Applications of Monte Carlo Methods in Charged Particles Optics - - PowerPoint PPT Presentation

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) = σνe−σ ν!

where ν is the number of secondary electrons produced,

σ is SEY, calculated according to the formula: σ = σm[ V Vm √ cos θ]βeα(1−cos θ)+β(1− V

Vm

√ cos θ),

The energy distribution is described by a Yakobson formula:

p(ε) = 2.1¯ ε−3/2√εexp(−1.5ε/¯ ε)

where ¯

ε is the mean energy.

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Monte Carlo Simulations (Cont.)

Each secondary electron is assigned two emission angles chosen from Lambert’s law:

p1(θ) = sin 2θ p2(ϕ) = 1/2π

The trajectories of the electron motion inside the channel are calculated from the equations of motion in the uniform field.

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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 :

d2z dt2 = e m ∂U ∂z d2r dt2 = e m ∂U ∂r + r2

0V 2 ϕ0

r3 dϕ dt = r0 r2Vϕ0

(1)

where t is time, U = U(z, r) is the potential distribution, r0 is the initial electron coordinate, Vϕ0 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:

∂2U ∂z2 + 1 r ∂U ∂r + ∂2U ∂r2 = 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.

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Theorem of Serial Amplification Stages

Let pk(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 pk(ν) is:

gk(u) =

∞

• ν=0

uνpk(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:

GN(u) = GN−1[gN(u)]

• r

GN(u) = g0(g1(g2(...(gN(u))...)))

(3)

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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 PN(ν) after the N-th stage can be obtained:

M = m0m1...mk...mN =

N

• k=0

mk

(4)

D =

N

• k=0

dk

k−1

• i=0

mi

N

• j=k+1

m2

j

(5)

where mk and dk 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.

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Theorem of Parallel Amplification Paths

Let the primary particle be multiplied along one of n possible parallel paths, and pk be the probability of choosing the k-th path. If each path gives an average of gk particles at the output with a variance of dk, then the mean G and the variance D of this multiplication process can be obtained. Let ϕk(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

pkϕk(ν)

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Theorem 2 (Cont.)

Then the mean G of such a multiplication process is equal to:

G =

∞

• ν=0

Φ(ν)ν =

n

• k=1

pk

∞

• ν=0

ϕk(ν)ν =

n

• k=1

pkgk

(6)

After some work the variance D of the distribution at the

• utput of the system can be written as:

D =

n

• k=1

pkdk +

n

• k=1

pkg2

k − G2

(7)

Equations (6) and (7) can be used for discrete and for continuous systems, where sums should be changed to integrals.

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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 vr has an error δ compared with the relative variance of the amplitude distribution at the output of the entire channel.

n < ln(1 + m 2mδ )/ ln m

The effective length leff of the channel can be evaluated as

leff = λn where λ is the average free path of electrons in

the channel. For δ = 0.01, for typical values of the multiplier parameters, leff corresponds to half the channel length. The numerical experiment, using the MC methods, completely confirms this result.

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Effective Length (Cont.)

The figure shows the relative variance vr 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 dk is its diameter.)

• The effective length can be defined as a part of the channel

where the amplitude distribution is stabilized, and the shape

• f the distribution is close to a negative exponential

function.

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Effective Length (Cont.)

The figures show the amplitude distributions calculated by MC methods for the length of the channel z/dk = 1 and

z/dk = 22 (half of the channel).

• Sydney 13-17 February 2012 – p. 15/3

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.

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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.

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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.

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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

• ut

(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

• btained. The expressions depend on how the entire

process is split into a sequence of amplification stages. For example:

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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

m0 = γ, d0 = γ(1 − γ).

• 2. The collision of the primary electrons with the wall of the
• channel. The mean m1 and the variance d1 of the

distribution of the number of electrons knocked out by one primary electron:

m1 = d1 = σ1,

• 3. Further amplification of the electrons in the channel is

regarded as the third stage with the mean gain m2 = m(L) and the variance d2 = d(L).

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Noise Factor of a Single Channel (Cont.)

Taking into account the contribution of each stage to the

• verall process of amplification and with the help of the

theorems we obtain:

( S N )2

• ut = M2

D , where M = neγm1m(L), and D = ne[γm1m(L)]2 + γ(1 − γ)ne[m1m(L)]2 + d1neγm2(L) + d(L)neγm1. F = γ−1(1 + vr1 + vr2/m1),

where vr1 and vr2 are the relative variances.

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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

Rmin

ψ(R)g(R)dR, D = Rmax

Rmin

ψ(R)d(R)dR + Rmax

Rmin

ψ(R)g2(R)dR − G2,

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.

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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 σx √ 2πexp[−(R − ¯ R)2 2σ2

x

]

(10)

where σ2

x is the variance, and ¯

R is the mean.

After some work the expression for ψ(R) can be written as:

ψ(R) = R2e−( 300(R− ¯

R) √ 2δ ¯ R

)2

¯ R( δ ¯

R 300)2e−( 300R

√ 2δ ¯ R)2 +

√ 2πδ ¯ R 300 [ ¯

R2 + ( δ ¯

R 300)2]

.

(11)

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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.

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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.

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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.

G =

• s

ψ(s)g(s)ds,

(12)

V =

• s

ψ(s)v(s)ds +

• s

ψ(s)g2(s)ds − G2,

(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.

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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.

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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).

• Sydney 13-17 February 2012 – p. 26/3

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 × 104.

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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.

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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 30 - 60 seconds.

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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 30 - 60 seconds.

• 2. It would require about 20 days and nights to find the
• ptimal 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.

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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 30 - 60 seconds.

• 2. It would require about 20 days and nights to find the
• ptimal 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.

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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 30 - 60 seconds.

• 2. It would require about 20 days and nights to find the
• ptimal 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.

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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.

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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.

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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.

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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.

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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.

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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.

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Appendix

The time needed to calculate the electron pulse on the channel length x = z/d can be declared as

τ = τ0 x αeαtdt = τ0(eαx − 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, τ0 = 0.44 msec and

α = 0.12.

• 2

4 6 8 5 10 15 20 25 30

τms z/d

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