Applications of Monte Carlo Methods in Charged Particles Optics

Transcription

1 Sydney February 2012 p. 1/3 Applications of Monte Carlo Methods in Charged Particles Optics Alla Shymanska School of Computing and Mathematical Sciences Auckland University of Technology Private Bag Auckland 1142, New Zealand International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing February 2012

2 Sydney February 2012 p. 2/3 Introduction 1. Amplification of charged particles is a complicated stochastic process. This work is devoted to a theoretical investigation of stochastic processes of an electron multiplication in electronic devices.

3 Sydney February 2012 p. 2/3 Introduction 1. Amplification of charged particles is a complicated stochastic process. This work is devoted to a theoretical investigation of stochastic processes of an electron multiplication in electronic devices. 2. The essence of the approach proposed here consists of separating the amplification process into serial and parallel stages. The developed method is based on Monte Carlo (MC) simulations and theorems about serial and parallel amplification stages proposed here.

4 Sydney February 2012 p. 2/3 Introduction 1. Amplification of charged particles is a complicated stochastic process. This work is devoted to a theoretical investigation of stochastic processes of an electron multiplication in electronic devices. 2. The essence of the approach proposed here consists of separating the amplification process into serial and parallel stages. The developed method is based on Monte Carlo (MC) simulations and theorems about serial and parallel amplification stages proposed here. 3. The use of the theorems provides a high calculation accuracy with minimal cost of computations. The MC simulations are used once for one simple stage.

5 Sydney February 2012 p. 2/3 Introduction 1. Amplification of charged particles is a complicated stochastic process. This work is devoted to a theoretical investigation of stochastic processes of an electron multiplication in electronic devices. 2. The essence of the approach proposed here consists of separating the amplification process into serial and parallel stages. The developed method is based on Monte Carlo (MC) simulations and theorems about serial and parallel amplification stages proposed here. 3. The use of the theorems provides a high calculation accuracy with minimal cost of computations. The MC simulations are used once for one simple stage. 4. Splitting a stochastic process into a number of different stages, allows a contribution of each stage to the entire process to be easily investigated.

6 Sydney February 2012 p. 3/3 Introduction (Cont.) Here the method is used to minimize a noise factor of microchannel electron amplifiers. Microchannel plates, as arrays of single channels, have found wide applications in different areas of science, engineering, medicine etc. However, the loss of information caused by the statistical fluctuations in the gain of the channels, and by loss of primary electrons when they strike the closed area of a channel plate increases a noise factor.

7 Sydney February 2012 p. 4/3 Introduction (Cont.) The following physical picture was considered in the modelling. The electrons of a parallel monochromatic beam are incident on the input plane of a microchannel multiplier. Electrons entering the channel have different incidence coordinates and hit the walls at different angles, producing secondary electrons with different emission energy and directions. The secondary electrons are multiplied until they leave the channel. primary electrons secondary electrons

8 Sydney February 2012 p. 5/3 Monte Carlo Simulations The number of secondary electrons generated by the particular collision is defined by the Poisson distribution: P(ν) = σν e σ where ν is the number of secondary electrons produced, σ is SEY, calculated according to the formula: σ = σ m [ V V m cosθ] β e α(1 cosθ)+β(1 V Vm cos θ), The energy distribution is described by a Yakobson formula: where ε is the mean energy. ν! p(ε) = 2.1 ε 3/2 εexp( 1.5ε/ ε)

9 Sydney February 2012 p. 6/3 Monte Carlo Simulations (Cont.) Each secondary electron is assigned two emission angles chosen from Lambert s law: p 1 (θ) = sin 2θ p 2 (ϕ) = 1/2π The trajectories of the electron motion inside the channel are calculated from the equations of motion in the uniform field.

10 Sydney February 2012 p. 7/3 Motion of electrons in the Potential Field The trajectories of the electrons in a nonuniform electrostatic field with axial symmetry are calculated by solving the system of differential equations : d 2 z dt = e 2 m U z d 2 r dt = e 2 m U r + r2 0Vϕ 2 0 r 3 dϕ dt = r 0 r 2 V ϕ0 (1) where t is time, U = U(z,r) is the potential distribution, r 0 is the initial electron coordinate, V ϕ0 is the initial azimuthal component of the electron velocity, e, m are electron charge and mass respectively. Classical Runge-Kutta method is used to solve the system of ODEs.

11 Sydney February 2012 p. 8/3 Motion of electrons in the Potential Field Determination of the potential field is a matter of finding a solution to the Laplace s partial differential equation expressed in cylindrical coordinates as follows: 2 U z r U r + 2 U r 2 = 0 (2) It is the classical mixed problem for the equation of Laplace with Dirichlet and Neumann boundary conditions. To find a solution, numerical finite-difference methods are used. The figure shows the nonuniform electrostatic field at the entrance of the channel.

12 Sydney February 2012 p. 9/3 Theorem of Serial Amplification Stages Let p k (ν) be the probability distribution of the number of particles at the output of the k-th stage, produced by one particle from the (k 1)-th stage. Then the generating function of the probability distribution p k (ν) is: g k (u) = ν=0 u ν p k (ν) where u 1. It can be shown that the generating function for the probability distribution of the number of particles after the last (N-th) stage can be constructed as: G N (u) = G N 1 [g N (u)] or G N (u) = g 0 (g 1 (g 2 (...(g N (u))...))) (3)

13 Sydney February 2012 p. 10/3 Theorem 1 (Cont.) If the expression (3) is converted to the logarithmic generating function, then after some work, the expressions for the mean M, and variance D of the amplitude distribution P N (ν) after the N-th stage can be obtained: M = m 0 m 1...m k...m N = N k=0 m k (4) D = N k 1 d k m i N m 2 j (5) k=0 i=0 j=k+1 where m k and d k are the mean and variance of the distribution of the number of particles at the output of the k-th stage for one particle at its input.

14 Sydney February 2012 p. 11/3 Theorem of Parallel Amplification Paths Let the primary particle be multiplied along one of n possible parallel paths, and p k be the probability of choosing the k-th path. If each path gives an average of g k particles at the output with a variance of d k, then the mean G and the variance D of this multiplication process can be obtained. Let ϕ k (ν) be the probability distribution of the number of particles ν at the output of the k-th path produced by one particle at its input. Then the probability distribution Φ(ν) of the number of particles at the output of the entire system of n parallel paths will be: Φ(ν) = n k=1 p k ϕ k (ν)

15 Sydney February 2012 p. 12/3 Theorem 2 (Cont.) Then the mean G of such a multiplication process is equal to: n n G = Φ(ν)ν = p k ϕ k (ν)ν = p k g k (6) ν=0 k=1 ν=0 k=1 After some work the variance D of the distribution at the output of the system can be written as: D = n p k d k + n p k g 2 k G2 (7) k=1 k=1 Equations (6) and (7) can be used for discrete and for continuous systems, where sums should be changed to integrals.

16 Sydney February 2012 p. 13/3 Effective Length of the Channel The theorem about series amplification stages enables one to evaluate the number of stages n, after which the relative variance v r has an error δ compared with the relative variance of the amplitude distribution at the output of the entire channel. n < ln( 1 + m )/ ln m 2mδ The effective length l eff of the channel can be evaluated as l eff = λn where λ is the average free path of electrons in the channel. For δ = 0.01, for typical values of the multiplier parameters, l eff corresponds to half the channel length. The numerical experiment, using the MC methods, completely confirms this result.

17 Sydney February 2012 p. 14/3 Effective Length (Cont.) The figure shows the relative variance v r as a function of the length of the channel. It is calculated for a single electron emitted at the beginning of the channel (z is the length of the channel, and d k is its diameter.) Vr The effective length can be defined as a part of the channel where the amplitude distribution is stabilized, and the shape of the distribution is close to a negative exponential function. z/dk

18 Sydney February 2012 p. 15/3 Effective Length (Cont.) The figures show the amplitude distributions calculated by MC methods for the length of the channel z/d k = 1 and z/d k = 22 (half of the channel) n G n

19 Sydney February 2012 p. 16/3 Computational Algorithm 1. The multiplication of a single electron emitted at the beginning of the channel is simulated by MC methods along half the channel length. Functions g(z), the mean, and d(z), the variance, are calculated on this length. For n electrons leaving the first half of the channel, the incidence coordinates and the values of the SEY (σ) are determined.

20 Sydney February 2012 p. 16/3 Computational Algorithm 1. The multiplication of a single electron emitted at the beginning of the channel is simulated by MC methods along half the channel length. Functions g(z), the mean, and d(z), the variance, are calculated on this length. For n electrons leaving the first half of the channel, the incidence coordinates and the values of the SEY (σ) are determined. 2. The amplification in the second half of the channel is considered to consist of n parallel paths. Each path has two sequential stages: first collision and multiplication of a single electron until it leaves the channel. Using the theorems the functions g(z) and d(z) along the entire channel length are calculated.

21 Sydney February 2012 p. 16/3 Computational Algorithm 1. The multiplication of a single electron emitted at the beginning of the channel is simulated by MC methods along half the channel length. Functions g(z), the mean, and d(z), the variance, are calculated on this length. For n electrons leaving the first half of the channel, the incidence coordinates and the values of the SEY (σ) are determined. 2. The amplification in the second half of the channel is considered to consist of n parallel paths. Each path has two sequential stages: first collision and multiplication of a single electron until it leaves the channel. Using the theorems the functions g(z) and d(z) along the entire channel length are calculated. 3. Further investigations and optimizations can be done without any additional MC simulations with high degree of accuracy.

22 Sydney February 2012 p. 17/3 Noise Factor of the Channel Multiplier The noise factor F, which is a measure of the loss of available information can be written as F = (S/N)2 in (S/N) 2 out (8) where (S/N) in and (S/N) out are ratios of the input signal to the noise and the output signal to the noise respectively. Using the definition of the noise factor (8) and the theorems about serial amplification stages and parallel amplification paths expressions for calculating the noise factor can be obtained. The expressions depend on how the entire process is split into a sequence of amplification stages. For example:

23 Sydney February 2012 p. 18/3 Noise Factor of a Single Channel 1. The first observation of electrons, incident at the input of the multiplier. If γ is the fraction of the front surface of the multiplier exposed to electrons, then the average number of particles entering the channel and the variance can be given by m 0 = γ, d 0 = γ(1 γ). 2. The collision of the primary electrons with the wall of the channel. The mean m 1 and the variance d 1 of the distribution of the number of electrons knocked out by one primary electron: m 1 = d 1 = σ 1, 3. Further amplification of the electrons in the channel is regarded as the third stage with the mean gain m 2 = m(l) and the variance d 2 = d(l).

24 Sydney February 2012 p. 19/3 Noise Factor of a Single Channel (Cont.) Taking into account the contribution of each stage to the overall process of amplification and with the help of the theorems we obtain: ( S N )2 out = M2 D, where M = n e γm 1 m(l), and D = n e [γm 1 m(l)] 2 + γ(1 γ)n e [m 1 m(l)] 2 + d 1 n e γm 2 (L) + d(l)n e γm 1. F = γ 1 (1 + v r1 + v r2 /m 1 ), where v r1 and v r2 are the relative variances.

25 Sydney February 2012 p. 20/3 Noise Factor of an Array of the Channels For the system of n parallel channels F = 1 γ (1 + D G2), where (9) G = Rmax R min ψ(r)g(r)dr, D = Rmax R min ψ(r)d(r)dr + Rmax R min ψ(r)g 2 (R)dR G 2, where ψ(r) is the probability density function, g(r) and d(r) are the mean and the variance of the amplitude distribution at the output of the channel with the radius R.

26 Sydney February 2012 p. 21/3 Variations in Channel Diameters Variations of the channel diameters as a result of technological distortions of a channel s geometry lead to the variations of the amplitude distributions at the outputs of different channels, and increase the noise factor. Such variations are defined by the normal distribution: ϕ(r) = 1 (R R) 2 exp[ σ x 2π 2σx 2 ] (10) where σ 2 x is the variance, and R is the mean. After some work the expression for ψ(r) can be written as: ψ(r) = R( δ R 300 ) 2 ( 300R e 2δ R) 2 + R 2 R) (300(R e 2δ ) 2 R 2πδ R 300 [ R 2 + ( δ R 300 ) 2 ]. (11)

27 Sydney February 2012 p. 22/3 Variations in Diameters (Cont.) Figure shows the results of numerical experiments where δ is the variations of the channels diameters. The results obtained here can be used to calculate the noise factor F for the given values of δ and R, to calculate δ which provides the required value F, and also to optimize parameters of the channel plate in terms of the minimum F. Calculations of F(δ) using only MC simulations would take about 3 days and nights of constant computer calculating. The use of the theorems reduces this time to 1 minute.

28 Sydney February 2012 p. 23/3 Spread in Incidence Coordinates The electrons of the primary monochromatic parallel beam, directed into a cylindrical channel, have different angles and coordinates for their collision with the channel walls. The portion of the channel from an elementary area at its input, where the collision occurred, to the output of the channel can be considered as the amplification path.

29 Sydney February 2012 p. 24/3 Spread in Incidence Coordinates (Cont.) For variations in the collision coordinates of the electrons of the primary beam, the variance V and the average gain G at the output of the multiplier can be defined using the theorem of parallel amplification paths, where sums should be replaced by integrals. V = s G = s ψ(s)v(s)ds + ψ(s)g(s)ds, (12) s ψ(s)g 2 (s)ds G 2, (13) where s is the surface area stroked by particles; ψ is the probability density for the particle to strike the elementary surface ds; g(s) is the average number of particles with variance v(s) at the output of the path.

30 Sydney February 2012 p. 25/3 Spread in Incidence Coordinates (Cont.) In order to evaluate the effect on noise characteristics caused by the spread in the collision coordinates of input electrons two models have been used: a model with a fixed incidence coordinate of the input electrons and the model with the spread in the incidence coordinates. The numerical experiments have shown that the spread in the collision coordinates of primary electrons significantly affects the average gain and the noise factor, and must be taken into account in theoretical models. It has been shown, that maximum differences in calculations using two models are: 50% for the gain and 25% for the noise factor.

31 Sydney February 2012 p. 26/3 Optimization of the Channel Multiplier The figures show the dependence of the noise factor and the average gain on the energy of the input electron beam. The theoretical results (solid curves) are compared with the experimental data (dashed curves). F E, ev

32 Sydney February 2012 p. 27/3 Optimization (cont.) The figure shows the dependence of the average gain on the energy and the incidence angle of the input electron beam. The numbers on the curves refer to the values of the gain, G 10 4.

33 Sydney February 2012 p. 28/3 Optimization (cont.) The figure shows the dependence of the noise factor on the energy and the incidence angle of the input electron beam. The numbers on the curves refer to the values of the noise factor.

34 Sydney February 2012 p. 29/3 Efficiency of the Method 1. For the direct MC simulations calculations of F(E) and G(E) would take about 3 days and nights of the constant work of the computer (Pentium 4) for one characteristic. The use of the proposed theorems reduces the cost of calculations to seconds.

35 Sydney February 2012 p. 29/3 Efficiency of the Method 1. For the direct MC simulations calculations of F(E) and G(E) would take about 3 days and nights of the constant work of the computer (Pentium 4) for one characteristic. The use of the proposed theorems reduces the cost of calculations to seconds. 2. It would require about 20 days and nights to find the optimal combination of the energy and the angle of the input electron beam which provides the minimal noise factor and about 1-2 minutes if the proposed theorems are used.

36 Sydney February 2012 p. 29/3 Efficiency of the Method 1. For the direct MC simulations calculations of F(E) and G(E) would take about 3 days and nights of the constant work of the computer (Pentium 4) for one characteristic. The use of the proposed theorems reduces the cost of calculations to seconds. 2. It would require about 20 days and nights to find the optimal combination of the energy and the angle of the input electron beam which provides the minimal noise factor and about 1-2 minutes if the proposed theorems are used. 3. For the nonuniform electrostatic field the cost of calculations will be increased significantly for the direct MC simulations.

37 Sydney February 2012 p. 29/3 Efficiency of the Method 1. For the direct MC simulations calculations of F(E) and G(E) would take about 3 days and nights of the constant work of the computer (Pentium 4) for one characteristic. The use of the proposed theorems reduces the cost of calculations to seconds. 2. It would require about 20 days and nights to find the optimal combination of the energy and the angle of the input electron beam which provides the minimal noise factor and about 1-2 minutes if the proposed theorems are used. 3. For the nonuniform electrostatic field the cost of calculations will be increased significantly for the direct MC simulations. 4. For this application of the method, the MC simulations should be conducted only once on the effective channel length for one electron emitted at the beginning.

38 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages.

39 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages. 2. The theorems for the multistep sequential processes and for the parallel amplification paths have been proved.

40 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages. 2. The theorems for the multistep sequential processes and for the parallel amplification paths have been proved. 3. For the application here, it has been shown that the amplitude distribution at the output of the channel is determined by the effective length of the channel.

41 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages. 2. The theorems for the multistep sequential processes and for the parallel amplification paths have been proved. 3. For the application here, it has been shown that the amplitude distribution at the output of the channel is determined by the effective length of the channel. 4. The method provides high accuracy and significantly reduces the cost of calculations.

42 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages. 2. The theorems for the multistep sequential processes and for the parallel amplification paths have been proved. 3. For the application here, it has been shown that the amplitude distribution at the output of the channel is determined by the effective length of the channel. 4. The method provides high accuracy and significantly reduces the cost of calculations. 5. The contribution of different amplification stages to the entire stochastic process can be easily investigated.

43 Sydney February 2012 p. 30/3 Conclusion 1. The method for calculation of the stochastic processes has been developed where the entire process is represented in the form of the sequence of several stages. 2. The theorems for the multistep sequential processes and for the parallel amplification paths have been proved. 3. For the application here, it has been shown that the amplitude distribution at the output of the channel is determined by the effective length of the channel. 4. The method provides high accuracy and significantly reduces the cost of calculations. 5. The contribution of different amplification stages to the entire stochastic process can be easily investigated. 6. The method can be used for many stochastic processes which require computer simulations.

44 Sydney February 2012 p. 31/3 Appendix The time needed to calculate the electron pulse on the channel length x = z/d can be declared as τ = τ 0 x 0 αe αt dt = τ 0 (e αx 1) (14) Computational experiments show the average time needed for MC simulations of one electron pulse as a function of the channel length. From the graph, τ 0 = 0.44 msec and α = τ ms z/d