Fast Monte Carlo Method in Stochastic Modelling of Charged Particle Multiplication

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1 Fast Mote Carlo Method i Stochastic Modellig of Charged Particle Multiplicatio Alla V. Shymaska *, Vitali A. Babakov School of Computer ad Mathematical Scieces, Aucklad Uiversity of Techology, Private Bag 96, Aucklad 114, New Zealad. * Correspodig author. alla.shymaska@aut.ac.z Mauscript submitted Jauary 6, 15; accepted Jue 4, 15. doi: /ijapm Abstract: Developmet of electroic devices, where the secodary electro emissio is used to amplify iput sigals, requires simulatio of complicated stochastic processes. The essece of the method proposed here cosists of separatig the amplificatio process ito serial ad parallel stages. The mea ad variace of the amplitude distributio at the output of the etire system are obtaied usig the mea ad variace of each stage. Theorems about serial amplificatio stages ad parallel amplificatio paths costitute the key part of the method with partial Mote Carlo simulatio for oe simple stage. Here, the method is used to ivestigate the effect of the cotact coductig layer o the gai ad the oise factor of the chael amplifier. The case of image coverters ad itesifiers with a ivertig electro-optical system ad microchael plate as a amplifier is take for the cosideratio. Key words: Cotact coductig layer, microchael electro amplifier, Mote Carlo simulatios, oise factor, theorems about serial, parallel amplificatio stages. 1. Itroductio Amplificatio by secodary electro emissio has may advatages ad makes it possible to detect eve sigle photos. If electros or other particles strike the solid surface, secodary electros are emitted from it whose umber ca be much higher tha the umber of primary particles. Accordig to the form, i which the successive amplificatio of curret is beig obtaied, we ca distiguish photomultipliers ad chael multipliers. The priciple of a photomultiplier is show i Fig. 1. Electros emitted from the photocathode are focused by the iput focusig system ad reach the first dyode. The material of the dyode has a secodary emissio yield (SEY) σ > 1, so that the umber of electros startig from the first dyode is σ times greater. These electros are focused to the ext dyode, ad the process is repeated so that the umber of electros rises as a avalache. Chael multipliers have similar to the photomultipliers pricipals of multiplicatio but have o discrete multiplicatio stages (Fig. ). The whole multiplier cosists of cylidrical dyode havig a small diameter. The acceleratig voltage is applied to the eds of this cylider. If the ier surface has SEY σ > 1, the electro escapig from the surface at the etrace of the chael will provoke a avalache developig i the directio towards the ed of the chael. Chael multipliers are developed as sigle devices (chael tros) ad as microchael plates (MCP), which is a array of sigle parallel chaels (Fig. ). They are widely used for the detectio ad amplificatio of electros, ios ad eutros, as well as UV ad X-ray 18 Volume 5, Number 3, July 15

photos. They are very attractive for use i particle physics, astroomy, i medical X-ray imagig, image coverters ad itesifiers etc. Fig. 1. Photomultiplier. Fig.. Chael amplifier ad MCP.

Despite a umber of remarkable properties, poor oise characteristics are the mai drawback of systems with chael amplificatio.

2 photos. They are very attractive for use i particle physics, astroomy, i medical X-ray imagig, image coverters ad itesifiers etc. Fig. 1. Photomultiplier. Fig.. Chael amplifier ad MCP. Both types of amplifiers cause a icrease of oise ad therefore a deterioratio of the sigal-to-oise ratio. Despite a umber of remarkable properties, poor oise characteristics are the mai drawback of systems with chael amplificatio. Particularly, MCP is the mai oise source of low-level light image coverters ad itesifiers. Developmet of such devices require simulatio of complicated stochastic processes of the secodary electro emissio. It is usually doe by the use of Mote Carlo (MC) methods [1]-[6]. However, the direct simulatio of the etire amplificatio process by the MC methods requires cosiderable computer time. Optimizatio of a device s parameters i terms of gai, oise ad quality of image makes the simulatio especially difficult or eve impossible. Moreover, it is difficult to evaluate a cotributio of differet system's parameters to the etire amplificatio process, ad their effect o output characteristics. The essece of the method proposed here cosists of separatig the amplificatio process ito serial ad parallel stages. The mea ad variace of the amplitude distributio at the output of the etire system are obtaied usig the mea ad variace of each stage. Theorems about serial amplificatio stages ad parallel amplificatio paths [7] costitute the key part of the method. Combiatio of the theorems ad MC simulatios, which are used oce for oe simple stage, eables oe to coduct complicated ivestigatios ad optimizatios. The method preserves all advatages of the MC simulatios: represets real physical processes fully ad adequately, ad uses experimetal characteristics completely i the model. Moreover, splittig a stochastic process ito a umber of differet stages, allows a cotributio of each stage to the etire process to be easily ivestigated. The method provides high calculatio accuracy with miimum expeditures of the computer time. Here, the method is used to ivestigate the effect of the cotact coductig layer o the gai ad the oise factor of the chael amplifier. The case of image coverters ad itesifiers with a ivertig electro-optical system (EOS) ad MCP as a amplifier is take for cosideratio.. The Theorems 19 Volume 5, Number 3, July 15

3 .1. Theorem about Serial Amplificatio Stages The mea ad variace of the etire multiplicatio process ca be calculated usig the mea ad variace of the separate sequetial stages. Let p k ν be the probability distributio of the umber of particles at the output of the k-th stage, produced by oe particle at its iput. Let m k be the mea ad d k be the variace of the p k ν. The the geeratig fuctio of the probability distributio p k ν is q k u = ν = u ν p k ν, where u 1. Based o the approach used i[8] we ca costruct the geeratig fuctio for the probability distributio of the umber of particles after the last (N-th) stage: Q N u = q q 1 q q N u (1) To fid the mea M, ad variace D of the amplitude distributio P N (ν) after the N-th stage, we covert the expressios (1) to the logarithmic geeratig fuctios, itroducig ew variables: v = l u h k v = l ν= ν= e vν p k ν H N v = l e vν P ν The the expressios (1) ca be writte as H N v = h h 1 (h ( (h N v ) )), where H N v is the logarithmic geeratig fuctio of the distributio P N (ν);h k v is the logarithmic geeratig fuctio of the distributio p k (ν). Differetiatig H N v with respect to v oce ad usig the propertiesof the logarithmic geeratig fuctios, with v= we obtai the mea value of P N ν : M = m m 1 m k m N = N k= m k () Differetiatig H N v with respect to v twice, with v= we obtai the variace D after the N-th stage of this multistep sequetial process. D = d (m 1 m m N ) + d 1 m (m m 3 m N ) + d m m 1 (m 3 m 4 m N ) + + d k m m 1 m k 1 m k+1 m k+ m N + d N m m 1 m N 1 or: D = N k 1 N k= d k i= m i j=k+1 m j (3) The expressios () ad (3)costitute the theorem of serial amplificatio stages [7]... Theorem of Parallel Amplificatio Paths The mea ad variace of the amplitude distributio at the output of the system with some parallel amplificatio paths ca be calculated usig the mea ad variace of each path. Let the primary particle be multiplied alog oe of possible parallel paths, ad ρ k be the probability of choosig the k-th path. If each path gives a average of g k particles at the output with a variace of v k, the the mea G ad the variace V of this multiplicatio process ca be obtaied. Volume 5, Number 3, July 15

4 Let φ k (ν) be the probability distributio of the umberof particles ν at the output of the k-th path produced by oe particle at its iput. The the probability distributio Φ ν of the umber of particles at the output of the etiresystem of parallel paths will be Φ ν = k=1 G = Φ ν ν = ρ k φ k ν ν = ρ k g k ρ k φ k (ν). The the mea G is: ν= k=1 ν= k=1 (4) The variace V of the distributio at the output of the system ca be writte as V = ν=, ν= Φ ν ν Φ(ν)ν where the first sum ca be trasformed to Φ ν ν = ρ k v k + ρ k g k ν= k=1 k=1 Takig ito accout that ν= Φ(ν)ν = G the fial expressio is: V = k=1 ρ k ν k + k=1 ρ k g k G, (5) where G is determied by (4). Equatios (4) ad (5) costitute the theorem of parallel amplificatio paths [7] ad ca be used for discrete ad for cotiuous systems, where sums should be replaced by itegrals. 3. Electro Multiplicatio i Chael Amplifier 3.1. Physical Picture The theorems above with combiatio of partial MC simulatio eable oe to optimize the chael amplifier i terms of the oise factor, which is a measure of the loss of available iformatio. Particularly, the effect of the cotact coductig layer o the oise factor is uder ivestigatio. The followig real physical picture is cosidered i the modelig. Electros of a primary parallel moochromatic beam hit the walls of the chael at differet icidece coordiates ad agles (Fig. 3). Differet icidet agles icrease the divergece of SEY after the first collisio of the primary electros. Due to the spread of the icidet coordiates, the legths of the chael, alog which the multiplicatio of primary electros occur, are differet. After the first collisio each primary electro produces secodary electros with differet emissio eergy ad directios. The secodary electros are multiplied util they leave the chael. Whe all electros have emerged from the chael, the yield of the idividual pulse is kow. Fig. 3. Iput of the primary electro beam. 1 Volume 5, Number 3, July 15

5 The gai of idividual pulses is fluctuated cosiderably, ad the pulse amplitude distributio chages from Poisso distributio at the begiig of the chael to a egative expoetial form at its output. The legth of the chael, where the amplitude distributio chages ad stabilizes, is determied i [7] ad called "the effective chael legth". The mea gai ad the variace of the amplitude distributio at the output of a system defie the oise factor of a amplifier, which is greater for the egative expoetial distributio tha for the peaked oe. I this work the case of image coverters ad itesifiers with a ivertig EOS ad MCP as a amplifier is take for cosideratio. The MCP is placed behid the aode diaphragm close to the scree [9]. To accelerate the secodary electros toward the exit of the chael, the voltage is applied by depositig a cotact coductig layer at its eds. The electrostatic field of the EOS, the field iside the chael ad cotact coductig layer create a ouiform electrostatic field at the etrace of the chael. Sice there is o potetial drop alog the coductig layer, the coditios for the movemet of secodary electros i this regio are differet from motio of electros i a uiform field iside the chael. Existig models, based oly o full MC simulatios, eglect the effect of the ouiform field ad the spread i icidet coordiates of the primary electro beam, but all these factors are take ito accout i the computatioal model, proposed here. 3.. Computatioal Model A computatioal model for simulatio of stochastic processes of a electro multiplicatio i chael amplifiers is based o 3D MC simulatios, ad theorems about serial amplificatio stages ad parallel amplificatio paths. The etire multiplicatio process i the chael amplifier is split ito a umber of sequetial stages ad parallel multiplicatio paths. Multiplicatio of a sigle electro, emitted at the begiig of the chael, is cosidered as a separate stage. It is simulated by 3D MC methods i the homogeeous field alog the "effective chael legth" which is equal half of the chael [7]. The fuctios g(z), the mea, ad d(z), the variace (z is the coordiate directed alog the chael axis ad measured from its begiig), are calculated for z L/, where L is the coordiate of the ed of the chael. The process of MC simulatios uses a radom umber geeratig procedure [4], [7] to sample the various distributios such as: the distributio of the actual yield of secodaries after each collisio, the emissio eergy, ad the directio of each secodary electro. For electros, leavig the first half of the chael, the icidece coordiates ξ k > L/ ad SEY ς k are determied. The amplificatio i the secod half of the chael is cosidered to cosist of parallel paths. Each path has two sequetial stages: first collisio ad multiplicatio of a sigle electro util it leaves the chael. The fuctios g(z) ad d(z) alog the etire chael legth ( z L) are calculated usig expressios ()-(5).The fuctios g(z) ad d(z),ad the theorems allow us to coduct further ivestigatios ad optimizatios without ay additioal MC simulatios, what sigificatly reduces the cost of computatios, ad provides highly accurate results. The part of the chael with ouiform field is cosidered as a separate stage. If the iitial coditios of the collisio of the primary electros with the chael walls are chaged (icidet agle or eergy, legth of the coductig layer or a high-efficiecy emitter etc.), MC simulatios are coducted oly i the regio of ouiform electrostatic field. The fuctios g(z) ad d(z) are used to fid the mea ad variace of the distributio at the output of the chael, usig the theorems. The ouiform field at the etrace of the chael is calculated by solvig the Laplace's partial differetial equatio i cylidrical coordiate system. The trajectories of the electros i the ouiform field are calculated by the Ruge-Kutta method. Fig. 4 shows the ouiform field at the etrace of the chael. Computatioal results have bee obtaied for the chael diameter d=1 µm, the voltage o the chael U=8 V, ad the sputterig depth of the cotact layer at the chael iput h=.5d. Volume 5, Number 3, July 15

6 Fig. 4. Nouiform field at the etrace of the chael. The portio of the chael from a elemetary area at its iput, where the collisio occurred, to the output of the chael (Fig. 3) ca be cosidered as the amplificatio path whe the spread i icidece coordiates of the iput electros was take ito accout [1]. I such case, the variace D ad the average gai G at the output of the multiplier ca be defied usig (4) ad (5), where the sums should be replaced by itegrals over the surface of the chaels, bombarded by the electros of the primary beam: G = s ψ s g s ds, D = ψ s s d s ds + ψ s g s ds G, s where ψ is the probability desity for the particle to strike the elemetary surface ds; g(s) is the average umber of particles with variace d(s) at the output of the path. Usig the theorems, fially we obtai [1]: G = πr cotθ π/ ς(φ, V, θ) cosφdφ Rtaθcosφ g z, L dz (6) D = πr cotθ πr cotθ π π/ ς φ, V, θ cosφdφ ς (φ, V, θ) cosφdφ Rtaθcosφ Rtaθcosφ [g z, L + d(z, L])dz + g z, L dz G, (7) where φ is the azimuthal coordiate i the cylidrical system; z = Rtaθcosφ is the border of the area of the cylidrical chael, bombarded by the electros of the primary beam (Fig. 3). The oise factor F ca be defied as [4, 7]: F = γ 1 (1 + D G ), where G ad D are defied by (6) ad (7) Computatioal Results The computatioal results, obtaied here, reflect the ifluece of the ouiform electrostatic field ad the spread of the icidet coordiates of the primary electros o the gai ad the oise factor. The area covered by the iput electro beam depeds o the directio agle θ (Fig. 3). At small agles θ the collisio of the primary electros with the chael wall occurs ear the chael edge i the regio of the week electrostatic field. The secodary electros i this regio are ot accelerated 3 Volume 5, Number 3, July 15

7 sufficietly to cotiue the multiplicatio process. As a result, the gai is less tha i the case of the uiform field at the etrace of the chael. Moreover, a cosiderable part of the secodary electros with a egative velocity does ot tur back because of the absece of the field, ad leaves the chael. At large agles θ the large spread i the collisio coordiates of the iput electros with the chael walls icreases the variace of the amplitude distributio at the output of the chael. Whe the peetratio depth of the iput electros is icreasig the umber of multiplicatio stages is decreasig, ad the gai is decreasig as well. For the agle θ, where the effects, described above, are i balace, a miimum oise factor is observed, ad such agle θ ca be cosidered as a optimum for the give coverage depth. I order to icrease gai ad reduce the oise factor, a layer with icreased SEY is deposited o the top of the cotact coductig layer at the etrace of chaels. However, due to the effect of ouiform electrostatic field ad spread i icidece coordiates of the iput electro beam, the oise factor ca eve icrease [11]. Fig. 5 shows the depedece of the mea gai g(z, L) o the emissio coordiate z of a secodary electro exitig the high efficiecy emitter of the legth h=5d, deposited o the top of the cotact coductig layer. Curve 1 has bee calculated for the uiform field at the etrace of the chael, curve is relevat to ouiform field formed by the cotact coductig layer (what is the practical case). Whe the high efficiecy emitter is deposited o the top of a cotact coductig layer (curve ) oly electros, emitted from the border of the high efficiecy emitter (here, it is z>4d), take part i multiplicatio. It is see that the loss of secodary electros, emitted i the area of week field, is sigificat. g(z,l) x Fig. 5. Mea gai g as fuctio of emissio coordiatez (1 uiform field; - ouiform field). z µm G x Fig. 6. Depedece of gai G o legth h of cotact coductig layer for differet SEY. h µm 4 Volume 5, Number 3, July 15

8 Fig. 6 ad Fig. 7 show depedece of the gai G ad the oise factor F at the output o the legth of the cotact coductig layer h for differet values of SEY. It is see that the gai is decreasig ad the oise is icreasig regardless of the value of SEY. F Fig. 7. Depedece of oise F o legth h of cotact coductig layer for differet SEY. h µm Fig. 8. Depedece of oise factor o icidece agle θ ad legth h of high-efficiecy emitter. The summary results are show o the Fig. 8 as the fuctio F(θ, h), where θ is the icidece agle of the primary electro beam, ad h is the legth of the high-efficiecy emitter deposited o the top of the cotact coductig layer. The eergy of the primary electro beam was V = 1 kv. These results make it possible to determie the optimum combiatio of the parameters, providig the miimal oise factor ad the regio where it is stable. Thus, here θ 7 ad h 5d are optimum values, which provide the miimum oise factor F = 1.4. It must be poited out that the rage of parameter values 5 < θ < 75 ad h d - 5d esures the smallest value of the oise factor which practically does ot deped o variatios i θ ad h. Refereces [1] Bikley, D. M. (1994). Optimizatio of scitillatio-detector timig system usig Mote Carlo aalysis. IEEE Tras. Nucl. Sci., 41, [] Choi, Y. S., & Kim, J. M. (). Mote Carlo simulatios for tilted - chael electro multipliers. IEEE Tras. Electro. Devices, 47, [3] Giudicotti, L. (). Aalytical, steady-state model of gai saturatio i chael electro multipliers. Nucl. Istr. ad Meth. A, 48, [4] Guest, A. J. (1971). A computer model of chael multiplier plate performace. Acta Electro, 14, [5] Price, G. J., & Fraser, G. W. (1). Calculatio of the output charge cloud from a microchael plate. Nucl. Istr. ad Meth. A, 474, Volume 5, Number 3, July 15

[6] Shikhaliev, P. M., Ducote, J. L., Xu, T., & Molloi, S. (5). Quatum efficiecy of the MCP detector: Mote Carlo Calculatio. IEEE Tras. Nucl. Sci., 5, 157-16. [7] Shymaska, A. V. (1).

[9] Shymaska, A. V. (11). Numerical aalysis of electro optical system with microchael plate. J. Comput. Electro, 1, 91-99. [1] Shymaska, A. V., & Babakov, V. A. (14).

9 [6] Shikhaliev, P. M., Ducote, J. L., Xu, T., & Molloi, S. (5). Quatum efficiecy of the MCP detector: Mote Carlo Calculatio. IEEE Tras. Nucl. Sci., 5, [7] Shymaska, A. V. (1). Computatioal modelig of stochastic processes i electro amplifiers. J. Comput. Electro, 9, [8] Yaoshi, L. (1955). Statistical problems of a electro multiplier. Zh. Eksp. Teor.Fiz., 9, [9] Shymaska, A. V. (11). Numerical aalysis of electro optical system with microchael plate. J. Comput. Electro, 1, [1] Shymaska, A. V., & Babakov, V. A. (14). Computatioal models for ivestigatio of chael amplifier's optimal parameters. J. Comput. Electro., 13, [11] Shymaska, A. V. (15). Effect of high efficiecy emitter o oise characteristics of electro amplifiers. J. Comput. Electro. Alla Shymaska has her BEg i electroics ad PhD i mathematics ad physics (St Petersburg). She has a log research ad teachig experiece i areas of physical electroics ad charged particle optics at differet research ceters ad uiversities i Russia ad Ukraie. Curretly she cotiues her research ad teaches rage of papers i mathematics ad computer sciece at School of Computer ad Mathematical Scieces of Aucklad Uiversity of Techology, Aucklad, New Zealad. Vitali Babakov has his PhD ad doctor of sciece degrees i mathematics (Novosibirsk). His major research areas are mathematics ad ewtoia mechaics. For may years he teaches calculus for egieerig studets ad statistics for busiess studets at Aucklad Uiversity of Techology (New Zealad). His arrow field of activity is theory elasticity ad plasticity, where V. Babakov ivestigates mathematical problems that ca be applied for practical eeds. 6 Volume 5, Number 3, July 15

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