Application of a cellular automaton for the evolution of etched nuclear tracks

Radiation Measurements 50 (2013) 201e206

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Application of a cellular automaton for the evolution of etched nuclear tracks Leonardo de la Cruz-Trujillo a, C. Hernández-Hernández b, C. Vázquez-López c, *, B.E. Zendejas-Leal c, I. Golzarri d, G. Espinosa d a

Universidad Nacional Autónoma de México, Apartado Postal 20364, 01000 México, D.F., Mexico Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Ave. IPN 2508, Col. San Pedro Zacatenco, México 07360, D.F., Mexico c Departamento de Física, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Ave. IPN 2508, Col. San Pedro Zacatenco, México 07360, D.F., Mexico d Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20364, 01000 México, D.F., Mexico b

h i g h l i g h t s < We model the evolution of an etched nuclear track using cellular automata (ca). < A cellular automaton of a conical track has 4 states and 16 transition rules. < The ca of general tracks require a not regular mesh and the L(t) and Vb parameters.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 December 2011 Received in revised form 30 July 2012 Accepted 23 November 2012

In the present work, it is demonstrated the first application of cellular automata to the growing of etched nuclear tracks. The simplest case in which conical etched tracks are gradually formed is presented, as well as a general case of time varying etching rate Vt. It is demonstrated that the cellular automata elements consist in an image pattern of the latent nuclear track input cells, 16 rules for updating states, the Moore neighborhood and an algorithm of four states. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Nuclear tracks Cellular automata Mathematica

1. Introduction Cellular automata (CA) arose from numerous attempts to create a simple mathematical model describing complex biological structures and physical processes (Wolfram, 1986). The simplest case of a discrete dynamical system is the cellular automaton, in which the behavior is totally dependent upon the local interconnections between its elementary parts (Toffoli and Margolus, 1987). Schematically, a cellular automaton may be visualized as a regular spatial net, in which each cell is capable of assuming a number of discrete states. Time is varied in discrete steps, and evolution of the system obeys certain a priori fixed rules determining the new state of each individual cell at each successive step in accordance with the states of its nearest neighbors. Such a simple model has turned out to be very fruitful and has been widely applied in describing various complex structures and processes in physics, biology, chemistry, etc. At present, two different kinds of application of

* Corresponding author. Fax: þ52 55 5061 3386. E-mail address: cvlopez@fis.cinvestav.mx (C. Vázquez-López). 1350-4487/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radmeas.2012.11.021

cellular automata are known in high energy physics: Denby (1988) reported the revealing of clusters in a honeycomb calorimeter, and Ginzov et al. (1993) described the track filtration in cylindrical multiwire proportional chambers. The evolution of etched nuclear tracks is one of the most interesting issue in the nuclear tracks nanotechnology. The understanding of the influence of the incident ionizing particles with the polymeric materials in the formation of tracks is also an active issue in the field. The track growing parameters: etching time t, the bulk etch rate Vb, and the time dependent pit length L(t), determine the etched track profile and vice versa (Hermsdorf and Hunger, 2009). The track etch rate Vt is given by:

Vt ¼ dXðtÞ=dt ¼ Vb þ dLðtÞ=dt;

(1)

Where X(t) is the track vertex position, measured from the original surface of the detector. Ditlov (1995) considered the detector as a discrete structure of regularly packed bricks. Each brick, when is in contact with the solvent has a time life given by:

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Tði; j; kÞ ¼ Vol=AVb

(2)

Where Vol is the brick volume and A is the corresponding area in contact with the solvent. If the brick is damaged by the radiation, the lifetime is reduced, since Vb is replaced by Vt. A kind of finite element calculation is then performed, giving rise to the evolution of the etched track. This paper presents another attempt in this direction, though with the emphasis on the simplicity (and, in turn, interpretability) of the method employed. Here, we employ a cellular automaton to simulate the shape evolution of etched nuclear tracks. Cellular automata (CA) are an ideally decentralized and massively parallel computation system. It involves an input of initial cells configuration where each unit, or “cell”, can have n different states. Next, a neighborhood is defined such that each cell can only interact with the cells in its vicinity. How the cell interacts with its neighbors depend on the rules defined for the particular automaton. This interaction happens for each cell in the automaton at each time instant. In most implementations, these interactions are calculated at discrete time steps or iterations. The state of a cell in the next iteration depends upon the rules involving the current state of the cell and the cells belonging to the defined neighborhood. These rules decide how the cellular automata will behave. A well known example of one of such rules is the popular Game of Life (Toffoli and Margolus, 1987). The implementation of this work is by means of the Mathematica programming language (Wolfram, 2002). 2. Simplest case A conical etched track evolution. A conical etched track corresponds to a linear behavior of the time dependent pit length L(t), which normally occurs at the initial stages of the etching process. As a first approximation L(t) will be assumed linear in the interval [0, tsat], where tsat is the saturation etching time in which the solution reaches the range of the incident particle. For t > tsat, L(t) ¼ Lmax, a constant corresponding to the maximum depth of the track cavity. 2.1. Elements of the CA For this case, the input is the latent track configuration, which will be established as Step 1. The possible states of each cell are listed in Table 1. The neighborhood consists of the 8 neighbors bordering each cell (known as the “Moore neighborhood”. It was found that this neighborhood is the most reliable arrangement in this work, which is illustrated in Table 2 relative to cell X. The horizontal size of the latent track is of the order of 10 nm, according to the latent track core dimensions in plastics exposed to light ions (Hermsdorf, 2011). 2.2. CA evolution The CA should satisfy Equation (2) in the sense that the rate of dissolution of the healthy cells is proportional to the exposed area

Table 1 Possible states of the cells.

Table 2 The Moore neighborhood of cell X. The small letters are abbreviations of the cardinal directions. nw w sw

n X s

ne e se

to the etching solution, and the latent track cells have higher rate of dissolution than the bulk cells. In order to fulfill this requirement, the transition rules shown in Fig. 1 were designed. In Fig. 2 the evolution of the etched track is shown, for the first 12 steps. The Step 1 corresponds to the input configuration. In Step 9 the latent track has been completely etched. During these 9 steps the growing of a conical cavity whose angle is sin1(Vb/Vt) is observed. Etching is isotropic starting in Step 10, giving rise to the over-etched track stage. In order to use Mathematica language, it is desirable to arrange the transition rules as indicated in Table 3. Symbol * means any state of the cell. The neighbor cells are denoted with the nomenclature indicated in Table 2. 2.3. Programming details 2.3.1. Mathematica code The structure of a CA is very simple when written in the Mathematica programming language. A matrix is set up, the transition rules are defined as a function, and the function is repeatedly applied to the matrix. The formalism of Gaylord and Nishidate (1996) is used. The matrix is defined by a function called NTrace [kp, a, mp, t], where kp is the range of the latent track (vertical depth denoted by the number of cells), a is the diameter of the latent track (in number of cells), mp is the dimension of the square matrix, and t is the number of steps in the evolution. The principal function NTrace [kp, a, mp, t] is formed by the following functions: initCon_g, which is the mp  mp matrix with the cells in the initial states of the CA. The transition rules of the CA are indicated by the evolution function. The Moore function takes into account the neighborhood of each cell. The latent nuclear track is constructed by a vertical set of cells centered in the matrix. In order to obtain the initial condition, we append equal numbers of columns to each side of the track to preserve symmetry (m ¼ 2 mp  1). The range consists on j < k cells which is also odd (k ¼ kp  1). Function if is used to satisfy initial conditions using the Boolean “&&”. First we create a centered table (Table[If[m/2 <¼ i < (m þ 2)/2 && j < k, i ¼ 3, i ¼ 0], {j, 1, m}, {i, 1, m}]]) with range j < k. For each “i” we append to the transpose table, a cell with the state “2” (Map[Join [{2}, #]). This constitutes the matrix that represents our initial condition (initConfig). initConfig ¼ Transpose[ Map[Join[{2}, #] &, Transpose[Table[If[m/2 <¼ i < (m þ 2)/2 && j < k, i ¼ 3, i ¼ 0], {j, 1, m}, {i, 1, m}]]]]; The transition rules are established by Table 3 and the function evolution consisting in 15 rules:

Cell description

Abbreviation

State Xij(t)

Healthy Sick Dead Latent

H S D L

0 1 2 3

Rule Rule Rule Rule

1: 2: 3: 4:

evolution[2, n_, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 2; evolution[3, 3, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 3; evolution[0, 1, 0, s_, 0_, 0_, se_, sw_, 0_]:¼ 0; evolution[0, 3, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 0;

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Fig. 1. Transition rules of the central cell, according to the neighbor states. The * symbol denotes no matter the state.

Rule Rule Rule Rule Rule Rule Rule Rule Rule Rule Rule Rule

5: evolution[0, 0, 0_, s_, 0_, ne, se_, sw_, nw_]:¼ 0; 6: evolution[0, 2, 0_, s_, 0_, ne_, se_, sw_, nw_]:¼ 1; 7: evolution[0, 1, 0_, s_, 0_, 2, se_, sw, nw_]:¼ 1; 8: evolution[0, 1, 0_, s_, 0_, ne_, se, sw_, 2]:¼ 1; 9: evolution[0, 0, e_, s_, 2, 0_, se_, sw_, nw]:¼ 1; 10: evolution[3, 2, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 2; 11: evolution[1, 2, e_, s_, w_, ne, se_, sw_, nw]:¼ 2; 12: evolution[1, n, 2_, s_, w_, ne, se_, sw_, nw]:¼ 2; 13: evolution[1, n, e_, s_, 2, ne, se_, sw, nw]:¼ 2; 14: evolution[0, 0, 1, s_, 0_, 2, se_, sw, nw_]:¼ 1; 15: evolution[0, 0, e, s_, 1, ne, se_, sw, 2_]:¼ 1; 16: evolution[0, 0, 2, s_, 0, ne, se_, sw, nw_]:¼ 1;

Moore function takes two parameters: The rules of the automaton (func) and the matrix where the rules will be applied (lat). Ordered pairs represent the position of the cell with respect to the X position which is (0, 0) as shown in Table 2. Moore[func_, lat_]:¼ MapThread[func, Map[RotateRight[lat, #] &, {{0, 0}, {1, 0}, {0, 1}, {1, 0}, {0, 1}, {1, 1}, {1, 1}, {1, 1}, {1, 1}}], 2]; Finally, we apply “t” times Moore function and get the automaton evaluated at t. This is done by the “Nest” function.

Fig. 2. Evolution of an etched track. The healthy cells are white color (State 0). The sick cells are light gray (State 1). The dead cells correspond to the etching solution and/or the dissolved material and they are black color. (State 2). The latent track is represented as dark gray color cells (State 3).

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Table 3 The 16 transition rules from states x(t) to x(t þ Dt), where Dt is the elapsed time between one step and the next. Rule

x(t)

n

e

s

w

ne

se

sw

nw

x(t þ Dt)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 3 0 0 0 0 0 0 0 3 1 1 1 0 0 0

* 3 1 3 0 2 1 1 0 2 2 * * 0 0 0

* * 0 * 0 0 0 0 0 * * 2 * 1 * 2

* * * * * * * * * * * * * * * *

* * 0 * 0 0 0 0 2 * * * 2 0 1 0

* * 0 * * * 2 * * * * * * 2 * *

* * * * * * * * * * * * * * * *

* * * * * * * * * * * * * * * *

* * 0 * * * * 2 * * * * * * 2 *

2 3 0 0 0 1 1 1 1 2 2 2 2 1 1 1

Nest[Moore[evolution, #] &, initConfig, t] In order to plot the automata, “ArrayPlot” or “ListDensityPlot“ may be used. 2.3.2. List of the entire program NTrace[kp_, mp_, t_]:¼ Module[{initConfig, Moore, evolution, m, k}, m ¼ 2 mp  1; k ¼ 2 kp  1; initConfig ¼ Transpose[Map[Join[{2}, #] &, Transpose[Table[If [m/2 <¼ i < (m þ 2)/2 && j < k, i ¼ 3, i ¼ 0], {j, 1, m}, {i, 1,m}]]]]; evolution[2, n_, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 2; evolution[3, 3, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 3; evolution[0, 1, 0, s_, 0_, 0_, se_, sw_, 0_]:¼ 0; evolution[0, 3, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 0; evolution[0, 0, 0_, s_, 0_, ne, se_, sw_, nw_]:¼ 0; evolution[0, 2, 0_, s_, 0_, ne_, se_, sw_, nw_]:¼ 1; evolution[0, 1, 0_, s_, 0_, 2, se_, sw, nw_]:¼ 1; evolution[0, 1, 0_, s_, 0_, ne_, se, sw_, 2]:¼ 1; evolution[0, 0, e_, s_, 2, 0_, se_, sw_, nw]:¼ 1; evolution[3, 2, e_, s_, w_, ne_, se_, sw_, nw_]:¼ 2; evolution[1, 2, e_, s_, w_, ne, se_, sw_, nw]:¼ 2; evolution[1, n, 2_, s_, w_, ne, se_, sw_, nw]:¼ 2;

Fig. 3. Transition rules to be used in case of non constant Vt. X represents the state of the corresponding transition.

Fig. 4. A typical example of etched track length as a function of the etching time, as reported by Yamauchi et al. (2001). Vb ¼ 1.8 mm/h.

evolution[1, n, e_, s_, 2, ne, se_, sw, nw]:¼ 2; evolution[0, 0, 1, s_, 0_, 2, se_, sw, nw_]:¼ 1; evolution[0, 0, e, s_, 1, ne, se_, sw, 2_]:¼ 1; evolution[0, 0, 2, s_, 0, ne, se_, sw, nw_]:¼ 1; Moore[func_, lat_]:¼ MapThread[func, Map[RotateRight[lat, #] &, {{0, 0}, {1, 0}, {0, 1}, {1, 0}, {0, 1}, {1, 1}, {1, 1}, {1, 1}, {1, 1}}], 2]; Nest[Moore[evolution, #] &, initConfig, t] ] Animate[ArrayPlot[NTrace [r, n, t], ColorRules -> {0 -> White, 2 -> Black, 1 -> LightGray, 3 -> Gray}, Mesh -> True], {a, 1, 5}, {t, 0, 100, 1}, {r, 1, 20, 1}, {n, 10, 50}, AnimationRepetitions -> 1, AnimationRunning -> False, AnimationRate -> a] Animate[ListDensityPlot[Reverse[NTrace [r, 30, t]], ColorFunction -> GrayLevel], {a, 1, 5}, {t, 0, 100, 1}, {r, 1, 20}, AnimationRepetitions -> 1, AnimationRunning -> False, AnimationRate -> a]

Fig. 5. The input CA configuration.

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Fig. 6. Etching track simulation by AC formalism applied to L(t) shown in Fig. 4. The elapsed time between one step and the next is quarter hour.

3. General case The regular spatial net should be generalized to an irregular mesh, in which the latent track cells are not longer of the same size. Baran et al. (2010) have reported a possible evolutionary algorithm for these kind of irregular structures. For the sake of simplicity, it is possible to use the same states and transition rules shown in Fig. 1 and in Table 3, except that transition rules number 6, 9 and 16 have to be replaced by those shown in Fig. 3. It is also necessary the time dependent pit length L(t), as well as Vb. In order to illustrate the procedure, a typical result of L(t) reported by Yamauchi et al. (2001) was used, as shown in Fig. 4, with a bulk etching rate Vb ¼ 1.8 mm/h. The experimental conditions were: incident ions: Li-7; incident ion energy: 6.75 MeV; Polycarbonate detector: CR-39 Tastrack. The chemical etching conditions were: NaOH 7.25 N, 70  C. In order to take into account L(t) a central column of latent track cells of different heights was included. The cell heights were obtained from the etched track vertex position given by:

XðtÞ ¼ Vb t þ LðtÞ

(3)

X(t) is measured from the original surface of the detector. In Fig. 5 the CA input configuration is shown. The etched track vertex position for etching times of multiples of 0.25 h is marked with horizontal segments to determine the latent cell heights. The health cells have dimensions given by the rows determined by lines:

hðtÞ ¼ Vb t

(4)

Also with time in multiples of 0.25 h. The columns are equally spaced as the rows. The latent track column is not at scale, but is supposed to measure 10 nm in width, approximately. The CA evolution for time steps of 15 min is shown in Fig. 6. In Fig. 6 (Eight step) the etching solution has just dissolved the whole

latent track. The configuration was calculated following the transition rules of Fig. 1 with the modified transition rules shown in Fig. 3. 4. Conclusion For the special case of etched conical tracks, the cellular automaton elements are very simple: using Moore neighborhood, an initial configuration of latent track cells, with a set of initial dead cells corresponding to the etching solution, four possible states, and 16 transition rules, the evolution of the etched track has been determined. The Mathematical programming language is convenient to calculate the CA. This is the first step for understanding the track formation process in terms of a CA formalism. The CA formalism has been also generalized to the case of depth variation of the etching rate Vt. This methodology may be applied to determine complex etched nuclear track profiles useful in nanotechnology of nuclear tracks (Waheed et al., 2009). Acknowledgments This work was sponsored by Instituto de Ciencia y Tecnología del D.F. (México), Project 325-2009 and by UNAM-DGAPA-PAPIIT Project 1N101910. LCT thanks ICyT-D.F. for the scholarship to work in the software and to attend the CINVESTAV prerequisite courses. We thank Karina Vázquez for her technical support. Fruitful discussions with Prof. Valery Ditlov are acknowledged. References Baran, J., Petrovic, P., Schoenauer, M. 2010. Cellular automata with irregular structure: a compact representation. In: SASOW10 Proceedings of the 2010 IEEE International Conference on Self-Adaptative and Self-Organizing Systems Workshop, pp. 85e90. Denby, B., 1988. Neural networks and cellular automata in experimental high energy physics. Comp. Phys. Commun. 49, 429. Ditlov, V.,1995. Calculated tracks in plastics and crystals. Radiat. Meas. 25 (1e4), 89e94. Gaylord, R.J., Nishidate, K., 1996. Modelling Nature. Springer-Verlag.

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Ginzov, A., Kieel, I., Konotopskaya, E., Ososkov, G., 1993. Filtering tracks in discrete detectors using a cellular automaton. Nucl. Instr. Meth. A329, 262. Hermsdorf, D., 2011. Physics aspects of light particle registration in PADC detectors of type CR-39. Radiat. Meas. 46, 396e404. Hermsdorf, D., Hunger, M., 2009. Determination of track etch rates from wall profiles of particles tracks etched in direct and reversed direction in PADC CR39 SSNTDs. Radiat. Meas. 44, 766e774. Toffoli, T., Margolus, N., 1987. Cellular Automata Machines: a New Environment for Modeling. MIT Press, Cambridge, MA.

Waheed, A., Forsyth, D., Watts, A., Saad, A.F., Mitchell, G.R., Farmer, M., Harris, P.J.F., 2009. The track nanotechnology. Radiat. Meas. 44, 1109e1113. Wolfram, S. (Ed.), 1986. Theory and Applications of Cellular Automata. World Scientific. Wolfram, S., 2002. A New Kind of Science. Wolfram Media, Inc., Champaign, IL. Yamauchi, T., Ichijo, H., Oda, K., Doerschel, B., Hermsdorf, D., Kadner, K., Vaginay, F., Fromm, M., Chambaudet, A., 2001. Inter-comparison of geometrical track parameters and depth dependent track etch rates measured for Li-7 ions in two types of CR-39. Radiat. Meas. 34, 37e43.