Calculation of Track Parameters of Alpha Particles with Various Energies and Incident Angles in CR-39 Detector

In this paper, the bulk etch rate, VB of the CR-39 detector employed, was found experimentally to be equal to 1.49μmh. Detector samples were irradiated with alpha particles energies from 1 to 5MeV and incident angles from 20 to 90. After etching in a 6.25N NaOH solution kept at (70±1) C, the lengths of major and minor axes of the alpha track openings were measured, and the track etch rate, VT for normal and oblique incidence was calculated. The relationship between VT and the track depth, x of the alpha particle in the detector for normal incidence was studied. Track length, L and track length growth rate, dL/dt were also calculated. These data were used altogether to derive a V function, which took the functional form of the Brun’s function with new constants (a1=0.091, a2=1.2, a3=1.6, a4=1.61, a5=1.425). Experimental V function was compared with those computed by TRACK-VISION program, and a good agreement with three of them was found.


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INTRODUCTION Solid state nuclear track detectors (SSNTDs) are widely used in many fields of science and technology. Nearly almost these applications concern the detection of alpha particles emitted by natural radioactivity; therefore, the response of SSNTDs detectors against alpha particles is of essential interest (Nikezic and Yu, 2004). Among their applications are radiation protection, environmental radiation monitoring, accelerators, nuclear reactors, neutron monitoring, space research, geology, geophysics and radon studies (Nikezic et al., 2005. Immè et al., 2013. One of the most widely used as SSNTD is made of polyallyldiglycol carbonate, which is commercially available as CR-39 detector in which visible tracks can be formed after ion irradiation and suitable chemical etching condition (Durrani and Bull, 1987). The basic model describing the track geometry considers two different and specific etch velocities: the bulk etch rate, V B (i.e., the rate of chemical etching of the undamaged surface), and the track etch rate, V T (i.e., the rate of chemical etching along the ion trajectory). Both etch velocities can be influenced by the chemical structure of the detector material and several additional external conditions such as storage, etching temperature, etchant type, concentration and purity. Moreover, the track etch velocity depends sensitively on particle's type and energy (Hermsdorf, 2012). The sensitivity function V (which is also called the etch rate ratio V=V T /V B , or V function in the literature) is the most important quantity in the theoretical modeling of the track formation mechanism (Fleisher et al., 1975. Mheemeed, et al., 2013. While the measurements of track-opening diameters are relatively straightforward, direct measurements of track lengths are relatively difficult (Hermsdorf, 2009). However, track etch rate, V T and sensitivity function V, are usually determined through measurements of track-opening diameters (Hermsdorf and Reichelt, 2010). Analysis of tracks produced by protons, alpha particles or nuclear fission fragments in SSNTDs constitutes a very important tool in various areas (Ilic et al., 2003). For example, the geometrical properties of the cone formation in the SSNTDs can give information about the energy, charge, and the angular distribution of the interacting particles. Direct measurements of track parameters (major and minor axes, depths, etc.) are usually performed by optical methods (Palacios et al., 2011). Independently, two theoretical models for predicting track parameters and profile revealing have been developed by (Fromm et al., 1991, Nikezic andKostic, 1997). The equivalence of both models has been verified theoretically and experimentally by Dörschel for normal and oblique particle incidence (Dörschel et al., 2003a, and2003b).Two computer codes (TRACK_TEST and TRACK_VISION) for track parameter and wall profile calculations have been developed and published Yu, 2006, Nikezic, andYu, 2008).
The aim of this work was to present new experimental data on: The bulk etch rate, V B , angular dependence of major and minor track axes induced by alpha particles in CR-39 in various energies, and track etch rate V T , estimate the sensitivity function V for perpendicular case. Calculating track length, L and track length growth (dL/dt) for different alpha particle energies and etching times. Study the angular dependence of experimental data of V function for different alpha particle energies, and finally to compare and discuss experimental V function for the present work with those for other researchers.

EXPERIMENTS Bulk etch rate measurement
One of the well-known method to determinate the bulk etch rate, V B is the 'gravimetric' method (Kocsis et al., 1997), which is based on the measurements of detector mass before and after etching. A piece of detector material with dimension (10×10) mm 2 was weighted first and etched several times in time intervals 2 h, After every etching process the sample was dried and weighted. Based on the mass difference and the known density of the detector, it is possible to calculate V B .

Sample irradiation, etching and track axes measurements
CR-39 detectors samples of dimensions (5×5) mm 2 and a thickness 1.5 mm were irradiated with alpha particles energies from 1 to 5 MeV in steps of 1 MeV through a collimator, and incident angles varied from 20˚ to 90˚ in steps of 10˚ for a reasonable irradiation time of few minutes. The alpha source employed in the present study was a planer 241 Am source (activity 1 µCi and main alpha energy = 5.485 MeV under vacuum). In order to obtain different alpha particle incident energies, the relationship between alpha particle energy and distance, given by (Mahesh and Mustafa, 1976), was used with normal air (at NTP) as energy absorber. The desired incident angle was restricted by placing the detector on stand with a hole of diameter 1 mm confront the collimator guiding the alpha particles. Before starting the track analysis, the optical microscope calibration was performed using a micro-grid with known mesh. The detectors were etched by a standard etching condition, 6.25N NaOH solution at (70±1) C˚, in a water bath for different time intervals. During etching process, concentration and purity of etching solution were controlled regularly and refreshed every 3 h usage. After etching, detectors were cleaned in distilled water to remove etchant and etch products from detector surface. Diameters of different alpha-particle energies and incidence angles were measured by means of a semiautomatic system, consisting of an optical microscope (Nikon with magnification of 400x) equipped with (MDCE-5A) digital camera connected to a personal computer for images storage and analyses.

CALCULATIONS Bulk etch rate measurement
The bulk etch rate, V B can be calculated using the relation given by (Kocsis et al., 1997).
where A, is the detector surface area in (cm 2 ).
ρ is the density of the detector material in (g.cm -3 ).
∆m/∆t, is the slope of the relation between mass difference and etching time. Track etch rate V T For perpendicular incident particle a circular track is produced, in which the time dependent major and minor axes are coincide with each other, (i.e. D maj =D min =D). In this case, track etch rate, V T is related to track diameter, D(t) (Hermsdorf and Reichelt, 2010).

…………………………..(2)
where h, is the removed layer thickness (h=V B .t). For oblique incidence the circular track opening changes to elliptical contours with a major axis D maj and a minor axis D min where D maj excites D min usually. The track etch rate, V T can be calculated according to the formula (Hermsdorf and Reichelt, 2010).

Track length L, track depth x, and residual range R '
Track length L, track depth x and residual range R ' , of alpha particles in detector material can be calculated using the following relations (Dörschel et al., 1996).
(6) where R, is the range of alpha particles in detector material before etching. Track length growth can be written as (Fromm et al., 2003).

V function
The V function (sensitivity function, or etch rate ratio) was given as (Hermsdorf and Reichelt 2010).

RESULTS AND DISCUSSION Determination of bulk etch rate V B
( Fig.1) shows the relationship between mass difference and etching time. The bulk etch rate, V B was found experimentally using (Eq.1). From the slope shown in (Fig.1), the known density (1.32 g.cm -3 ) and area (1 cm 2 ) of the detector, V B was found equal to 1.49 µmh -1 .

Track diameter measurements
The track diameters for normal and oblique incident angles and various energies have been measured using the microscope-camera system. (Fig. 2) presents the relationship of track opening diameter, D with etching time, t for normal incidence and alpha particle energy range (1-5) MeV. In general, the important notice indicates exponentially increasing which is in agreement with other studies (Amero et al., 2001). Alpha energy Normal incidence Fig. 2: Track diameter and etching time for different alpha particle energies at normal incidence.
( Fig.3) presents the angular dependence of major axis of the track's opening and alpha particle energy at various incident angles for 8 h etching time. It is interesting to note that the maximum major axes at 4 MeV alpha particle energy are found for incident angles from (20°-40°) but for incident angles from (50°-70°) the maximum major axes were found at 3 MeV, and for incident angles between (80°-90°) the maximum major axes were found at 2 MeV. ( Fig. 4) shows the angular dependence of minor axis of the track's opening and alpha particle energy at various incident angles for 8 h etching time. It is interesting to note that the maximum minor axes for all incident angles occur at 2 MeV alpha particle energy. Incide numbers=incident angles in degrees Etch.t=8h

Fig. 4: Minor axis with alpha particle energy for various angles.
An example of track contours in dependence on the incidence angle is given in (Fig. 5)   Track etch rate V T versus etching time, t and depth, x (Fig. 6) presents track etch rate, V T determined from its relation with track diameter according to (Eq. 2), for normal incidence. The resulting curves differ in the slope for different alpha energies, which reflects the non-linearity of track diameters, D and they show a distinct spread of the V T curves as alpha energy increased. (Fig. 7) presents V T values calculated at a given depth, x from experimental data (Eq. 5). At certain depth, x where V T falls to V B , agrees quite well with the range, R of the alpha particles. The maxima occur approximately at the depths where the Bragg peaks are situated. The positions of the maximum heights increase systematically at higher initial alpha energy. This effect was already found for 4 He, 7 Li, 11 B and 12 C ions by (Dörschel et al., 2002b). Track length, L and length growth, (dl/dt) versus etching time, t (Fig.8) shows the calculated track lengths as a function of the etching time for normal incidence and various energies (Eq. 4). The resulting curves differ in the slope for different alpha energies, which reflects the non-linearity of track etch rate, V T . After the time needed for etching the total range of incident particle, the track length does not grow further but a saturation behavior occurs. The position of the curves and their maximum heights depend on the particle energy. With increasing energy a shift to longer etching times is visible and the saturation value becomes higher. For etching times larger than the saturation time, the length is determined by L= L max + V B t. This indicates that further etching proceeds with the bulk etch rate. (Fig. 9) shows the track length growth, which represents the curve slope of track lengths (Eq. 7). Track length growth increases with increasing etching time, reflecting the growing track etch rate along the particle trajectory, and then tends to zero value when the end of the particle range was reached. V function (Sensitivity function) (Fig.10) presents angular variations of experimental data (Eq.3) of V function for alpha particle energy range (1-5) MeV, in which it indicates the dependence of sensitivity function V on the energy loss of the ion along its trajectory (i. e as the energy loss increases the sensitivity function V increases also). Exp. data Etch.t=8 h Alpa energy Fig. 10: Experimental V functions versus incident angles for alpha particle energy ranges (1-5) MeV.
( Fig.11) shows the angular variations of V function for alpha particle of 5MeV incidence energy. Experimental data were compared with those computed by the four functions provided in the TRACK_VISION code, which was sited between Durrani-Bull function (1987) and Brun function (1999), and it is nearest to the second one.  Fig.11: V functions versus incident angles of 5MeV alpha particle energy, experimental data, with calculated data using TRACK_VISION program.
In the case of perpendicular incidence, the experimental data of sensitivity function V (Eq.8) was plotted with the residual range, R' (Eq.6) to determine a V function, which took the functional form of Brun et al., 1999, with a new set of parameters (a 1 =0.091, a 2 =1.2, a 3 =1.6, a 4 =1.61, a 5 =1.425) with R 2 =1. The four expressions for V function available in the TRACK_VISION software (Eqs.9, 10, 11 and 12), were programmed and plotted in (Fig.12) with our experimental one, for comparison. A maximum appears close to the end of the particle range, which corresponds to the Bragg peak in the stopping power curve. A large difference is observed between Durrani-Bull (1987), and others. The discrepancy is very pronounced in the low energy region around the Bragg peak, at which our function is closer to Brun and Yu functions, but it is about half the value of Durrani and Bull function. At larger residual ranges, the functions of Durrani-Bull, Brun, and ours are convergent but they are divergent with the two functions of Yu et al., (2005a, and2005b).