Applications of Monte Carlo Methods in Charged Particles Optics Alla Shymanska alla.shymanska@aut.ac.nz School of Computing and Mathematical Sciences Auckland University of Technology Private Bag 92006 Auckland 1142, New Zealand International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing 13-17 February 2012 Sydney 13-17 February 2012 – p. 1/3
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. Sydney 13-17 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. Sydney 13-17 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. Sydney 13-17 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. Sydney 13-17 February 2012 – p. 2/3
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. Sydney 13-17 February 2012 – p. 3/3
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. ������������������ �������������������� � Sydney 13-17 February 2012 – p. 4/3
Monte Carlo Simulations The number of secondary electrons generated by the particular collision is defined by the Poisson distribution: P ( ν ) = σ ν e − σ ν ! where ν is the number of secondary electrons produced, σ is SEY, calculated according to the formula: √ σ = σ m [ V √ cos θ ] β e α (1 − cos θ )+ β (1 − V cos θ ) , Vm V m The energy distribution is described by a Yakobson formula: ε − 3 / 2 √ εexp ( − 1 . 5 ε/ ¯ p ( ε ) = 2 . 1¯ ε ) where ¯ ε is the mean energy. Sydney 13-17 February 2012 – p. 5/3
Monte Carlo Simulations (Cont.) Each secondary electron is assigned two emission angles chosen from Lambert’s law: p 1 ( θ ) = sin 2 θ p 2 ( ϕ ) = 1 / 2 π The trajectories of the electron motion inside the channel are calculated from the equations of motion in the uniform field. Sydney 13-17 February 2012 – p. 6/3
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 2 = e ∂U m ∂z ∂r + r 2 0 V 2 d 2 r dt 2 = e ∂U ϕ 0 (1) m r 3 dϕ dt = r 0 r 2 V ϕ 0 where t is time, U = U ( z, r ) is the potential distribution, r 0 is the initial electron coordinate, V ϕ 0 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. Sydney 13-17 February 2012 – p. 7/3
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 ∂r + ∂ 2 U ∂z 2 + 1 ∂U ∂r 2 = 0 (2) r 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. Sydney 13-17 February 2012 – p. 8/3
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: ∞ � u ν p k ( ν ) g k ( u ) = where | u | ≤ 1 . ν =0 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) Sydney 13-17 February 2012 – p. 9/3
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: N � M = m 0 m 1 ...m k ...m N = m k (4) k =0 N k − 1 N � � � m 2 D = d k m i (5) j i =0 k =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. Sydney 13-17 February 2012 – p. 10/3
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: n � Φ( ν ) = p k ϕ k ( ν ) k =1 Sydney 13-17 February 2012 – p. 11/3
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 ν =0 k =1 k =1 After some work the variance D of the distribution at the output of the system can be written as: n n p k g 2 k − G 2 � � D = p k d k + (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. Sydney 13-17 February 2012 – p. 12/3
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 2 mδ ) / ln m 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. Sydney 13-17 February 2012 – p. 13/3
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.) �� � � � � � � � ��� � � � � �� �� �� � 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. Sydney 13-17 February 2012 – p. 14/3
Recommend
More recommend
