domain: spi phys

text:
Student Member, IEEE, C. Inguimbert M C Ursule

T Nuns

J Morio

Comparison of methods to calculate the Dark Current Non Uniformity

Keywords: Space environment, Displacement damages, DCNU, Image sensors, Monte Carlo

Analytical and Monte Carlo DCNU prediction methods are compared. The difference is studied according to the different assumptions used for the calculations. The analysis is performed as a function of the incident proton fluence. The convergence of the DCNU toward a Gaussian distribution predicted by the central limit theorem is also investigated.

INTRODUCTION

he space environment is hostile for satellites and their payload. They can suffer from significant degradations leading potentially to critical failures. Cosmic rays, radiation belts, solar flare and solar wind are sources of particles able to initiate various types of degradation in the electronic components of satellites. Energetic particles are able to reach and degrade solid state sensors (CCD, CIS) used in various applications such as star trackers, sun sensors or earth observation systems. By depositing energy in the sensor arrays, space particles cause interactions with electrons of the medium (SEE, ionizing dose) or atomic displacements (Displacement Damage Dose) [START_REF] Hopkinson | Proton Effects in Charge Coupled Devices[END_REF]. Here, the creation of atomic displacements is only considered. Those displacements lead to some specific degradation in image sensors such as the increase of the dark current. This intrinsic parasitic current creates a reading noise affecting the images. The dispersion of the dark current from pixel to pixel called the Dark Current Non Uniformity (DCNU) is known to be critical for some applications that require a low noise level such as star trackers. The prediction of this degradation is essential for space missions. The calculation of the DCNU consists in evaluating, consecutively to an irradiation by incident particles (protons, electrons, neutrons), the dark current induced by the amount of damage produced in each pixel of an array. For each pixel the damage is the sum of the damage produced by different interactions. Each interaction initiates a damage cascade depending on the energy of the Primary Knock on Atom (PKA). The probability to get a given amount of damage is a combination of the probability to get a recoil atom of a given energy, by the amount of atomic displacements produced by this PKA. This probability is known as the individual Probabiliy Density Function (individual PDF). At the array of pixels level, the convolution of the damage produced by different PKAs can be solved by a Monte Carlo algorithm. That means, randomly selecting the energy of the PKA in the differential interaction cross section [START_REF] Dale | A Comparison of Monte Carlo and Analytic Treatments of Displacement Damage in Si Microvolumes[END_REF][START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF], and calculating for each PKA the amount of produced damage. This calculation is 1) M.C. Ursule (PHD student), C. Inguimbert, T. Nuns and J. Morio are with ONERA-DESP, 2 avenue E. Belin, 31055 Toulouse, France (tel: 33-562252734, email: marie-cecile.ursule@onera.fr). repeated for each pixel, according to the number of interactions that have occurred in each pixel. But this Monte Carlo algorithm [START_REF] Dale | A Comparison of Monte Carlo and Analytic Treatments of Displacement Damage in Si Microvolumes[END_REF][START_REF] Inguimbert | Modeling the Dark Current Non Uniformity of image sensors with GEANT4[END_REF][START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF] is CPU time consuming, and this convolution can be performed [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF][START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF][START_REF] Virmontois | Displacement damage effects due to neutron and proton irradiations on CMOS image sensors manufactured in deep submicron technology[END_REF][START_REF] Belloir | Pixel pitch and particle energy influence on the dark current distribution of neutron irradiated CMOS image sensors[END_REF] by a direct numerical integration [START_REF] Schlenker | Methods for calculating the probability distribution of sums of independent random variables[END_REF]. Such alternative way of calculation has been developed first, and was very successful over the years. This method has the benefit to be very fast, if the product of convolution leads to a known function, avoiding this way to perform the convolution product itself. It is why, originally, Gaussian functions [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] were used to describe the differential damage distribution of the individual interactions (individual PDF). But, Gamma functions have also been adopted [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF], because at low fluence level, they better represent the differential interaction cross sections. Analytical and Monte Carlo calculation methods are thus equivalent in the limit of the assumptions made to simulate the differential interaction probabilities. The impact of the use of different probability density functions, on the DCNU calculation shall be investigated. This paper aims at evaluating this impact by comparing the full Monte Carlo calculation with some methods based on the use of approximated PDF. The differences will be analyzed according to the incident fluence. In addition, at high fluence level, the DCNU converge towards a Normal distribution having a known mean and standard deviation. But this speed of convergence is closely dependent on the chosen differential interaction probability. This will also be analyzed. However, it was noted that, all these methods do not take into account some physical processes such as the Electric Field Enhancement effect (EFE), or the impact of the Total Ionizing Dose (TID) on the DCNU. These degradation mechanisms, that are neglected by the different methods will, therefore, not be discussed in this work. For this study, a method based on the same assumption as the methods of references [START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF][START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF] have been developed and compared to the full Monte Carlo calculations. The experimental data of the Jade device [14] from e2v will serve as reference for comparison. In the first part, both the analytical and Monte Carlo DCNU models will be presented. Then, the second part will be devoted to the description of the method implemented in our toolkit. The third part shows a comparison of these different methods of calculation. The limits beyond which the DCNU converge to a normal distribution having a known mean and standard deviation will also be established.

II. DCNU METHODS OF PREDICTION

A. Monte Carlo approach

A priori, the most straightforward way to predict the DCNU consists in a Monte Carlo algorithm [START_REF] Dale | A Comparison of Monte Carlo and Analytic Treatments of Displacement Damage in Si Microvolumes[END_REF][START_REF] Inguimbert | Modeling the Dark Current Non Uniformity of image sensors with GEANT4[END_REF][START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF] that directly reproduces the damage process. But historically, because the Monte Carlo method is CPU time consuming, faster alternate T ways were preferred [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF][START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF][START_REF] Virmontois | Displacement damage effects due to neutron and proton irradiations on CMOS image sensors manufactured in deep submicron technology[END_REF][START_REF] Belloir | Pixel pitch and particle energy influence on the dark current distribution of neutron irradiated CMOS image sensors[END_REF]. The principle of these two approaches is similar but the methods of calculation differ slightly [START_REF] Schlenker | Methods for calculating the probability distribution of sums of independent random variables[END_REF]. The method of prediction based on the Monte Carlo algorithm used in this study has been presented and compare to experimental data in detail in [START_REF] Inguimbert | Modeling the Dark Current Non Uniformity of image sensors with GEANT4[END_REF][START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF], but for the sake of clarity and the understanding of the study, the key points are reminded briefly here (cf. Fig. 1, Monte Carlo method). According to the incident fluence a different number of interactions (Coulombic, nuclear elastic and nuclear inelastic) is produced in each pixel. The distribution of this number of interactions follows a Poisson distribution at low fluence levels and then tends to a normal distribution for higher fluences with an average :

= Φ (1)
Where is the atomic density of the target material ( / ), the volume of the pixel ( ) and the proportion of interaction produced in the pixels that lead to a degradation of the depleted region. Each interaction will produce a Primary Knock on Atom PKA of a given energy [keV, MeV] that will initiate a damage cascade resulting in a given number of atomic displacements. The DCNU can be estimated by defining randomly for each pixel the number of interactions , and then evaluating the number of atomic displacements generated by each interaction with a second random draw. The number of atomic displacements generated by each interaction is then evaluated randomly. Thanks to the characteristics of the recoil nuclei of coulombic interactions (low energy ~ , short range), the number of atomic displacements can be deduced with the Lindhard energy partition function after estimation of the recoil energy of the PKA [START_REF] Insoo Jun | Proton non ionising energy loss (NIEL) for device applications[END_REF]. For nuclear interactions, because of the recoil nuclei range, a GEANT4 application [START_REF] Agostinelli | GEANT4 a simulation toolkit[END_REF][START_REF] Allison | GEANT4 Developments and Applications[END_REF] has been developed to model the non-negligible number of displacement cascades entering in the depleted volume after being initiated outside. This application models and records in a database the displacement cascades that spread over neighbor pixels created by thousands of interactions [START_REF] Agostinelli | GEANT4 a simulation toolkit[END_REF]. In a given pixel, the number of atomic displacements created will be randomly selected in this database. This method allows taking into account border crossing effects. But, previous works show that the border crossing effect is negligible when the ranges of recoil ions are much smaller than pixel dimensions [START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF]. However, if the dimensions of the depleted volume are in the order of the micrometer, this effect needs to be considered in the simulations [START_REF] Ursule | Impact of the Border Crossing Effects on the DCNU for Pixel Arrays Irradiated With High Energy Protons[END_REF]. Finally, the dark current is estimated by assuming its proportionality to the number of atomic displacement thanks to the use of the Universal Damage Factor (UDF) [START_REF] Srour | Universal Damage Factor for Radiation-Induced Dark Current in Silicon Devices[END_REF]. Mechanisms such as Electric Field Enhancement effect (EFE), or the Total Ionizing Dose (TID) is not taken into account in this study.

B. "Analytical" method

As described in the previous section, many random draws, in both the interaction cross sections and the Poisson's law are required for the full Monte Carlo algorithm calculation. But this convolution product can be performed thanks to a direct numerical integration [START_REF] Schlenker | Methods for calculating the probability distribution of sums of independent random variables[END_REF]. The DCNU is the result of the convolution by itself of the probability density function of the number of displacements generated by a single interaction (individual PDF). The individual PDF is a function of both the interaction cross section and the Lindhard energy partition function. The distribution of damage of a population of pixels undergoing λ i interactions is the λ i convolution of the individual PDF. As the distribution of the population of pixels undergoing λ i interactions is governed by a Poisson's law, the final DCNU is simply the sum of the distributions previously calculated for different λ i values, weighted by this Poisson's law. Historically this way of calculation has been developed first [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] and have been very successful along the years [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF][START_REF] Virmontois | Displacement damage effects due to neutron and proton irradiations on CMOS image sensors manufactured in deep submicron technology[END_REF][START_REF] Belloir | Pixel pitch and particle energy influence on the dark current distribution of neutron irradiated CMOS image sensors[END_REF]. A detailed description can be found in ref. [START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] (Fig. 1a & 1b in [START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF]). The salient points of this method are reminded here. It differs from the Monte Carlo approach only by the assumption made on the shape of the individual PDF (Gamma, Gaussian) used to calculate the number of displacements generated by a single interaction. The analytical convolution product also does not take into account the existing statistical noise of this stochastic damage process. But, the convolution product leads to an exact distribution neglecting any statistical noise, toward which the DCNU tends at very high fluence level. In return, this method is significantly faster than the Monte Carlo one, explaining its success over the years.

If the number of interaction per pixel λ i is big enough, the damage produced becomes proportional to the Non Ionizing Energy Loss (NIEL) of the incident particle. The average number of interactions in a pixel is given by the following formula:

= Φ ! " (2)
Where is the interaction cross section ( # ), the density of silicon ($/

), Φ the incident fluence (% & ' ( / # ), the depletion volume (

), ! Avogadro's number ( ( )* ) and " the atomic mass of the target material (" = 28 $/ ( for silicon).

The mean and the standard deviation of the individual PDF will be called respectively . and in the rest of this paper. Each interaction (Coulombic, nuclear elastic and inelastic) are treated separately with different individual PDF.

In the case of elastic Rutherford scattering, considering a high number of coulombic events per pixel, Marshall et al. approximated the individual PDF by a Gaussian distribution of mean .

and variance # . As we will see further in the paper, this approximation is only valid when the average number of interactions per pixel is large enough so that the final DCNU tends also to a Normal law (appendix A & B). In that case where a large number of interactions is produced, Marshall [START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] shows that the convoluted damage distribution can tend to a Gaussian distribution defined by a mean which is the product of the average number of interactions and the mean of this individual PDF ( .

), and a variance which is the product of the average number of interactions and the variance of this individual PDF ( # ) (Demonstration in appendix A & B) [START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] (appendix A & B). But, before converging towards a Normal law the DCNU is driven by the shape of the Rutherford scattering density probability function. Robbins [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF] proposed to simulate the individual PDF with a gamma distribution, which closer reproduce the Coulombic case. The individual PDF of Coulombic interactions has a very sharp dissymmetric shape that converges very slowly toward a Gaussian function. Robins [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF] chose to describe by this gamma function both inelastic and elastic interactions, in order to increase the accuracy of the prediction for low fluences, but also because of the existence of the negative dark current with Gaussian distribution [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF]. For its part, Germanicus reevaluated the parameters of the gamma distribution of damage for a better fit of the prediction [START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF]. The demonstration is provided in the appendix of this paper (appendix A & B).

III. IMPLEMENTATION OF THE DIFFERENT DCNU PREDICTION METHODS

A similar method to the one of Marshall et al. [START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] has been developed in this work (method 1 of Fig. 1). The difference of this method with the Monte Carlo approach is that, in the latter case, the convolution is performed by a random process, while in the former case, the characteristics of the convoluted distribution is determined according to the NIEL and the standard deviation of the NIEL. The convergence toward the normal distribution (method 2) is also investigated. The speed of convergence have been studied as a function of the fluence level, for the three different interaction types (Coulombic, nuclear elastic and inelastic). Indeed, it is closely dependent on the shape of the energetic spectra of the recoil nuclei which are quite different from a type of interaction to another. The method based on the approximated PDF (method 1 of Fig. 1) is described in the following section III-A The DCNU calculation for large fluence levels (method 2 of Fig. 1) is presented in section III-B.

A. Method1, based on approximated PDF

The population of pixels undergoing λ i interactions is given by the Poisson's law (D1) governing the statistics of λ i . λ i ranges from 0 up to a maximum value that can reach ten for nuclear reactions, but can overpass a thousand for coulombic interactions. The damage distribution produced by these λ i interactions will be denoted D2 in this paper. As mentioned before, D2 can been chosen as a Gaussian or a Gamma function [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF][START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF]. The asymmetric tail of the gamma distribution has the advantage to better describe the Rutherford scattering cross section. In this work, the DCNU calculated assuming both Gaussian and Gamma distributions have been compared. This method 1 developed here is based on the same assumption as the analytical approaches proposed by Marshall [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF] and Robbins [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF] on the basis of a convolution calculation. These methods are equivalent, exception that the method 1 described here, takes into account the intrinsic statistical noise of the stochastic damage process, while the "convolution" methods [START_REF] Dale | The Generation Lifetime Damage Factor and Its Variance in Silicon[END_REF][START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF][START_REF] Marshall | Proton-Induced Displacement Damage Distributions and Extremes in Silicon Microvolumes[END_REF][START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF][START_REF] Bogaerts | Enhanced dark current generation in proton-irradiated CMOS active pixel sensors[END_REF][START_REF] Germanicus | Evaluation and prediction of the degradation of a COTS CCD induced by displacement damage[END_REF][START_REF] Virmontois | Displacement damage effects due to neutron and proton irradiations on CMOS image sensors manufactured in deep submicron technology[END_REF][START_REF] Belloir | Pixel pitch and particle energy influence on the dark current distribution of neutron irradiated CMOS image sensors[END_REF] provide the theoretical distribution towards which the DCNU tends at very high fluence level. According to that simplification, compare to the full Monte Carlo algorithm, the necessary number of random draws is then reduced here to two per pixel (Fig. 1, method 1). In brief, as can be seen in Fig. 1, for each pixel, the number of interactions is randomly drawn in the normal distribution /1. The mean degradation per interaction is then randomly drawn in the gamma distribution /2 defined by its mean . and its standard deviation . The total number of displacements S created in the pixel ', is obtained by multiplying the number of interactions with the mean number of displacements created per interaction.

The first and second moment of the individual PDF (. , ) are proportional respectively to the mean deposited energy 8

to create atomic displacements and its standard deviation. For coulombic interactions, it has been evaluated thanks to the NIEL of NEMO code [START_REF] Inguimbert | NEMO: A Code to Compute NIEL of Protons, Neutrons, Electrons, and Heavy Ions[END_REF]. The number of atomic displacements is simply deduced knowing the threshold displacement energy 9 according to the formula 1. For nuclear interactions, these momentums that correspond to the nuclear NIEL are estimated thanks to a database evaluated with GEANT4 (DBG4, cf. equation 2).

. = E ;<= > > 29 ?@ = A 2B5 > > 29 (1)
. =< /DE4 > ?@ = HIJ (2) Where > is the depth of depleted volume ( ) and 9 the threshold energy (K ).

B. method 2, limit at high fluence level

can show (appendix VI.B) that the result of the method 1 is a DCNU of a known mean λ µdisp , and standard deviation

L # + . #
. But the shape of this distribution is not a priori known. It is affected by the shape of both the chosen individual PDF and the Poison's law. But, when the incident fluence increases, or in a large geometry, the number of interaction per pixel λ i reaches a level for which some statistical simplifications can be applied. In that case, for each pixel, according to the central limit theorem and regardless of the shape of the individual PDF, the damage distribution on the pixel array tends to be normally distributed around this average value .

, and having a standard deviation

L # + . #
( [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF] and proof in the appendix VI.A). This distribution is noted D3 in Fig. 1 (method 2). The method 2 is simply a random draw within the D3 distribution, to take into account the statistical noise which otherwise does not appear in this theoretical distribution. For lower fluences, below the convergence level of the DCNU towards a Gaussian distribution, the discrepancy between the DCNU and a Gaussian distribution has been investigated. Thus, for the same reasons than in the alternative method 1, both Gamma and Gaussian distributions have been studied. In the next section, both methods 1 & 2 will now be compared to the Monte Carlo approach. Some measurements performed on the Jade device will also serve as reference.

IV. COMPARISONS OF THE PREDICTION METHODS

A comparison was made between the Monte Carlo method, and the two simplified approaches 1 and 2. Comparisons with experimental measurements performed on JADE (E2V) devices [START_REF] Martin | Prediction of CMOS Image Sensor Dark Current Distribution and Noise in a Space Radiation Environment[END_REF] are also made. The fluence has been adapted to vary the average number of interaction per pixel λ in the range [0.381, 5531]. Four energy and five fluences per energy were then considered (cf. 1: Characteristics of the simulations realized in this study. For an energy, the fluence, the mean number of nuclear interactions (inelastic and elastic) and the mean number of coulombic interactions per pixel are presented. For coulombic interactions the cross sections have been calculated according to the ZBL model [START_REF] Ziegler | The stopping and range of Ions in solids[END_REF]. Both nuclear elastic and inelastic cross sections are extracted from GEANT4 [START_REF] Agostinelli | GEANT4 a simulation toolkit[END_REF][START_REF] Allison | GEANT4 Developments and Applications[END_REF].

As explained in the previous sections, the shape of the final DCNU is a compromise between the individual PDF chosen for the calculation and the speed of convergence of the DCNU towards a Normal distribution. Two types of functions (Gaussian, Gamma) have been tested for individual PDF to confirm the interest of gamma distribution raised by Robbins [START_REF] Robbins | High-Energy Proton-Induced Dark Signal in Silicon Charge Coupled Devices[END_REF]. The contributions of the three interactions have been considered but the paper presents only the results of Coulombic and inelastic interactions. Indeed, the case of nuclear elastic is quite identical to the degradation caused by nuclear inelastic interactions. Moreover, only DCNU generated by 60 K protons are presented because the trend is the same for other irradiation energies. All these prediction methods have a completely different CPU time. Simplified methods 1 and 2 estimate the DCNU of a sensor array of 500 000 pixels in a few minutes regardless the fluence, when the Monte Carlo approach may need several hours or days for the most critical configurations (i.e. large fluences or large geometries). The Coulombic interactions are the most CPU time consuming, as several thousand of interactions can occur per pixel. The results are presented according to three groups: large fluences generating in average thousands of interactions per pixel, intermediary fluences generating in average between 1 and 1000 interactions per pixel, and weak fluences generating in average less than a single interaction per pixel.

A. High fluence regime

the largest fluences considered in this study (60 K , 7.28 * 10 *# % +/ # , Y Z = 19.7, [\ []^ = 4820), the three methods converge toward the same Gaussian distribution (cf. Fig. 2). Those largest fluences were considered in order to work with close to those used in the literature with analytical methods. The use of a normal distribution instead of a gamma distribution for both methods (1 and 2) provides, at low degradation level, worse results than the use of a gamma distribution. This phenomenon can be observed for both the Coulombic and nuclear inelastic interactions (Fig. 3 & Fig. 4). Surprisingly, even if the average number of nuclear interactions per pixel remains relatively small ( Y Z = 19.7), the final degradation distribution tends to a Gaussian (Fig. 4). The statistical noise at low damage level (Fig. 4) is due to this relatively low value of λ inelastic . for different . The more this term tends to 0, the more the distribution tends to a Gaussian distribution (0=Gaussian). The calculation of this limit (cf. Table 2) proves that the nuclear inelastic and elastic interactions converge faster than coulombic interactions towards a normal distribution. In order to obtain a coulombic limit equivalent to a nuclear limit, []^ needs to be a thousand time higher than Y\

. The speed of convergence is closely dependent on the asymmetry of the individual PDF. The differential Rutherford interaction cross section is a very steep decreasing function while the energy distribution of inelastic recoil nuclei looks more like a Gaussian. That explains the good speed of convergence of the inelastic interactions. For this level of fluence, the simplified methods 1 and 2 give an accurate prediction of the DCNU faster than the Monte Carlo approach.

B. Intermediary fluence regime

When the fluence and accordingly the number of interactions per pixel decrease ( Y Z = 0.197 and [\ []^ = 48.2 at 60 K , 7.28 * 10 *• % +/ # ), the use of these two different PDF (Gaussian , Gamma) lead to different results. But, methods 1 and 2 give results relatively close to each other (Fig. 5). These two methods experience difficulty to predict both Coulombic and nuclear contributions (Fig. 6 & Fig. 7). In particular this is true for the low degradation regime (Δ• < 1?"/ # ). The methods based on normal distributions underestimate the number of hot pixels in regards to the other methods. The sampling of simplified methods seems to be too small to predict the degradation at this level of fluence. The Kullback-Leibler (KL) divergence method [START_REF] Kullback | On information and sufficiency[END_REF] has been used to compare these three estimations with experimental data (Fig. 5). The criterion of comparison chosen for the comparison and which highlight the dissimilarity between the different distributions is the distance between the reference case h(o) (here the experimental results) and the modeled distribution '(o):

/ ƒ| (h||') = " h(o)( $ h(o) '(o) (5)
Two identical distributions will lead to a comparison criterion / ƒ| (h||') = 0. The smaller the / ƒ| (h||') parameter is, the closer the simulated '(o) distribution is close to the experimental h(o) distribution. The Fig. 8 compares the divergence for each prediction method proposed in our study.

The four first points of the simulated distributions have not been considered for the calculation as they are very badly estimated (Fig. 5). At this damage level, the ionizing dose is significant and affects the DCNU. The analysis is focused on the tail of the DCNU. As can be seen in Fig. 8, the Monte Carlo approach with an average value of / ƒ| (h||') < 0.16 provides the results in best agreement with experimental data. This is true whatever the incident energy for a fluence level of 7.28 10 % +/ # that corresponds to ~50. At this level of fluence, the Monte Carlo approach gives the most relevant prediction.

C. Low fluence regime

For weak fluence (i.e. < 1, cf. Fig. 9) and despite an average number of interactions per pixel relatively small ( [\ []^ 0.482 at 60 K , 7.28 * 10 ` % +/ # ), the Monte Carlo approach and the simplified methods based on gamma distribution give surprisingly quite similar DCNU predictions. At this low fluence level, when λ<1, the final DCNU tends towards the individual PDF. For example, when λ Coulombic = 0.482, the number of nuclear interactions becomes very low (λ Nuclear = 0.00197), and the shape of the DCNU is driven by the shape of the Coulombic PDF (Fig. 9 & Fig. 10). For coulombic interactions the individual PDF can calculated according to the ZBL interaction cross section [START_REF] Ziegler | The stopping and range of Ions in solids[END_REF] and the Lindhard energy partition function [START_REF] Agostinelli | GEANT4 a simulation toolkit[END_REF]:

∝ 1 † ‡ˆ ‰ • ‰ I(‰) ( 9
)
Where Q is the recoil energy of the PKA, dσ ZBL /dQ the differential ZBL cross section, and G(Q) the Lindhard energy partition function. The ZBL differential distribution is a strongly decreasing function comparable to a gamma function. That explains why using a gamma function in the analytical methods (1&2) gives results close to the full Monte Carlo algorithm (Fig. 9, Fig. 10). For nuclear interactions (Fig. 11), the agreement is better when using a Gaussian PDF that better represents the shape of the differential nuclear distribution. In order to be compared, the individual PDF and the total degradations predicted by the three approaches have been converted into integrated probabilities (cf. Fig. 12 and Fig. 13). As we can see on the Fig. 12 and Fig. 13, the final DCNU and individual PDF are superimposed on most of the dark current range. It shows the very good agreement between the full Monte Carlo algorithm and the simplified approaches based on the use of a gamma function. On the contrary, the use of a Gaussian distribution is not relevant (Fig. 12). In conclusion, the good agreement between the method 1 and the Monte Carlo approach shows that a Gamma function is well suited to represent the individual PDF for Coulombic scattering (Fig. 10).

V. CONCLUSION

Analytical DCNU calculation methods, that have been widely used in the past has the benefit to be faster than Monte Carlo algorithm. But they make some assumptions on the used damage probability density function that impacts the final result. In other words, such kind of methods are equivalent to the Monte Carlo approach in the limit of the made assumption. In this work, the impact of the use of different probability density functions on the DCNU calculation has been studied. ]^ < 1 the shape of the DCNU is driven by the Coulombic probability density function. In between the DCNU can be well simulated using a Gamma function as a probability density function. This explains in particular the decreasing exponential often observed experimentally on the tail of the DCNU. A difference of convergence speed towards a Gaussian distribution is observed as a function of the interaction type. The nuclear interactions need a value a thousand times smaller than the coulombic interactions to converge towards a Gaussian distribution. But when the fluence decreases (i.e. 1 < < 1000), some discrepancies between methods appear. At this fluence level, the Monte Carlo approach provides the best results according to the calculation of the Kullback-Leibler divergence. The simplified methods based on gamma distribution underestimate the low degradation levels, and the simplified methods based on normal distribution overestimate the pixels lightly impacted and underestimate the hot pixels. At weak fluences (i.e. < 1), the prediction made using a gamma PDF are relatively close to the Monte Carlo simulations showing that Gamma functions are well suited to represent Coulombic differential interaction probabilities.

VI. APPENDIX

A. Demonstration of the method 2

This demonstration determines the shape, the mean value and the variance of the distribution /3 used in the alternative method 2. The goal is to estimate the distribution of the random variable S defined as S = ∑ y •

7 3 •Ž*
where the random variable follows a normal distribution with mean and y • are independent and identically distributed (i.i.d.) random variables as y with mean . and variance # . and y • are also independent random variables for all •. Let us determine the mean and the variance of also the asymptotic form of the distribution of k when → +∞. If we apply the law of total expectation, the mean of k, w(k) is given by w(S) = wmw(S| )n = " w(S| = ?)ℙ( = ?) X' YŽ• [START_REF] Marshall | Displacement Damage Extremes in Silicon Depletion Regions[END_REF] Where w(S| ) is the mean of k conditionally to . As we have also

w(S| = ?) = w "" X • Y •Ž* -= ?w(y) = ?. (7)
We can conclude that the expression of the mean of k is w(S) = " ?. ℙ( = ?) = .

It follows that

K ™ ( ) = exp i j K ¤ i j L # + . # p q l l . L # + . # p q , ∈ ℝ (19)
The series expansion of K £ can be used here if the different moments of y can be defined

K £ ( ) = 1 + w(y) + # w(X # ) 2! + ⋯ + • w(X • ) •! + ⋯ (20)
When → +∞, we get

K £ i j L # + . # p q = 1 + . L # + . # + # 2 + 0 § 1 ¨, ∈ ℝ (23)
The MGF of the random variable ˜ of Eq. ( 20) is then simplified when → +∞ in the following way

K ™ ( ) = exp © # 2 + 0(1)ª , ∈ ℝ (24)
The term exp

B. Relation between method 1 and 2

This demonstration proves that the method 1 and 2 have the same mean and the same variance. In the method 1, the random variable k does not follow a usual law. But in the method 2, the random variable follows a Gaussian law. Let us determine the mean and the variance of as used in the method 1, equal to × y where follows a Poisson distribution with mean and the random variable y follows a normal distribution with mean .

and variance

30) In the method 1, the random variable k is defined by × y, with drawn in the distribution /1 and y drawn in the distribution /2. We have shown here that the mean and the variance of the distribution of this random variable are the same as those of the distribution /3.

Fig. 1 :

 1 Fig. 1: Flowchart of the different methods presented in this study. For each pixel, the degradation is computed with one of the three presented methods.

Fig. 2 :

 2 Fig. 2: Total degradation generated by an incident flux of protons (60 K ; 7.28 * 10 *# % +/ # ).

Fig. 3 :

 3 Fig. 3: Degradation caused by coulombic interactions for an incident flux of protons (60 K ; 7.28 * 10 *# % +/ # ) with [\ []^ = 4 820.

Fig. 4 :

 4 Fig. 4: Degradation caused by nuclear interactions for an incident flux of protons (60 K ; 7.28 * 10 *# % +/ # ) with Y Z = 19.7.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Total degradation generated by an incident flux of protons (60 K ; 7.28 * 10 *• % +/ # ). Experimental data are from ref. [20].

Fig. 7 :

 7 Fig. 7: Degradation caused by nuclear inelastic interactions for an incident flux of protons (60 K ; 7.28 * 10 *• % +/ # ) with Y Z = 0.197.

Fig. 8 :

 8 Fig. 8: Calculation of the Kullback-Leibler divergence of the predicted results with experimental data at four proton energies (30 K , 60 K , 120 K , 185 K ) and for an incident flux of 7.28 * 10 *• % +/ # .

Fig. 9 :

 9 Fig. 9: Total degradation generated by an incident flux of protons (60 K ; 7.28 * 10 ` % +/ # ).

Fig. 10 :

 10 Fig. 10: Degradation caused by coulombic interactions for an incident flux of protons (60 K ; 7.28 * 10 ` % +/ # ) with [\ []^ = 0.482.

Fig. 11 :

 11 Fig. 11: Degradation caused by nuclear inelastic interactions for an incident flux of protons (60 K ; 7.28 * 10 ` % +/ # ) with Y Z = 0.00197.

Fig. 12 :

 12 Fig. 12: Comparison between DCNU and individual PDF caused by coulombic interactions ( [\ []^ = 0.482). Those distributions have been converted, normalized and integrated for the comparison.

Fig. 13 :

 13 Fig. 13: Comparison between DCNU and individual PDF caused by nuclear interactions ( Y\ = 0.00197). Those distributions have been converted, normalized and integrated for the comparison.

  mean of . The law of total variance states that the variance of k, -(k) is equal to -(k) = wm-(S| )n + -mw(S| )n (9) Where -(k| ) is the variance of k conditionally to . The first term wm-(S| )n of equation 9 is easily derived as y • are i.i.d. random variables, -(S| = ?) = -"" wm-(S| )n = wm # n = # (11) The second component of equation 9, -mw(S| )n, is just -mw(S| )n = -m . n moment-generation function (MGF) K , K ™ ( ) = w(exp( ˜)), ∈ ℝ (15) The MGF of k is defined by K Ÿ ( ) = w(exp( k)) = w wmexpm (k| )nn¡ , ∈ ℝ(16)Conditionally to = ?, we have wmexpm (k| )nn = w "exp " are i.i.d. random variables and where K ¤ is the MGF of the random variable y. The MGF of S is then K Ÿ ( ) = wmK ¤ ( ) 7 3 n = " K £ ( ) Y ℙ( = ?)

  to the MGF of a standard normal distribution. Consequently, when → +∞, ˜ converges to a standard normal distribution and k converges to a normal distribution of mean .

  with the law of total expectation:w(k) = wmw(k| )n = wmw( × y| )n = wm × w(y| )n = wm × . n = . (25)where w(k| ) is the mean of k conditionally to . The law of total variance states that the variance of k, -(k), is equal to-(k) = wm-(k| )n + -(w(k| ))(26)where -(k| ) is the variance of k conditionally to . The first term w(-(k| )) corresponds to-(k| ) = -( × y| ) = # -(y| ) = #(27)and thus wm-(k| )n = wm # n = # (28) The second component of variance, -(w(k| )), is just -mw(k| )n = -m . n = . # (29) It follows that -(k) = # + . #

Table 1

 1

	).

Table 2 :

 2 Right-hand term of the Berry Esseen inequality for the three interactions in function of the mean number of interactions per pixel, .

  The analysis has been performed in term of different incident fluence levels. It is demonstrated that the DCNU is a distribution having a known mean and standard deviation (resp. λ µdisp ,

	L # + . # ). This distribution tends towards a Gaussian function at high fluence level ( Y Z ~10,

[\ []^ ~ 1000). At low fluence regime ( [\ [


Min Zhang

Yuhua Wu
email: awuyuhua@163.com

Ting Yang

Shijun Li

Control System Design of Soy Sauce koji-making Based on ARM

Keywords: S3C2440, soy sauce, koji-making, control system

Koji-making is an important part in the production of soy sauce. The traditional means of yeast pool ventilation starter depends on the experience of staff to control the environment of yeast antrum, which koji way automation level is not high, koji product stability is not guaranteed. This paper proposes a design method of soy sauce koji-making control system based on ARM processor S3C2440. It mainly consists of embedded microprocessor S3C2440, power conversion circuit, data storage circuit, display circuit, communication circuit composition. The Linux operating system is used to achieve the automatic control of the traditional process of making koji. The staff can judge the bending machine, fan working time directly through the soy sauce making koji control system, which improves the traditional production of soy sauce production level, and ensures the stability of the quality of the product.

Introduction

With the technical progress of China's soy sauce brewing industry, koji equipment are constantly improved. In the 1960s, traditionally disk and plaque were used for koji-making. In the 1970s, plane ventilation system is adopted to make Koji, which made some progress than the traditional plate, plaque. In the 1980s, koji box appeared, such as the fixed koji box with a cover, tilting koji box with a cover and chained koji box. During the end of the 1980s and the beginning of the 1990s, there appeared the box-type cranes, disc-type machine-made koji and other forms [START_REF] Guanzongmiao | Research on the selection of three kinds of production equipment for 4.5 tons of soy sauce workshop[END_REF]. Because of its high degree of mechanical automation, the mechanical processing of discs was supported by large enterprises. However, because of its high price, some small and medium enterprises still adopt the way of ventilation and koji [START_REF] Wanli | Discussion on optimizing the preparation of soy sauce[END_REF][START_REF] Tianxing | Comparison of thick -layer ventilation koji making and disk ventilation kimako processing equipment in soy sauce production[END_REF]. Koji-making process for turning koji, ventilation time rely entirely on the experience of technical staff regulation, to regulate the temperature of yeast pool. While this conventional koji process of making products with good taste, unique flavor and other characteristics, but the quality is hard to achieve the best and output below the industry standard, the poor stability of the product. Retain the traditional soy sauce manufacturing advantages and characteristics of process based, while improving the quality of the products, to maintain the stability of the product, combining advanced automation technology and communication technology, the soy sauce koji process of real-time monitoring, in order to achieve the optimization of management and reliable operation [START_REF] Xiaowei | The application of networking technology in Fen Daqu in production management[END_REF]. In this paper, the design of the control system of a soy sauce koji making workshops, the system based on S3C2440 processor as the core, based on embedded operating system Linux, combined with multithreading processing technology, to show the real-time workshop environment parameters, and then according to the experts to give the best of environment parameters, the working personnel can through the regulation of the system in a timely manner to control turn song and ventilation equipment of the time, to ensure the stability of yield and quality of soy sauce koji in the production process of koji making workshops.

The whole design scheme of the control system of soy sauce koji workshop is shown in Fig. 1. It is mainly composed of S3C2440 processor, voltage conversion circuit, data storage circuit, control interface circuit, communication interface circuit [START_REF] Xiuze | Design of liquid crystal display control system based on AM335X[END_REF]. Core processor S3C2440 chip, which is mainly responsible for data processing. The voltage conversion circuit converts 5V voltage for the voltage of the system. Control interface circuit comprises LCD and touch screen, the LCD is used to display real-time data, and staff can find real-time environment parameters koji workshop by touching the screen.

Communication interface circuit comprises a serial port, USB interface, network interface, serial port for transplantation debugging information display system, USB interface with the network interface to download the bootloader, Linux kernel image at the same time and the koji making car for interactive [START_REF] Wen | Based on S3C2440 embedded Linux system migration[END_REF].

All the parts of the system using an independent modular design, any of debugging, maintenance, upgrading will not affect other modules work. Power module power is stable and reliable work of the whole system is stable and reliable. System power supply to 5V input, using different modules in the system power regulator chip level conversion, isolated between each module is effectively reduced harass [START_REF] Wenxin | Embedded sensor measurement and control system based on S3C2440 and WinCE[END_REF].

Power Conversion Circuit

S3c2440 microprocessor including ARM core and the memory, the kernel needs a 1.2V power supply, memory is required to 3.3V power supply, external I/O need 3.3V power supply. In order to meet these needs, uses the voltage conversion circuit, the 5V voltage through a voltage conversion circuit is converted to 1.2V and 3.3V.

Power conversion circuit is mainly composed of ltc3406 chip, ltc3406 is a constant frequency current mode architecture, monolithic synchronous buck regulator using and operating input voltage for the 2.5V~5.5V, ltc3406 can be selected adjusting output or fixed 1.5V, 1.8V output. LTC3406 operation process of the power supply current is only 20uA, in shutdown state dropped to 1uA or lower. Due to these characteristics of ltc3406. Through ltc3406 to voltage conversion 1.2V to the kernel power supply, can maintain stable kernel operation. Core voltage conversion circuit as shown in Fig. 2. TFT-LCD with high resolution, rich color, fast response, contrast and brightness are higher, the screen can be a visual angle, easy to achieve a series of features such as large area [START_REF] Gao Hongjin | LCD and flat panel display technology[END_REF]. The system in the AT043TN24 and resolution for 272 TFT LCD screen, interface circuit as shown in Figure 4, including Vsync is a touch screen vertical synchronous signal, Hsync for horizontal synchronous signal, VLCK for the clock signal, led and LED is 3.3V touch screen back photo voltage, R, G, B port transmission of data signals.

Has been in the real life in the extensive use of the touch screen, such as mobile phones, flat, ATM machines and other, with the push of a button compared to the more convenient and more comfortable to use the touch screen. At present on the market touch the type of screen there are many, such as ultrasonic touch screen, capacitive touch screen, resistive touch screens. The resistive touch screen due to the low cost and the electrical can direct access to the user's system and get a lot of use. But it essentially is a resistance divided voltage regulator, the rectangular area in touch the physical location of the point (x, y) conversion on behalf of the X and Y coordinates of the voltage [START_REF] Wei Dongshan | Embedded Linux application development manual[END_REF]. As shown in Fig. 4 directly through the TSYP, TSXP, TSYM, TSXM and touch screen YP, XP, YM, XM connections to achieve human-computer interaction. Software design is implemented in the operating system, the embedded operating system is the core of the whole system development. Because Linux is open source, a high degree of modularity, extensive hardware support, efficient, stable and easy to develop a graphical interface, support the advantages of network transmission and file system, so system software platform by multi task embedded Linux operating system to achieve. In data acquisition system, the Linux system can also handle multiple sensors are transmitted to the more complex control tasks, so as to improve the overall system response speed and parallel processing ability. Regulation system startup processes such as shown in Fig. 5.

Display Driver

Koji workshop control system is tailored to the Linux-2.6.22 kernel based on the display driver in the form of the module is loaded into the Linux kernel, the Linux operating system, device driver is in the form of documents, each device driver are stored in the dev directory, called the device file. The application can open and close read and write the driver files, to complete the operation of equipment operation, like ordinary data files. The user process of equipment file mode of operation is that some system calls. In order to system calls and device driver linked to Linux, all the equipment file provides a unified user interface, which is encapsulated in the operation file_operations this structure inside the. Linux kernel through the file_operations structure into the specific device file operation function, for example: when the upper application through the operating system Commission calls open( ) opens the device file is through the file operations data structure called open () function. Display driver used main function interfaces as shown in Fig. 6, mainly to complete set display palette, pixel display and other functions.

Conclusions

It mainly describes the soy sauce koji workshop control system hardware circuit design and software design and hardware circuit design are introduced in detail process of power conversion circuit, data storage circuit, control circuit, communication interface circuit of the chip selection and interface; software design specific start message processing system software were introduced in this paper, LCD display driver design. The hardware of the system adopts modular design, the software design using hierarchical structure, linux-2.6 embedded operating system as a platform, using multi thread processing control logic, construction of soy sauce koji making workshops regulation system, solves the traditional soy sauce koji making process rely on the experienced staff to control over bending machine, fan and other equipment to work time and improve the level of automation in the production of traditional soy sauce koji, improves the level of traditional soybean paste koji production, to ensure the stability of koji production and quality.

Fig. 1 .

 1 Fig. 1. The overall system framework

Fig. 2 .

 2 Fig. 2. S3C2440 Core voltage conversion circuit Ap3407 application is a kind of more extensive, inexpensive generic voltage converter, using this chip to the 5V voltage conversion to a 3.3V voltage, can save development costs, and to ensure the normal work of S3C2440 chip memory and external I/O. Such as shown in Fig. 3 is a 3.3V voltage conversion circuit.

Fig. 3 .

 3 Fig. 3. 3.3V Voltage conversion circuit

Fig. 4 .

 4 Fig. 4. LCD interface circuit

Fig. 5 .

 5 Fig. 5. Software design

Fig. 6 .

 6 Fig. 6. Display driver function

Acknowledgments

The authors wish to express their gratitude to the projects: Control System Design of Soy Sauce koji-making Based on ARM from Education Department of Jilin Province, Jilin Province Economic Structural Adjustment Leading Fund Special Project (No. 2014Y108) and Changchun City Science and Technology Plan Project (No. 14nk029), Key Tackling Item of Jilin Province Science & Technology Department (No. 20140204045NY), for their generous support of this work. At the same time, the other authors also thank the corresponding author, Shijun LI, for his assistance in submitting this paper.