STARDUST. A Code for the Simulation of Particle ...

STARDUST. A Code for the Simulation of Particle Tracks on Arrays of Sensitive Volumes with Substrate Diffusion Currents G. Rolland, L. Pinheiro da Silva, C. Inguimbert, J.P. David, R. Ecoffet, M. Auvergne.  Abstract—This paper describes STARDUST, a new

Monte Carlo code dedicated to the simulation of particle tracks on imaging detector arrays. The geometrical detector model, the particle parameters sampling method, the particle tracking and energy deposition and, finally, the carrier collection by diffusion currents in the substrate are described in detail. Some examples of simulations are given at the end, together with a comparison between COROT Satellite data and Simulation results. Index Terms—Simulation, detectors, GCR impacts, Proton Impacts, Electron Impacts

T

I. INTRODUCTION

HE imaging detectors used in observation satellites or the optical detectors inside the satellite star trackers may be submitted to high energy ionizing particle fluxes during a space mission. The impact of these particles (electrons, protons and ions) on the detectors induce transient tracks on the images. These transient tracks may unsettle in some cases the imaging processing algorithms. The STARDUST 1 Monte Carlo code presented hereafter is now available to assess this kind of effect. The goal of such a code is to compute realistic samples of images and to study the statistical properties of the induced particle tracks. In this field, there is famous pioneering work done by S.Kirkpatrick [1], G.R.Hopkinson [3] and T.S.Lohmeim [4 & 16]. More recently, the REACT code has been presented by J.C.Pickel et al. [15]. In comparison with these works, STARDUST uses new solutions of the diffusion equation and it takes into account: nuclear reactions, the shield anisotropy description via a thickness G. Rolland & R. Ecoffet are with CNES, 18 Avenue Edouard Belin, 31401 Toulouse Cedex 9, France. Emails : [email protected]., and [email protected] , Tel: +33561273664. L. Pinheiro da Silva is with CESR/CNRS, 7 avenue du colonel Roche, 31028, Toulouse, France; and is funded by the Brazilian Ministry of Science and Technology (CNPq grant #200334/2003-4). C.Inguimbert and JP. David are with ONERA / DESP, 2 Avenue Edouard Belin, 31055 Toulouse Cedex 4, France. Email: [email protected]., [email protected]. M. Auvergne is with LESIA/CNRS, 5 place Jules Janssen, 92195 Meudon, France. 1 STARDUST stands for “Simulation of particle Tracks on ARrays of sensitive volumes with Diffusion currents in Uniform Substrates”.

978-1-4244-1704-9/07/$25.00 ©2007 IEEE

table, the propagation of energetic electrons and a rough model for delta electrons. See also the work done by A.Claret et al. with the GEANT4 software to analyze the effect of particle impacts on the ISOCAM detectors [12], [22]. STARDUST is an engineering code. To do its work, the code needs some information about the detector structure, the environment particle spectra and the thickness table describing the shield around the detector. Section II of this paper describes the theoretical model implemented in the code (i.e.: the geometrical and physical detector model, the initial conditions particle sampling method, the deposited energy model and the diffusion model). Section III gives examples of simulations. The code validation is discussed in section IV. The conclusion is in V. II. THEORETICAL MODEL A. The physical & geometrical detector model: The detector model (see Fig. 1) is an array of contiguous semiconductor sensitive volumes (the pixel depletion zones) over a uniform neutral diffusive substrate. Semiconductor walls are surrounding this structure and an oxide passivation layer is added on the top. Other more complicated structures are possible, this would require additional programming in the Matlab Language. Owing to the fact that the trajectory duration for an energetic particle across the detector does not exceed some tens of picoseconds, it can be easily verified that such a static structure is sufficient to study the statistical distribution of the tracks (the sequencing problem being solved by image superposition). The effects induced by direct particles hitting on the readout amplifiers are neglected. The detector case and window are considered to be a part of the surrounding shield and must be considered in the thickness table computed in the sector analysis phase. The chemical composition associated to each element of this geometrical model can be arbitrarily chosen.



Let n f be the normal vector of the face number

Passivation (SiO2)

Pixels

(Si)

f  S f X Y

Z

Figure 1: Exploded view of the geometrical and physical model of the detector in the STARDUST Monte Carlo code. (In this example a silicon CCD detector).

B. The particle initial conditions The initial conditions for the particles incident to the detector are fully determined specifying the set of parameters (Z, A, E, θ, φ, f, Uf, Vf). The parameters Z and A, only necessary for the GCR (Galactic Cosmic Rays), are the atomic number and the atomic mass of the ion. E is the particle kinetic energy. θ and φ are the incoming direction angles. f is the number of the die face containing the entry point of the particle and (Uf, Vf) are the local coordinates of the entry point in this face (see Fig. 2).

Z

 n5

(Z,A,E)

entering across this face

Y

(Uf Vf)  f1

f5 f 2 f6

Figure 2: The six faces of the detector die and the angular definition of the incoming directions.

These parameters are sampled according to probability distribution laws during the Monte Carlo simulations. We give hereafter the method used to compute these laws. First of all, we need the thickness table describing the equivalent  aluminum thickness T in the directions  around the center of the die. This table may be built thanks to a sector analysis tool apart from STARDUST. For particles traveling in this direction, the cutoff energy of the shield is given, in the  CSDA 2 approximation, by R 1(T ( )) , where R is the range versus energy curve for that type of particle. Knowing this cutoff energy, the integral flux coming from that direction is  given by ( R 1(T ( ))) , where (E ) is the integral 3 environment isotropic flux curve for these kind of particles.



  ( R 1 (T ( )))    ( n f   ) d 4

(1)

  ( n f  )0

Where d is the differential solid angle. So, the probability law used to sample the entry face number f of the detector is ( T being the total particle current entering the detector): PrF  f  

f T

6

with : T 



f

(2)

f 1

 Once the entry face is chosen, the direction  is sampled according to the probability density law :  S f  ( R 1 (T ( )))     dP (3)    (n f   ) U (n f   ) d  f 4 where U is the unit step function.  Knowing  , the environment energy of the particle is sampled from a cumulated probability distribution law F(E) deduced from the integral energy spectrum ( E ) for this kind of particle: F (E )  1 

f3



X

f

whose area

in the detector die is given by : Substrate (Si)

f4

is S f . Then, the particle current

f

( E ) ( E min )

(4)

The sampling interval starts from the cutoff energy in the  direction  and extends toward the maximum energy of this spectrum. The energy at the detector level is then deduced from this sampled energy by subtracting the cutoff energy (according to the CSDA approximation). The coordinates (Uf,Vf) are sampled with uniform probabilities along the face plane. For the case of cosmic ray ions, the value of Z is sampled from the probability law: PrZ  z  T ( z )



T

( z)

(5)

z

The total number of particles in a frame may be deterministic or sampled with a distribution law supplied by the user. All these probability laws are tabulated once in the initializing phase of the code according to the thickness table, the particle range data and the particle spectra. During the Monte Carlo drawing phase, the height parameters are sampled in the order: Z, A, f, (θ,φ), E, and finally (Uf,Vf). For protons and electrons, the sampling sequence is obviously (f, (θ,φ), E, (Uf,Vf)). Notice that the A(Z) law in STARDUST is currently deterministic and given by the standard isotopic A(Z) relation. To introduce here a probabilistic distribution law is straightforward (A must be sampled 4 given Z). This possibility will be included in the next code version.

2

CSDA stands for “Continuous Slowing Down Approximation”.  (E ) Is computed thanks to environment codes [26], [27]. This determines the maximum energy used in all the physical models, except for the nuclear reaction data base, which is limited to 300 MeV (§ II.C). 3

4 The range versus (total) kinetic energy curves R(E) depend on A. Fluctuations in A will then introduce changes in these curves.

The code may be run with an external integral spectra computed by other transport codes such as GEANT4 [29] or NOVICE [28]. This may be useful if it is necessary to take into account the generation of secondary particles in the shield around the detector die (see [10], [11], [13] and [15]). In order  to do this, ( R 1(T ( ))) in equation (1), and (3) is replaced by  the externally computed spectrum  ext ( ) . Moreover, STARDUST is able to take into account a possible anisotropy of the radiation environment. To do so, we  multiply the isotropic flux ( R 1(T ( ))) in (1) and (3) by the  relevant normalized function of  (see this model in [6] and [14]). For the current version of the code, only the pitch angle distribution model for the radiation belts is implemented. The East-West effect will be implemented in the next version. C. The deposited energy: Once the particle travels inside the detector, it is supposed to lose energy along its trajectory according to the CSDA mechanism or the production of delta electrons or by nuclear reactions (for the protons and the ions). The trajectories are always supposed piecewise rectilinear. The mean deposited energy  Ed  in an element along such a trajectory segment of length L by the CSDA mechanism is:  Ed   E0  R 1 ( R( E0 )  L)

(6) ( E 0 being the energy at the beginning of the segment and R 1 the

reciprocal of the range versus energy function). For the ions and the protons, the R function is computed by using the same algorithm as in the SPAR code [24]. Tabulated values coming from the SRIM code [25] may also be chosen. For the electrons, the algorithm given by Chibani [5] is used. The value of the deposited energy E d may be deduced from  Ed  taking into account the energy straggling. To do so, distributions of energy losses are used. For the protons and the ions, the model in [9] has been chosen. In this model, the Vavilov distribution is computed according to the Chibani algorithm [8]. The Landau distribution is computed by the Findlay & Dusautoy algorithm [2] for the values of the Landau parameter less than 100 and by an asymptotic form for the values greater than 100. For the electrons, the FindlayBlunck-Chibani model is used [2], [5]. Delta electrons generation is simulated by using crosssections formulae coming from the GEANT4 software physics manual [29]. Nuclear reactions are simulated using the ONERA/DESP database [18]. This data base has been constructed by GEANT4 simulations using the binary cascade model. The total probability and the probabilities of the different reaction channels have been tabulated and interpolated using spline functions. During the simulations, a fast spline interpolation algorithm is used to sample the nature and the parameters of the secondary particles produced by a reaction. These parameters are sampled respecting the general conservation laws (energy-momentum etc. …). After a nuclear reaction, only the kernel fragments and the protons are propagated. The

neutron propagation is not taken into account in this version of the STARDUST code because the neutrons interact with the die only by nuclear reactions. So, due to the fact that the mean free path of these neutrons is generally greater than the maximum dimension of the die, the probability to have two consecutive nuclear reactions on the same trajectory inside the die is very small. Moreover, the production of neutrons inside the shield surrounding the die is neglected. This nuclear data base covers the energy range [1 to 300MeV], the projectile being a proton. Outside this interval, this model is switched off. So, this data base is mainly used to assess the effect of the protons radiation belt. In the next versions of the code, this energy range will be extended and other projectiles will be available in order to describe precisely the nuclear reactions induced by the GCR. The secondary particles generated inside the detector are propagated with the same procedures as the primary ones (i.e. they may generate other secondary particles etc. …). The number of deposited electron-hole pairs is finally obtained by dividing the deposited energy E d (or possibly  Ed  if the energy straggling model is switched off) by the energy necessary to create a thermalized electron-hole pair (3.6 eV in Silicon). Charges deposited in depleted zones are directly collected. Otherwise, they are subject to diffusion. D. The charge diffusion model: The carriers generated by the ionizing particles in the substrate are submitted to the thermal diffusion and may be collected in the pixels. A collection model 5 has been implemented in STARDUST. This model uses tabulated Green functions obtained by solving analytically the diffusion equation in a semiconductor infinite slab with null Dirichlet boundary conditions. This well known equation is: D  n  with : n

n





n  G t  0 and n

( z  0)

( zH )

0

(7)

Where D is the ambipolar diffusion constant,  the Laplacian operator,  the carrier lifetime, n the volumic carriers density and G the carriers generation rate. The bottom plane (z = H) of this slab is the rear Ohmic contact (or a layer with high conductivity) and the top plane (z = 0) is the boundary of the pixel depletion zones. The Green function is the solution of equation (7) when   the G right hand term is equal to the distribution  (r  z )   (t ) . This Green function fulfilling the boundary conditions of equation (7) may be obtained by a linear combination of Green functions of the same equation suitable for an infinite isotropic medium. For such an infinite medium, we have:

5 We give here a brief presentation of this model. For more details, see our companion paper [21] in this conference. The choice of an analytical diffusion model gives us the opportunity to decrease the computation time. Compare, for example, with the CPU time needed in device simulations reported in the previous study [19].

  (r  z ) 2 t  ) exp( 4 Dt 

 g (z , r , t )  U (t )

3

4 D t 

The charge (8)

2

of

    g ( k m , r , t )  g ( k m , r , t )

(9)

m  

  k m  z0  2m H  k

 r

 ρ

-g (-2H - Z0) + g(-2H + Z0) -g(-Z0)

(-2H - Z0)

(-2H + Z0)

g(Z0)

 k

- Z0

-g(2H - Z0)

2H - Z0

2H + Z0

rear contact

Z=H

The current density flowing across the z  0 plane is:   J (ρ)   D grad(GT ( z 0 , r , t ))  ρ (10)  ρ  z  0 The charge density collected in the half space z < 0 during the time period T is given by:



T 0

 k  grad(GT ( z 0 , r , t ))  dt

(11)

ρ

Examining the properties of equation (8), we can see that may be extended to infinity provided that the condition   T is fulfilled. This condition is generally true for detectors whose integration times are greater than or equal to a few milliseconds in silicon. According to this hypothesis,  Q( z 0 , ρ) is then given by: T

 Q ( z 0 , ρ) 

1

m  



a m 1   R m  exp( R m )

2  H 2   m  with :   ρ

z and : a m  0  2 m ; R m  H

Rm 3

(12) 2 H

2

 am 2

0

Sp

 v

(13)

z  0 point source

Finally, the total charge QT collected by a pixel is given  by the integral of Q p (w ) weighted by the LET of the particle

QT 

Figure 3 : The main point source located at z0, the associated virtual sources necessary to satisfy the boundary conditions and the associated weighted Green functions.

 Q ( z 0 , ρ)  D

 

 Q( z , (v  u)) dS

  with v and u 

+ g(2H + Z0)

Z

Z0

Z=0

of its depletion zone

along its trajectory:

substrate

Pixels depletion zone

Sp

   w  k z0  u being the position vector of the unit  and v the position vector of the dS v element.

m  



extended over the part

 Q p (w ) 

( z is the position of the instantaneous unit point source, r is the current point where g is evaluated and U is the unit step function). The linear combination is constructed using the method of images (see Fig. 3 ). The result is :  GT ( z0 , r , t ) 

collected in a given pixel is the integral

boundary lying in the z  0 plane:





 Q ( z0 , ρ)

 Q p (w )

; 

H Ld

Ld  D  is the diffusion length 6 .

6 For a semi-infinite substrate, equation (12) reduces to its m = 0 term which corresponds to equation A4 in the Smith et al. paper and then to the Kirkpatrick equation if Ld tends to infinity (see [7] and [1]).

Ep



S 0

 LET ( s ) Q p (w ( s)) ds Ep

(14)

is the energy necessary to produce a thermalized electron

hole pair and s is a linear coordinate along the trajectory (varying from 0 to S). In order to decrease the computer time, the following procedure is used : In the initializing phase (before the Monte Carlo  drawings), knowing H and Ld , values of Q( z0 , ρ) are computed according to equation (12) on a dense grid in the domain extending along X and Y to several diffusion lengths and for 0  z 0  H . These grid values are used to construct a 3D primary spline interpolant. For each layer z 0 of the grid, this primary spline interpolant is integrated along the X and Y coordinates of this grid from the (Xmin, Ymin) point to the (X, Y) current point. This results in a final 3D spline interpolant QI ( X , Y , z 0 ) representing the value of equation (13) with  u  0 and Sp  X min , X  x Ymin , Y  for each grid layer z 0 . The Monte Carlo drawing phase of the simulation gives a set of trajectory segments crossing the substrate. This set of segments is memorized. After the Monte Carlo phase, each one of these trajectory segments is broken into elementary segments of length s . Each one of these elementary segments is at a depth z(s) from the {z = 0} plane. The 3D interpolant is used to find the distribution corresponding to this depth. This (X,Y) distribution is moved to the elementary segment center and associated with an instantaneous point source of strength LET (s) s Ep . The contribution of this elementary source to the collected charges in a pixel of area X 1 , X 2  x Y1 , Y2  is given by a linear combination of four interpolant values (after an obvious relevant variable substitution associated to the translation) : (15) QI ( X 2 , Y2 , z )  QI ( X 2 , Y1, z )  QI ( X1, Y2 , z )  QI ( X1, Y1, z ) . With the Matlab facility in matrix manipulation, this gives rise to a concise and very efficient algorithm: a matrix of the interpolant values in each vertex of the pixels grid is first constructed. Then a matrix convolution with the kernel [1 -1 ; -1 1] gives simultaneously the contribution of this elementary

segment to the charge collected by each one of the pixels. Cumulating in a loop over all the elementary segments finally gives the matrix containing the value of QT for each one of the pixels according to (14). E. The postprocessor At the end of the simulations a postprocessor gives statistic distributions for the morphologic parameters of the tracks. The results may be, for example, the histogram of the number of pixels concerned by a track, the histogram of the amplitude of the pixels in the tracks and so on. Images of the particle tracks including or not the diffusion contribution are also given. III. EXAMPLE OF CCD SIMULATION:

Figure 6 : A zoom in the frame presented in Figure 4. We can recognize here the part of the trajectory crossing the depleted zone of some pixels (high amplitudes in red) and the halo due to the collection by diffusion in the substrate (blue to pale green amplitudes on this figure).

We give hereafter some results of CCD detector simulations. The detector is a 1000x1000 matrix of pixels of size 13.5µm x 13.5µm x 25µm on a 100µm substrate with a 1µm SiO2 passivation layer.

Figure 7 : Illustration of the effect of two energetic delta electrons emitted from the part of the trajectory lying in the substrate. Figure 4 : The frame resulting from 1 second integration time. The environment spectrum is AP8min. The thickness table of the satellite is given in Figure 5.

IV. CODE VALIDATION A. Comparison with known results: The STARDUST code has been carefully checked to minimize the occurrence of bugs. Moreover, simulations done on simple geometrical structures, for which the solutions are well known, give satisfying results. For example, the mean number of pixels disturbed by the impact of constant LET particles for the case of a dense and large set of contiguous pixels, without substrate diffusion, is given in [12] by the simple formula:

nN

Figure 5 : The thickness table (represented as an histogram) giving the shield thicknesses (in mm) due to the satellite around the CCD detector.

ab  bc  ac AB  BC  AC

(16)

Where a, b and c are the pixels dimensions, A, B and C, the dimensions of the die and N the number of pixels. The simulation of such a structure using STARDUST, with particles of constant LET, gives a result in very close agreement with the prediction of (16). Another example is the simulation of the chord length distribution of an isolated pixel which gives also the correct result (see Fig. 8).

Figure 9: The geometrical and physical radiation model used to compute the thickness table of Fig. 5 (longitudinal section of the COROT satellite obtained with the FASTRAD code).

0.016 0.014

Chord length distribution Probability density x 50

0.012 0.01 0.008 0.006 0.004 0.002 0

5000

10000

15000

Number of deposited electrons (for 100 MeV protons)

Figure 8: Chord length distribution given by a Stardust simulation for a single isolated pixel of dimensions 10 x 20 x 30µm. Particles are with constant LET (100 MeV protons) and the flux is isotropic (No diffusion).

B. Comparison with experimental results: Some results of simulations done using STARDUST may also be compared with in-flight measurements. These results come from dark images acquired during the commissioning phase of the COROT satellite. 1) The COROT satellite: The COROT satellite is a high precision photometry experiment dedicated to stellar seismology and the search for extra-solar planets. The instrument is based on an afocal telescope and a wide-field camera mounted on a recurrent platform. The focal plane is composed of four 4280-series CCD devices provided by E2V Technologies. Those are frame transfer, thinned, backside illuminated detectors and are operated at -40°C. The image size is 2048x2048 pixels. The size of each pixel is 13.5x13.5µm. The COROT satellite has been recently inserted in a polar orbit at an altitude of 896 km. 2) The COROT structural model: A geometrical and physical model of the COROT satellite has been built with the FASTRAD software [27] (Fig. 9), leading to the thickness table given in Fig. 5. Moreover, this model may be translated by the FASTRAD code in GEANT4 format to feed secondary radiation computations.

Secondary mirror

Camera

CCD

Protective lid Primary mirror

Satellite platform

3) The experimental radiation flux characterization: The incoming radiation flux, as a function of the satellite position, was empirically characterized in-flight before the opening of the protective shutter placed at the entrance of the telescope [23]. The objective was to measure the flux seen by the CCD detectors behind the effective shielding. With this purpose, a portion of a CCD image (100 x 100 pixels) was periodically acquired (at 1Hz) over 24 hours. The identification of particle impacts in an image requires an iterative approach, since there is a compromise between the detection of all disturbed pixels and the rejection of statistical outliers associated with readout noise. A method of double clipping has been used. Impacted regions are first identified through the application of a high threshold (e.g. 6noise). A lower threshold is then applied (e.g. 3noise) and those pixels contiguous to the first identification are also flagged as disturbed. To do so, a robust estimator of the standard deviation based on the median absolute deviation has been used. Hot pixels (dark current defects) are rejected. Particle impacts were characterized in terms of statistical distributions, after reduction of the input images (offset and background correction, gain conversion, correction for any instrumental artifacts). The analyzed features were the imparted energy per impact (in e-), the size of an impact trail (in number of disturbed pixels), along with the energy distribution of the brightest pixel of an impact (in e-). The corresponding histograms are given in Fig. 10, 11, and 12 (see data 1). 4) The STARDUST simulation conditions: These simulations are done on a 1000 x 1000 pixels detector structure. The pixels dimensions are 13.5 x 13.5µm. The thickness of the pixels depletion zone and of the quasineutral zone are adjusted to give the best fit between the experimental and simulated histograms. As we will see hereafter, the best fitting values are: 9µm for the depletion zone and 6µm for the quasi-neutral one. The data acquisition time for the dataset is chosen to reduce the fluctuations in the Monte-Carlo simulation results. The environment spectrum is computed using the OMERE code [27] at the coordinates (47°W, 30°S) in the South Atlantic Anomaly according to the AP8min model. The protons flux anisotropy is taken into account, with a loss cone angle equal to 72°. The electrons flux environment, at this point of the orbit, is neglected because the maximum energy found in this local spectrum is smaller than the cutoff energy associated to the minimum shield thickness around the CCD. The braking radiation due to the stopping of this electrons flux in the shield is also neglected (because the current version of the STARDUST code does not incorporate the photons propagation and interaction). The GCR flux is neglected. This is due to the weakness of the number of events induced, at this point of the orbit, by this kind of particles compared with the number of protons induced events. For the simulation versus experimental results comparison purpose, the secondary

particle flux of electrons and photons has been computed in the CCD neighborhood using the GEANT4 software. 5) The Number of Disturbed Pixels Histogram: Fig. 10 gives the, in-flight, and simulated, “Number of Disturbed Pixels” histograms. The experimental (in-flight) histogram presents a characteristic shape for the group of bars 2 to 4. It can be verified by simulation that this shape may be explained by the carrier diffusion in the substrate. The thickness values of the pixels depletion zone (z) and of the substrate quasi-neutral zone (H) are critical to reproduce this shape. The two effective values (z = 9µm and H = 6µm) guarantee an optimal reproduction of the bar 2 to 4. Notice that the choice is very limited around these values. For example: the two other couples (z = 10µm and H = 5µm) and (z = 8µm and H = 7µm) do not permit to reproduce this characteristic shape. So, these two effective parameters (z and H ) seem to be well defined 7 . An important discrepancy exists between these two experimental and simulated histograms for the bars of ranks greater than or equal to 5 (see Figure 10). As explained hereafter, this is mainly attributed to X-ray secondary photons impinging the backside face of the CCD, most of them having quasi-normal incidence angles. Moreover, this is corroborated by an examination on the experimental images which reveals that a lot of disturbed zones associated to impacts exhibit a nearly circular symmetry.

the histogram corresponds to events induced by small LET and long range particles (roughly protons with energies between 150 to 250 MeV). A second simulation has been done. It takes into account the secondary electrons flux spectrum previously computed by a GEANT4 simulation. In this new result, the unpopulated zone has been reduced from [0 e-, 2500 e-] to [0 e-, 1500 e-]. This is due to the great number of electrons of the secondary spectrum with long ranges and low LET. The minimum LET value ( 100 e- /µm) for the electrons is too large to generate a population of events with total amplitudes comprised in the remaining empty zone of the histogram (the interval between 0 and 1500 e-). To fill in this interval, in order to fit the experimental histogram, particles giving small energy deposition with moderate ranges are needed. We neglect the contribution of the protons and electrons having simultaneously very low energies and very low ranges. Good candidates seem to be low energy bremsstrahlüng and/or fluorescence photons (X-rays with energies less than 5400 eV for example). The mean free path in Silicon of such particles is lesser than 20µm, so the disturbed zones will frequently exhibit a nearly circular symmetry. 500 450

Total Amplitude Histogram 400

data1 data2 data3

Number of events

350 4000

Number of Disturbed Pixels per Track Histogram 3500

data1 data2

3000

300 250 200

Number of events

150 2500

100 50

2000

0

1500

5000

10000

15000

20000

25000

30000

Total Amplitude (in electrons)

Figure 11: Total Amplitude Histograms. In-flight histogram (data 1). STARDUST simulation without secondary electrons (data 3). Simulation taking into account the secondary electrons spectrum computed with a previous GEANT4 simulation (data 2).

1000

500

0

5

10

15

20

25

30

Number of disturbed pixels

Figure 10: Number of disturbed pixels Histograms. Experimental histogram (data 1). STARDUST Simulation taking into account the secondary electrons spectrum computed with a previous GEANT4 simulation (data 2).

6) The Total Amplitude Histogram: Fig. 11 gives the in-flight and simulated “Total Amplitude” histograms. The total amplitude is the sum of the amplitude of all the pixels disturbed by an impact. This is directly related to the total imparted energy per impact. A first simulation has been done taking into account only the proton environment flux. The result reported in Fig. 11 (data 3) exhibits a good agreement for the total amplitudes greater than or equal to  3800 electrons. For amplitudes smaller than this threshold, the number of events decreases rapidly to a null value for amplitudes lesser than 2500 electrons. This region of 7 Moreover, the “unpopulated” low amplitudes zone of the two others histograms (see next paragraphs) is nearly insensitive to this choice.

7) The Maximum Amplitude Histogram: Fig. 12 gives the in-flight and simulated “Maximum Amplitude” histograms. The maximum amplitude of a track is the amplitude of the brightest pixel of the track. The same analysis as in § IV.B.6) can be done, with the same conclusions. Moreover, we see in the experimental histogram in Fig. 12 that the maximum of the distribution is located near an amplitude value of 1500 electrons. These events may be induced by photons with an energy around 5400 eV, with a mean free path around 20µm, implying the necessity of quasinormal impacts. This suggests again that for these disturbed zones, a nearly circular symmetry is privileged.

T. Beutier and P.F. Peyrard from TRAD for all the work done to construct and tune the COROT model and the help on GEANT4 simulations.

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REFERENCES

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Figure 12 : Maximum Amplitude Histograms. In-flight histogram (data 1). Simulation taking into account the secondary electrons spectrum computed with a previous GEANT4 simulation (data 2).

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V. CONCLUSION The first version of the STARDUST code is now available. It is written in Matlab language and compiled. The use of Matlab language gives more flexibility and rapidity than the use of more powerful codes like GEANT4 and so on. It is able to run on PC computers or on Unix machines. It will be used at first to assess the effect of the radiation constraints on the imaging detectors but we hope that its field of applications will be extended to other domains like, for example, the studies of MBU and so on (see [7], [17] and [20]). In comparison with previous works, STARDUST uses new solutions of the diffusion equation and it takes into account: nuclear reactions, the shield anisotropy description via a thickness table, the propagation of energetic electrons and a rough model for delta electrons. A comparison with in-flight data coming from the COROT satellite has demonstrated the criticality of the knowledge of the detector incident secondary spectra, particularly in the low energy domain when a good description of the small amplitude events are needed. Similar facts have yet been found by several authors, see for example [12], [13] and [22]. These secondary spectra may be computed by external codes like GEANT4 or NOVICE and used in STARDUST simulations. To do so, a precise geometrical and physical model of the satellite is needed. The next version of the code will incorporate the photons treatment. The comparison with COROT data has also demonstrated the good qualitative agreement between measurements and simulations for events of amplitudes greater than 3800 eV, even without a knowledge of the secondary spectra.

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ACKNOWLEDGMENT

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The authors thank A. Claret, O. Boulade and P. Galais from CEA/DAPNIA, S. Duzellier, D. Boscher and S. Bourdarie from ONERA / DESP for helpful discussions. ( D. Boscher and S. Bourdarie have also computed the magnetic field orientation and the loss cone angles above the SAA). Thanks to J. Denis, student at the Rouen University, for his work on the first Unix / C version of the code. Thanks also to

[19]

[7]

[8] [9] [10] [11] [12] [13]

[14]

[15] [16] [17]

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