere studied experimentally and theoretically (Katz et al.,1972). Radial effects can be described with the delta ray the-ory in which the secondary electrons are responsible for theformation of the main defects. Both Bethe and delta ray the-ories are velocity dependent theories. It should be remem-bered that the electrical charge of the projectile is not con-stant; it varies with velocity, which is generally taken intoaccount using the concept of an effective charge (Barkas,1953). According to the delta electrons number, their energydistribution and to their effects on matter, the track structurecan be described using the concepts of track core (where thedensity of deposited energy is high, sometimes creating areal channel due to high ionisation and thermic effects) andthe track halo (where the density of defects decreases withrespect to the distance to the particles trajectory). The halodecomposes in several entities (short tracks, blobs, spurs).There are a number of Monte-Carlo track-structure codesthat have been developed independently these recent yearsfor radiobiology purposes (for a review, seeNikjoo et al.,1998). In the case of MeV/amu light ions slowed downin polymers, the track core radial extension remains gen-erally little in comparison to one micrometer and very lit-tle with light ions. Spectrometric measurements indicate in-creasing (power-type) behaviour of the core radius with LET(Yamauchi et al., 2003) in CR-39. The latent track halo ra-dial dimensions vary from a few nanometers to the scale ofthe thousands of nanometers in cellulose triacetate whenZ

varies from 2 (Helium) to 18 (Argon) (Vareille, 1982). Asan illustration,Fig. 2 shows the distribution of radial dose(Gy) in CR-39 calculated for a Li ion projectile with initialenergy 3MeV. In polymers, there is a rather satisfactory ac-cordance between calculated radial dose (Waligorski et al.,1986), or LET and measured latent track radial dimensions(Yamauchi et al., 2003; Vareille, 1982). Another quantity,the restricted energy loss (REL), is useful in track analy-ses, it has been and is still widely used, particularly at high

162 M. Fromm / Radiation Measurements 40 (2005) 160–169

8

7

6

54

3

2

100

24

6 0 500 1000 1500 2000 2500 3000

Energy (keV)

Radial distance(nm)

Dos

e (G

y)

× 106

Fig. 2. Radial delta-ray dose calculated for a lithium ion,Z = 3(initial energy 3MeV) slowed down in a PADC detector as afunction of the radial distance to the trajectory and the ion’s energy.Calculation are performed with the formalism byWaligorski et al.(1986)using the Matlab� software.

energies for cosmic ray analyses and nuclear physics. RELwas proposed by Benton (Benton, 1967), it uses a cut-offdelta ray energy,�, considering that the outer part of thetracks halo do not participate in the first-order etching pro-cesses and then do not contribute essentially to the tracketching rate. The work by Apel and coworkers (Apel et al.,1999) about the track pore membrane etch mechanism inpolyethylene terephtalate shows that a variable radial etchrate exists inside the latent track. The general trend ob-served in polymers for the radial track etch rate is a decreasefrom VT to a value lower thanVB then an increase fromthis minimum up toVB at distances covering 100nm fromthe ions (MeV/amu, gold) trajectory. Sub-micronic radialetch effects also have been observed by Francisco, Bernaolaand co-workers in heavy ion tracks etched in polycarbonate(Francisco et al., 2004). Such effects are signatures and theycan be observed for short etching times and seem to vanishat longer etching time durations. The authors report that for0.09 and 0.02MeV/amu light ion tracks in polycarbonate,the variations are smooth and the transitions between zones(core, halo, cross-linking, bulk) are not clear. This looks asa kind of limit for the obtention of precise information bymeans of etched track analyses at the micrometer scale.The topic of the present paper is mainly concerned with

this question. We particularly would like to focus on fewMeV/amu light ion tracks (say 1�Z�6) etched in CR-39. With such ions, the latent track can be considered asa line segment at the micrometer scale (optical observationof etched tracks). The track etching process can be welldescribed with a two etch rates model. This model willfurther be considered in details in the particular case of ionswhose stopping and further etching include the Bragg-peak(typically the case inFig. 2).

Fig. 3. Conical in shape track etching model.

2. Two etch rate models and detector response

Etched track analysis deals essentially with two final ques-tions. On the one hand, can we get the particles identity(Z/�, . . . .) by measuring etched track parameters and onthe other hand, what is the relation between the numberof etched tracks and the real number of particles that havecrossed the surface of a detector?These problems are generally treated in terms of response

function (qualitative analysis), as well as critical angle anddelayed-action time, called induction time by others or us-ing the concept of critical layer removal, under given etch-ing conditions (quantitative analysis). These concepts aresupported by track-etch models, generally based on the twodistinct etching velocitiesVT andVB, (Benton, 1968; Fleis-cher et al., 1969; Henke and Benton, 1971; Somogyi andSzalay, 1973; Paretzke et al., 1973; Fleischer et al., 1975;Durrani and Bull, 1987). VB stands for the bulk etch rate(isotropic) and theVT symbolises the specific track etch rate(anisotropic and longitudinal).The most obvious approach is to consider the simple

model of a conical in shape etched track. Based on constantbulk and track etch ratesVB andVT, this model neverthe-less leads implicitly to a paradoxe. We will first recall thebasics for this model.As illustrated inFig. 3, with this model, both etch rates

are linked by the simple relation:

sin� = VB

VT= V −1, (1)

where sin� is of course a constant value. Thus, we meetthe paradoxe, becauseVT has to be different with differentenergies of a same type of ion, which is implicitly impos-sible within the range of the particle. For example, if thismodel of a conical in shape etched track would always be therule for describing the etching process, then all alpha par-

M. Fromm / Radiation Measurements 40 (2005) 160–169 163

ticles (whatever is their energy) would give the same trackwith a same incidence angle and within given etching con-ditions. Models for varyingVT’s were developed since the1960–1980’s (Fleischer et al., 1975; Somogyi and Szalay,1973), in thesemodels, the variations of theVT are taken intoaccount with monotonically increasing or decreasing para-metric functions of the range or/and energy losses, ionisa-tion rates or they were provided using calibrations based onetched track measurements (generally ignoring the Bragg-peak region). Experimentally, conical in shape etched tracksare observed. This is particularly the case whenVT?VB orwhen the rangeR is large compared to the etch removal,VB.t , within a short etching time. This suggests that a conicalin shape model can nevertheless be applied in special casesor as a limit, particularly with heavy and very energetic ions(cosmic rays, etc.) with which the above-mentioned require-ments seem to be fulfilled. If applied, this model enablesV , the sensitivity or etch ratio, relations (1) and (2) as wellas the reduced etch rate(V − 1) to be determined (Benton,1967). A conical track with a diameterD and a lengthL,etched duringt time units has the following etch ratio

V = VT

VB= L + VB.t

VB.t= 1+ (D/2.VB.t)2

1− (D/2.VB.t)2. (2)

The third expression in (2) is obvious fromFig. 3. Thederivation of the last expression is described in details in apaper by Somogyi and Szalay (Somogyi and Szalay, 1973).Thus and whatever theV value, for one given conical in

shape track, the etch ratio or sensitivity is a unique real num-ber. These relations, strictly applicable to normal-incidentparticles could be computed in such a way that they includesthe effect of the incidence angle (Somogyi, 1980; Oda etal., 1992). Then, on a mathematical point of view, we dealwith ellipses and conical sections.It seems us interesting to look at what happens if the en-

ergy deposition rate varies notably over distances compa-rable to the etched removal;(VB.t), and particularly whentheVT includes a peak. Important and localised variationsof the energy dissipation must produce important variationsof the VT. If the VB can be considered constant and theVT variations are known, a model can be established whichwell matches experimental track parameter measurements inmost of the cases (Dörschel et al., 2003), including conicalin shape tracks and overetching without any discontinuity.This model takes theVT’s variations into account; in con-sequence, the cone angle� is replaced with a more realisticvariable angle�. Such an approach was discussed in sev-eral papers; see for example (Paretzke et al., 1973). Etchedtrack profiles can be calculated as soon asVT’s and theVBare available (Fleischer et al., 1982; Fromm et al., 1991;Dörschel et al., 2003).It is important to note that theVB can vary from one

experiment to another even if the same detector materialas well as etching conditions (C, T) and time duration areused. This is mainly due to the etching device conception,

Table 1MeanVB values(�m/h) obtained by three different teams (Kobe-Japan, Dresden-Germany, Besancon-France) (Vaginay, 2001)

Kobe Dresden Besancon

Tastrak 1.80 1.73 1.92Baryotrak 1.70 1.64 1.79

Etching conditions are NaOH 7.25M, 70◦C. Measurements wereperformed on 252-Cf tracks etched for 12 different etching timesup to 6h. Results are for both Tastrak (UK) and Baryotrak (Japan).

in which the samples are or not stirred and if yes, how arethey stirred? Statistical averagemeasurements illustrate sucha trend; seeTable 1which summarises the measurementsmade by three teams, located in Japan, Germany and France(Vaginay, 2001) using a same set of detectors.Let us have a glance at etched track profiles and model-

based simulations in different cases. We will use theVBvalue as measured in our laboratory with Tastrak CR-39etched in NaOH 7.25M at 70◦C;VB=1.95�m/h.Any otherexperimental value could whatever be used. Consider ameanVT = 6�m/h all along the particles range of 30�m, whichroughly corresponds to a particle withZ ≈ 2–3 and an initialenergyW ≈ 1–2MeV/amu, based on our observations.The mean〈VT〉 = 6�m/h could in principle be obtainedusing infinity of differentVT functions. We will considerthe four following cases with first a constant value (conicalin shape model,VT = 6�m/h), a decreasing linear value(from VT ∼ 10 toVB = 1.92�m/h, with an averageVT ∼6�m/h), a decreasing power function (with a higher averageVT ∼ 9�m/h) and a Bragg-type function (averageVT ∼6�m/h), maximum atu=27�m andVmax

T =16�m/h. Allthose functions fall down to theVB value at the end of therange, one of them; the constant etching rate, presents adiscontinuity which do not allow the over-etching processto be taken into account in a same step, revealing someweakness on the strict physical point of view. In previouspapers, most of the researchers overpassed this problem byusing different stages during the etching process: Latenttrack etching, transitional phase, spherical phase (see forexampleSomogyi, 1980).Here, the two later used track etch rates were obtained

using a flexible parametric function depending on the depthin the material,u, and on two parameters

VT(u) = VB + 2a(Vmax

T − VB)(R − u)

a2 + (R − u)2, (3)

R stands for the particles range(�m), (R − u) being theresidual range,Vmax

T is the maximumVT value, and “a” isthat particular value of(R−u)where themaximum tips. Thisfunction, which is not infinitely flexible, has no particularphysical meaning but it gave nevertheless good results foralpha particle track etching simulation, it has to be seen as atool for etching simulations due to its versatility. Simulationsare performed with a tailor-made computer code written in

164 M. Fromm / Radiation Measurements 40 (2005) 160–169

Fig. 4. Simulation of the sequential etching of a track with a rangeR = 30�m for 6 h (step= 1h). From left to right, the usedVT’s are aconstant value of 6�m/h (conical in shape model), a decreasing linear value (fromVT ∼ 10 toVB = 1.95�m/h) with an average value of6�m/h, a decreasing power-like function (with an average〈VT〉 9�m/h) and a Bragg-type function (average value of 6�m/h).

the visual basic language. For computing the co-ordinatesof the points that belong to the etched track walls at a givenetching timet we used the following expressions

X = u. cos(i) + ru. sin[�u ∓ i],Y = u. sin(i) ± ru. cos[�u ∓ i], (4)

where “i” denotes the incidence angle. The signs in (4)depend on the side of theX-axis whereY co-ordinates arecalculated,X stands for the normal to the surface of thedetector, the incidence angle “i” is measured with respectto theX-axis, theZ-axis points perpendicularly out of thesection that includes the particles trajectory in an etchedtrack. More details about how to compute these relationscan for example be found in (Dörschel et al., 2003).These expressions are equivalent to those proposed ear-

lier (Fleischer et al., 1975;Fujii and Nishimura, 1986), theyaccount for all phases of the etching process (including theover-etching or spherical phase) if the usedVT drops downto theVB at end of the trajectory. In the present form, theymight be able to describe accurately any incidence anglesetching processes of light ion tracks.Track longitudinal sections after 6 h etching with one hour

steps as well as the usedVT functions are presented inFig. 4. In a first glance, no very significant differences areobserved in the etched lengths and apertures; the etchedtrack shapes nevertheless appear as different. In order toget a better comparison, we can determine the responses ofsuch tracks in terms of sensitivities,V ; using the expressionsin (2), keeping in mind that these expressions strictly arederived from conical in shape etched tracks. The inputVTfunctions are displayed onFig. 4. We have then a base fora comparison of calculated track responses using diameters

0123456789

1011

0 1 2 3 4 5 6 7 8 9 10 11

VB

Etching time (h)

<V>

x > R

Re

spo

nse

s, V

V(D) conical

V(D) linearV(D) powerV(D) Bragg V(L) conicalV(L) linearV(L) powerV(L) Bragg

<V>=6/(1.92)=3.125

Fig. 5. ResponsesV obtained from the simulation of the sequentialetching of a track with a rangeR=30�m for 10h(step=1h). Themean expected value is indicated. The zone where the end of theparticles range is etched is indicated by dashed lines. Open symbolsare used for the responses determined by means of diameters(D) asfull symbols are for responses calculated with etched track depths(L).

(D), track lengths(L) and initial as well as “exact” or better,reference track etch rates.The main results obtained from this simple simulation

concerns the validity of the conical in shape track-etchmodelused for light and low energy ions in CR-39 in the par-ticular application where responses or sensitivities are de-termined. With a 6�m/h constant track etch rate, we haveV = VT/VB = 6/(1.92) = 3.1, based on a conical etchedtrack geometry. InFig. 5, we present the calculatedV val-ues for the four usedVT functions after a simulation of a10h etching period with 1 h steps. The decreasing linearfunction with an average 6�m/h gives rather unsatisfactoryV values, which is not so surprising. The decreasing power

M. Fromm / Radiation Measurements 40 (2005) 160–169 165

like function, used with a higher average value of 9�m/hleads to unacceptableV values when using the diametersand provides high values for the short etching times thattend to theVB value for longer etching times. The Bragg-type function exhibits first, as expected, lowV (D) that arenearly 80% of the mean value(6�m/h) but tends then tothe averageV = 6�m/h value after 10h etching withoutexceeding this average value. The Bragg-peak is totally ab-sent (smoothened) with such a measurement. As for theV (L) values calculated with the Bragg-type function, dueto the drastic variations of theVT, we observe first, as ex-pected, an increase, then a decrease of the response thattends to theVB. This confirms clearly that a model based onstrictly conical in shape tracks applied to etched tracks pro-duced by a varyingVT (Bragg function) is consistent onlyif average values are needed. These responses are neverthe-less mean values which do not conserve precise informationabout the realVT, notably the peak. At the contrary, whenthe responses are determined using etched track depths orlengths(L), we over- as well as under-estimate the expectedaverageV ’s depending on the used etching time. With amaximum of the inputVT function that peaks at 16�m/h(Vpeak= 8.33) we find a “measured”V nearly equal to 4,meaning we under-estimate the response with a 50% accu-racy. It has to be kept in mind that we used a meanVTvalue that is close to theVB one (V ∼ 3). More preciseresponses or equivalently specific track etch rates were as-sessed recently by the Dresden, Kobe and Besançon teams(Yamauchi et al., 2001). Several other authors did evaluateprecisely PADC sensitivity to low energy ions in the past(Zamani and Charalambous, 1978; Al-Najjar and Durrani,1984; Hafez et al., 1991; Golovchenko, 1992). The mainresult yield by such precise analyses shows Bragg-type be-haviours of theVT functions. The sensitivity no more is rep-resented by one number but by a whole data set (responsefunction). It has been shown thatVT and LET or REL�maxima, or radial dose do not necessarily pit at the sameresidual range co-ordinate. This might be caused by depthdependant effects due to a possible anisotropy of the chem-ical composition inside the CR-39 including dissolved gasconcentration gradients. Additional effects of statistical na-ture are due to straggling, scattering and multiple electronsscattering in relation with the particle–matter interactions.The detector response was calculated in the framework ofthe Poisson distribution as a function of the ions linear en-ergy transfer (LET) for light ions slowed down in CR-39(Fromm et al., 2004). Calculations were performed after theLET were shifted along the range in order to get a coinci-dence with theVT maxima. Following the particles propa-gation, the region before the Bragg-peak was named BetheBloch (BB) as the region after the Bragg-peak was calledthin down (TD) by authors studying nuclear tracks in emul-sions. With LET as an input parameter and a non linearfit to obtain the values�, LET0 and the normalisation con-stantC of the many hit theory that describes these regionsof theVT’s, we found that those parameter depend both on

the type of ion and their energy. In addition, in most of thecases, whenVT values (sets of values over a whole energyrange) from both regions, BB and TD, are studied versusLET, hysteresis-like behaviours were observed: curves looklike loops at the vicinity of the Bragg-peak. Such results arerather discouraging if dealing with track analysis but are inturn consistent with recent fundamental physics experiments(Neugebauer et al., 1999), where the yields of backscatteredand forward electrons produced are different before and af-ter the Bragg-peak: there is no bi-univocal relation betweenLET and measured angular and energetic�-electron distri-butions. It is therefore questionable if, in their simplest ex-pression, restricted energy loss or radial dose calculationsmight be applied with success to the description of experi-mental data using the multi- or many hit theories in condi-tions where a latent track is etched. Several papers presentedin this conference are linked to such a topic. It is importanthere to distinguish those studies were the responses weremeasured in CR-39 at high energies and with heavy ions(Heinrich et al., 1989; Bhattacharyya et al., 1995) and thosewhere light ions with A-MeV energies are used. With heavyenergetic ions, the validity of REL200 as a cut-off energyhas been demonstrated using detector calibrations thanks tohigh-energy ion beams. In the case of light ions, more in-vestigation is required, depending on the need for a roughaverage response or real response functions.

3. Two etch rate models and detection capabilities

It is expected that the angle of incidence of a particle ismemorised in the final geometry of an etched track aper-ture. With a conical in shape model, etched normal-incidentparticle tracks must be circular and sloped trajectories willproduce elliptical apertures in the surface plane of the de-tector. On the other hand, a variableVT(u) with depth pro-duces convex or concave etched track shapes depending ontheVT slope and whose apertures are circular at normal in-cidence but not necessarily elliptical, rather ovoidal; egg-or drop-shaped, as observed experimentally at larger inci-dence angles. This difference is illustrated inFig. 6, wherethe same inputVT functions as mentioned before are used inorder to determine track apertures shapes for an incidenceangle of 60◦. For the track perimeter computation we usethe following expressions (Fromm, 1990)

X = VBt ,

Y = 1

sini[u − VBt. cosi + ru sin�u],

Z = ±√

(VBt − ru)2 − (VBt − u cosi)2 − (Y − u sini)2.

(5)

These expressions were successfully tested with proton andalpha particle tracks and thus are suitable for light ions tracketching computer simulation. This set of relations includesall phases of the etching process (over-etching or spherical

166 M. Fromm / Radiation Measurements 40 (2005) 160–169

Fig. 6. Track mouth apertures calculated in the etched plane of the CR-39. FourVT’s are used as inFig. 4. The incidence angle is takenequal to 60◦ with respect to the normal to the initial surface.

Fig. 7. Critical angle and delayed-action time for two track-etch models, i.e. constant and Bragg-typeVT’s. The concept of delayed-actiontime, experimentally verified cannot be explained completely on the basis of a constant track etch rate. This particular time,td, only canbe determined using a numerical integration as it is the solution of a transcendental equation.

phase) as soon as the condition of a continuity between theVT and theVB exists. The interesting thing is that sometimes,when computing, no track appears before a given etchingtime (we call this delayed-action time when an etched trackhas finally a chance to appear after a given etching time).Different is the case where the etched track never will ex-ist over a given incidence angle (critical angle). These casescorrespond to well known analogues that were mainly stud-ied in the framework of a conical in shape track, where thecritical angle is not energy (or depth) dependant and the de-layed action time is depicted by means of a removed layerthat does not enable to take into account those etched tracksthat appear after a given etching time (it is in fact impossible

in the framework of a constant track etch rate). When deal-ing with a variableVT, these questions have been studied indetails using experimentally determinedVT functions; thedetection efficiency can be calculated precisely, includingresolution limits of the observation tool. There are for exam-ple possible applications for neutron dosimetry and radonmeasurements. The bases for such calculations have beenpresented in details in the next references (Membrey et al.,1993; Barillon et al., 1995; Fromm et al., 1996; Meyer et al.,1997). An illustration of the experimental evidence and de-termination of the variation of the critical angle with respectto alpha particle energy can in found in (Calamosca et al.,2003). The most important points are summarised inFig. 7

M. Fromm / Radiation Measurements 40 (2005) 160–169 167

where the differences between a constant track etch rate anda variable one in the framework of track etching models andtheir ability to predict the detection efficiency are compared.The relation between incidence angle and easy to measureetched track parameters (not diameter or length) was inves-tigated with fission fragments. In this case, there is a simplebi-univocal relation between the feel factor or shape factorof the etched track aperture and the incidence angle (Pussetet al., 2005). Another interesting approach was proposed bySkvarc (1999)) where the average grey level is measured inindividual etched tracks.

4. Open questions and experimental reality

Track etching models dedicated to light ions that use avariable track etch rate can be useful when applied to radonand its daughters dosimetry as well as neutron detection. Inthis later case, the whole energetic range covered by the neu-trons can be as huge as ten decades (typically 10−3–107 eV)which makes it particularly difficult to include all processesthat may occur for micro-dosimetry purposes. CR-39 seemswhatever to be a good candidate for such applications. Sev-eral other applications of fast ion etched tracks in CR-39with few MeV/amu ions exist (radiobiology, cold or “hot”nuclear fusion experiments, laser plasma ion jets, etc.). Itshould be noted that a well applied dosimetric method basedon CR-39 should include the use of calibrated sources orfields (national or international standards or references) andthe respective calibration curves determined in given etch-ing conditions.We would like to indicate some questions that still are

open and of interest for an accurate etched track analysis.The first one concerns the question of invariability or not

of the track etching process when the incidence angle ofa same particle varies. We generally consider that due toisotropic bulk etch properties of the CR-39 material a trackwith an incidence anglei has the same shape as a normal-incident track tilted by an anglei. It means we suppose thatan etched track with an incidence angle “i” is the same as theperpendicular (to the surface)—one, cut by a plane inclinedwith an angle “i”. This was nevertheless never confirmedexperimentally in our knowledge. InFig. 8, two alpha par-ticle tracks are shown. The 3-D snapshots were obtained bymeans of confocal microscopy (Meesen and Poffijn, 2001).The tracks were produced with 2.7MeV alphas etched 2.5 hwith 7.25M NaOH at 70◦C. The track on the left side (a)corresponds to a 60◦ incidence angle as the track on theright side (b) is at normal incidence. The track profile in“a” was extracted and superimposed on the picture “b”. Thesuperimposition is not perfect and a part of the extractedprofile moves away from the etched track walls at normalincidence (lower right side of picture b). TheVT’s were de-termined in both cases and compared. Such a trend could bedue to the statistical variations in the particle beams energyand trajectory fluctuations or to an angle dependant effect

Fig. 8. Effect of the incidence angle of the measured responsefunction. The contour of an etched alpha track(W = 2.7MeV)

with a 60◦ incidence angle is superimposed on an etched trackwith a normal incidence. Both tracks were etched in the very sameconditions (NaOH 7.25M, 70◦C) for 2.5 h.

specific to the etching process, at most to both of them. Anangle-dependant effect should affect the response functionat high incidence angle values. More experimental resultsare required.The second point concerns the so-called depth dependant

effects. Such effects were observed that lead to different re-sponse functions for same ions with different energies. Dif-ferent average response functions cannot be superimposed(Dörschel et al., 2002). This aspect was studied these re-cent years and needs more investigation. Another interestingpoint concerns the etching of end of ion ranges. This part ofthe track we call TD is not yet well understood on a physi-cal point of view. It might be interesting to investigate thisparticular part of a track in details using chemical etching.It is for example of a high interest for a good understand-ing of the physics of hadron therapy applications where theBragg-peak has to be located inside a tumour. The variationin forward and backward rates of scattered delta electronson both sides of the Bragg-peak (Neugebauer et al., 1999)seems to be one of the keys to a better understanding ofsuch phenomena.

5. Conclusion

In this paper we summarised our understanding of fewMeV/u light ions track etching in PADC giving a special at-tention to the most often used etching models. It was shownthat a model including a variable track etch rateVT(u) leadsto more accurate numerical simulations. Sensitivities de-termined by means of the well-known expressions derivedfrom a strictly conical in shape model provide estimates ofthe response that tend at least to the average of the responsefunction taken all over the particles range when using thediameters. When using the etched track depths, a variationof the determined responses with etching time can be ob-served that exhibits a maximum value linked to the Bragg

168 M. Fromm / Radiation Measurements 40 (2005) 160–169

peak of theVT. The drastic variation of theVT is never-theless attenuated when using this method. When a precisemeasurement of the specific track etch rate is performed, ac-curate simulations can be performed that make it possible todetermine the critical angle, delayed-action time and finallythe detection efficiency for any particle energy. It should bekept in mind that these problems can be overpassed experi-mentally by using calibration methods, which is for exam-ple the case for radon or neutron dosimetry where referencesources or fields can be used for a dose determination infixed etching conditions. If an etching model with a Bragg-type track etch rate provides undoubtedly a more accuratedescription of the chemical etching, several points remainunclear. Some of them were mentioned before and might beconsidered as an Ariadne thread for future studies aiming ata precise description of ion track etching in isotropic media.

Acknowledgements

Wewould like to thank Dr. David Pusset for refreshing ourcomputer codes. This paper is dedicated to the memory ofProfessor Birgit Dörschel and Professor Alain Chambaudetwith which I had many fruitful discussions, they both havepassed away in the two last years and played an importantrole for our progress in track physics. I could not finishthis paper without thanking the 22ICNTS local organisingcommittee who invited me to present this paper.

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