applications of monte carlo methods in charged …applications of monte carlo methods in charged...

Applications of Monte CarloMethods in Charged Particles Optics

Alla Shymanska

[email protected]

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. 1/31

Introduction

1. Amplification of charged particles is a complicatedstochastic process. This work is devoted to a theoreticalinvestigation of stochastic processes of an electronmultiplication in electronic devices.


Introduction

1. Amplification of charged particles is a complicatedstochastic process. This work is devoted to a theoreticalinvestigation of stochastic processes of an electronmultiplication in electronic devices.2. The essence of the approach proposed here consists ofseparating the amplification process into serial and parallelstages. The developed method is based on Monte Carlo(MC) simulations and theorems about serial and parallelamplification stages proposed here.


Introduction

1. Amplification of charged particles is a complicatedstochastic process. This work is devoted to a theoreticalinvestigation of stochastic processes of an electronmultiplication in electronic devices.2. The essence of the approach proposed here consists ofseparating the amplification process into serial and parallelstages. The developed method is based on Monte Carlo(MC) simulations and theorems about serial and parallelamplification stages proposed here.3. The use of the theorems provides a high calculationaccuracy with minimal cost of computations. The MCsimulations are used once for one simple stage.


Introduction

1. Amplification of charged particles is a complicatedstochastic process. This work is devoted to a theoreticalinvestigation of stochastic processes of an electronmultiplication in electronic devices.2. The essence of the approach proposed here consists ofseparating the amplification process into serial and parallelstages. The developed method is based on Monte Carlo(MC) simulations and theorems about serial and parallelamplification stages proposed here.3. The use of the theorems provides a high calculationaccuracy with minimal cost of computations. The MCsimulations are used once for one simple stage.4. Splitting a stochastic process into a number of differentstages, allows a contribution of each stage to the entireprocess to be easily investigated.


Introduction (Cont.)

Here the method is used to minimize a noise factor ofmicrochannel electron amplifiers.

Microchannel plates, as arrays of single channels, havefound wide applications in different areas of science,engineering, medicine etc. However, the loss of informationcaused by the statistical fluctuations in the gain of thechannels, and by loss of primary electrons when they strikethe closed area of a channel plate increases a noise factor.


Introduction (Cont.)

The following physical picture was considered in themodelling. The electrons of a parallel monochromatic beamare incident on the input plane of a microchannel multiplier.Electrons entering the channel have different incidencecoordinates and hit the walls at different angles, producingsecondary electrons with different emission energy anddirections. The secondary electrons are multiplied until theyleave the channel.

primary electrons secondary electrons


Monte Carlo Simulations

The number of secondary electrons generated by theparticular 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) = 2.1ε̄−3/2√εexp(−1.5ε/ε̄)

where ε̄ is the mean energy.Sydney 13-17 February 2012 – p. 5/31

Monte Carlo Simulations (Cont.)

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

p1(θ) = sin 2θ p2(ϕ) = 1/2π

The trajectories of the electron motion inside the channelare calculated from the equations of motion in the uniformfield.


Motion of electrons in the Potential Field

The trajectories of the electrons in a nonuniformelectrostatic field with axial symmetry are calculated bysolving the system of differential equations :

d2zdt2 = e

m∂U∂z

d2rdt2 = e

m∂U∂r +

r20V 2

ϕ0

r3

dϕdt = r0

r2Vϕ0

(1)

where t is time, U = U(z, r) is the potential distribution, r0 isthe initial electron coordinate, Vϕ0 is the initial azimuthalcomponent of the electron velocity, e, m are electron chargeand mass respectively. Classical Runge-Kutta method isused to solve the system of ODEs.


Motion of electrons in the Potential Field

Determination of the potential field is a matter of finding asolution to the Laplace’s partial differential equationexpressed 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 Laplacewith Dirichlet and Neumann boundary conditions. To find asolution, numerical finite-difference methods are used.The figure shows the nonuniform electrostatic field at theentrance of the channel.


Theorem of Serial Amplification Stages

Let pk(ν) be the probability distribution of the number ofparticles at the output of the k-th stage, produced by oneparticle from the (k − 1)-th stage. Then the generatingfunction 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 theprobability distribution of the number of particles after thelast (N -th) stage can be constructed as:

GN (u) = GN−1[gN (u)] or GN (u) = g0(g1(g2(...(gN (u))...)))(3)


Theorem 1 (Cont.)

If the expression (3) is converted to the logarithmicgenerating function, then after some work, the expressionsfor the mean M , and variance D of the amplitudedistribution 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

m2j (5)

where mk and dk are the mean and variance of thedistribution of the number of particles at the output of thek-th stage for one particle at its input.


Theorem of Parallel Amplification Paths

Let the primary particle be multiplied along one of npossible parallel paths, and pk be the probability of choosingthe k-th path. If each path gives an average of gk particlesat the output with a variance of dk, then the mean G and thevariance D of this multiplication process can be obtained.Let ϕk(ν) be the probability distribution of the number ofparticles ν at the output of the k-th path produced by oneparticle at its input. Then the probability distribution Φ(ν) ofthe number of particles at the output of the entire system ofn parallel paths will be:

Φ(ν) =n∑

k=1

pkϕk(ν)


Theorem 2 (Cont.)

Then the mean G of such a multiplication process is equalto:

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 theoutput of the system can be written as:

D =n∑

k=1

pkdk +n∑

k=1

pkg2k −G2 (7)

Equations (6) and (7) can be used for discrete and forcontinuous systems, where sums should be changed tointegrals.


Effective Length of the Channel

The theorem about series amplification stages enables oneto evaluate the number of stages n, after which the relativevariance vr has an error δ compared with the relativevariance of the amplitude distribution at the output of theentire channel.

n < ln(1 +m

2mδ)/ lnm

The effective length leff of the channel can be evaluated asleff = λn where λ is the average free path of electrons inthe channel. For δ = 0.01, for typical values of the multiplierparameters, leff corresponds to half the channel length.The numerical experiment, using the MC methods,completely confirms this result.


Effective Length (Cont.)

The figure shows the relative variance vr as a function ofthe length of the channel. It is calculated for a singleelectron emitted at the beginning of the channel (z is thelength of the channel, and dk is its diameter.)

0

1

2

3

4

5

6

0 6 12 18 24

Vr

z/dk

The effective length can be defined as a part of the channelwhere the amplitude distribution is stabilized, and the shapeof the distribution is close to a negative exponentialfunction.


Effective Length (Cont.)

The figures show the amplitude distributions calculated byMC methods for the length of the channel z/dk = 1 andz/dk = 22 (half of the channel).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.8 1.6 2.4 3.2 Gn

n


Computational Algorithm

1. The multiplication of a single electron emitted at thebeginning of the channel is simulated by MC methods alonghalf the channel length. Functions g(z), the mean, and d(z),the variance, are calculated on this length. For n electronsleaving the first half of the channel, the incidencecoordinates and the values of the SEY (σ) are determined.



1. The multiplication of a single electron emitted at thebeginning of the channel is simulated by MC methods alonghalf the channel length. Functions g(z), the mean, and d(z),the variance, are calculated on this length. For n electronsleaving the first half of the channel, the incidencecoordinates and the values of the SEY (σ) are determined.2. The amplification in the second half of the channel isconsidered to consist of n parallel paths. Each path has twosequential stages: first collision and multiplication of asingle electron until it leaves the channel. Using thetheorems the functions g(z) and d(z) along the entirechannel length are calculated.



1. The multiplication of a single electron emitted at thebeginning of the channel is simulated by MC methods alonghalf the channel length. Functions g(z), the mean, and d(z),the variance, are calculated on this length. For n electronsleaving the first half of the channel, the incidencecoordinates and the values of the SEY (σ) are determined.2. The amplification in the second half of the channel isconsidered to consist of n parallel paths. Each path has twosequential stages: first collision and multiplication of asingle electron until it leaves the channel. Using thetheorems the functions g(z) and d(z) along the entirechannel length are calculated.3. Further investigations and optimizations can be donewithout any additional MC simulations with high degree ofaccuracy.


Noise Factor of the Channel Multiplier

The noise factor F , which is a measure of the loss ofavailable information can be written as

F =(S/N)2in(S/N)2out

(8)

where (S/N)in and (S/N)out are ratios of the input signal tothe noise and the output signal to the noise respectively.Using the definition of the noise factor (8) and the theoremsabout serial amplification stages and parallel amplificationpaths expressions for calculating the noise factor can beobtained. The expressions depend on how the entireprocess is split into a sequence of amplification stages. Forexample:


Noise Factor of a Single Channel

1. The first observation of electrons, incident at the input ofthe multiplier. If γ is the fraction of the front surface of themultiplier exposed to electrons, then the average number ofparticles entering the channel and the variance can begiven by

m0 = γ, d0 = γ(1 − γ).

2. The collision of the primary electrons with the wall of thechannel. The mean m1 and the variance d1 of thedistribution of the number of electrons knocked out by oneprimary electron:

m1 = d1 = σ1,

3. Further amplification of the electrons in the channel isregarded as the third stage with the mean gain m2 = m(L)and the variance d2 = d(L).


Noise Factor of a Single Channel (Cont.)

Taking into account the contribution of each stage to theoverall process of amplification and with the help of thetheorems we obtain:

(S

N)2out =

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.


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) andd(R) are the mean and the variance of the amplitudedistribution at the output of the channel with the radius R.


Variations in Channel Diameters

Variations of the channel diameters as a result oftechnological distortions of a channel’s geometry lead tothe variations of the amplitude distributions at the outputs ofdifferent channels, and increase the noise factor.Such variations are defined by the normal distribution:

ϕ(R) =1

σx

√2πexp[−(R− R̄)2

2σ2x

] (10)

where σ2x is the variance, and R̄ is the mean.

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

ψ(R) =R2e−( 300(R−R̄)

√

2δR̄)2

R̄( δR̄300)2e−( 300R

√

2δR̄)2 +

√2πδR̄300 [R̄2 + ( δR̄

300)2]. (11)


Variations in Diameters (Cont.)

Figure shows the results of numerical experiments where δis the variations of the channels’ diameters.

The results obtained here can be used to calculate thenoise factor F for the given values of δ and R̄, to calculate δwhich provides the required value F , and also to optimizeparameters of the channel plate in terms of the minimum F .Calculations of F (δ) using only MC simulations would takeabout 3 days and nights of constant computer calculating.The use of the theorems reduces this time to 1 minute.


Spread in Incidence Coordinates

The electrons of the primary monochromatic parallel beam,directed into a cylindrical channel, have different angles andcoordinates for their collision with the channel walls.

The portion of the channel from an elementary area at itsinput, where the collision occurred, to the output of thechannel can be considered as the amplification path.


Spread in Incidence Coordinates (Cont.)

For variations in the collision coordinates of the electrons ofthe primary beam, the variance V and the average gain Gat the output of the multiplier can be defined using thetheorem of parallel amplification paths, where sums shouldbe replaced by integrals.

G =

∫sψ(s)g(s)ds, (12)

V =

∫sψ(s)v(s)ds+

∫sψ(s)g2(s)ds−G2, (13)

where s is the surface area stroked by particles; ψ is theprobability density for the particle to strike the elementarysurface ds; g(s) is the average number of particles withvariance v(s) at the output of the path.


Spread in Incidence Coordinates (Cont.)

In order to evaluate the effect on noise characteristicscaused by the spread in the collision coordinates of inputelectrons two models have been used: a model with a fixedincidence coordinate of the input electrons and the modelwith the spread in the incidence coordinates. The numericalexperiments have shown that the spread in the collisioncoordinates of primary electrons significantly affects theaverage gain and the noise factor, and must be taken intoaccount in theoretical models.It has been shown, that maximum differences incalculations using two models are: 50% for the gain and25% for the noise factor.


Optimization of the Channel Multiplier

The figures show the dependence of the noise factor andthe average gain on the energy of the input electron beam.The theoretical results (solid curves) are compared with theexperimental data (dashed curves).

1.5

2.0

2.5

3.0

3.5

4.0

0 400 800 1 200 1 600 2 000

F

E, eV


Optimization (cont.)

The figure shows the dependence of the average gain onthe energy and the incidence angle of the input electronbeam. The numbers on the curves refer to the values of thegain, G× 104.


Optimization (cont.)

The figure shows the dependence of the noise factor on theenergy and the incidence angle of the input electron beam.The numbers on the curves refer to the values of the noisefactor.


Efficiency of the Method

1. For the direct MC simulations calculations of F (E) andG(E) would take about 3 days and nights of the constantwork of the computer (Pentium 4) for one characteristic.The use of the proposed theorems reduces the cost ofcalculations to 30 - 60 seconds.



1. For the direct MC simulations calculations of F (E) andG(E) would take about 3 days and nights of the constantwork of the computer (Pentium 4) for one characteristic.The use of the proposed theorems reduces the cost ofcalculations to 30 - 60 seconds.2. It would require about 20 days and nights to find theoptimal combination of the energy and the angle of theinput electron beam which provides the minimal noise factorand about 1 - 2 minutes if the proposed theorems are used.



1. For the direct MC simulations calculations of F (E) andG(E) would take about 3 days and nights of the constantwork of the computer (Pentium 4) for one characteristic.The use of the proposed theorems reduces the cost ofcalculations to 30 - 60 seconds.2. It would require about 20 days and nights to find theoptimal combination of the energy and the angle of theinput electron beam which provides the minimal noise factorand about 1 - 2 minutes if the proposed theorems are used.3. For the nonuniform electrostatic field the cost ofcalculations will be increased significantly for the direct MCsimulations.



1. For the direct MC simulations calculations of F (E) andG(E) would take about 3 days and nights of the constantwork of the computer (Pentium 4) for one characteristic.The use of the proposed theorems reduces the cost ofcalculations to 30 - 60 seconds.2. It would require about 20 days and nights to find theoptimal combination of the energy and the angle of theinput electron beam which provides the minimal noise factorand about 1 - 2 minutes if the proposed theorems are used.3. For the nonuniform electrostatic field the cost ofcalculations will be increased significantly for the direct MCsimulations.4. For this application of the method, the MC simulationsshould be conducted only once on the effective channellength for one electron emitted at the beginning.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.2. The theorems for the multistep sequential processes andfor the parallel amplification paths have been proved.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.2. The theorems for the multistep sequential processes andfor the parallel amplification paths have been proved.3. For the application here, it has been shown that theamplitude distribution at the output of the channel isdetermined by the effective length of the channel.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.2. The theorems for the multistep sequential processes andfor the parallel amplification paths have been proved.3. For the application here, it has been shown that theamplitude distribution at the output of the channel isdetermined by the effective length of the channel.4. The method provides high accuracy and significantlyreduces the cost of calculations.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.2. The theorems for the multistep sequential processes andfor the parallel amplification paths have been proved.3. For the application here, it has been shown that theamplitude distribution at the output of the channel isdetermined by the effective length of the channel.4. The method provides high accuracy and significantlyreduces the cost of calculations.5. The contribution of different amplification stages to theentire stochastic process can be easily investigated.


Conclusion

1. The method for calculation of the stochastic processeshas been developed where the entire process isrepresented in the form of the sequence of several stages.2. The theorems for the multistep sequential processes andfor the parallel amplification paths have been proved.3. For the application here, it has been shown that theamplitude distribution at the output of the channel isdetermined by the effective length of the channel.4. The method provides high accuracy and significantlyreduces the cost of calculations.5. The contribution of different amplification stages to theentire stochastic process can be easily investigated.6. The method can be used for many stochastic processeswhich require computer simulations.


Appendix

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

τ = τ0

∫ x

0αeαtdt = τ0(e

αx − 1) (14)

Computational experiments show the average time neededfor MC simulations of one electron pulse as a function ofthe channel length. From the graph, τ0 = 0.44 msec andα = 0.12.

0

2

4

6

8

0 5 10 15 20 25 30

τ ms

z/d


applications of monte carlo methods in charged …applications of monte carlo methods in charged...

Documents