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Multiparticle Trajectory Simulation for Ion Trap

Mass Spectrometers

A Thesis

Submitted for the Degree of

Master of Technology

By

Neeraj Kumar Verma

Supercomputer Education and Research Center

Indian Institute of Science

Bangalore - 560 012, India

July 2008

Acknowledgements

During my stay at IISc, I have met many wonderful people who finally have become an

integral part of my professional and personal life. I take this opportunity to thank all

those people. Their help, support and guidance, have been invaluable for me.

I thank my project guide Dr. A. K. Mohanty. His problem solving approach accom-

panied with knowledge base and commitment has been a great source of inspiration for

me. Under his guidance not only I have completed my project work, but have also learnt

much which will help me in my future academic and professional life.

I thank Prof. A.G. Menon for his continuous support and encouragement. This project

work would not have been completed without his support. I would also like to express my

sincere gratitude to Prof. A. Chatterjee for his comments and suggestion on the work.

I am grateful to Prof. R. Govindrajan, Chairman, Supercomputer Education and

Research Centre , for allowing me to use all the facilities of the department. I also thank

all the faculty and staff members of the department for their assistance.

I would like to thank Mrs. Sandya who has been a good friend. I am also thankful to

my labmates for their help and support. They have been a part in making my lab stay a

wonderful experience. I would like to thank my classmates for their help and support.

I would thank my friends Madhurima, Krishna, Ganesh, Ganapathy, Mario, Rakesh,

Sumit, Mehul, Krishnakant and Ghouse. I will always cherish their friendship. My inter-

action with them has taught me many principles of life.

Last but not the least, I would like to render my sincere gratitude to all those who

have directly or indirectly helped in making this happen.

Preface

The project aims at development of numerical simulation software which will generate

mass spectrum of a given sample by characterising ion dynamics inside the trap. The

simulator has been designed to handle Paul like traps generating spectrum by the mass

selective boundary ejection experiment.

The first step in simulation is to compute electric field distribution for a given geometry

parameters. It is two step process. First, charge on the surface of the electrodes is calcu-

lated using Boundary Element Method (BEM). Second, field at any point is computed by

accumulating the contribution of all electrode elements. User defined experimental con-

ditions mark the starting point of ion trajectory simulation. Initial position and velocity

are sampled from standard/user-specified probability distribution. Interaction between

particles will be captured using viscous/collision model.

This simulation will be useful in exploring characteristic performance parameters of

traps with a wide range of possible geometries. It will help in ongoing attempts of minia-

turising ion trap spectrometer.

The thesis consists of four chapters. Chapter 1 provides the necessary background

for the study. Chapter 2 gives the system overview of the simulation package. Chapter

3 presents the details of modules and submodules of the simulator. Chapter 4 provides

results and its verification as obtained by the simulation studies.

References in the text have been given by quoting the author’s name and year of

publication. Full references have been provided, in an alphabetic order, at the end of the

thesis.

i

Contents

Preface i

List of Figures vi

List of Tables ix

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Trap Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Axially Symmetric Traps . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Linear Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Equation of Motion and Stability Plot . . . . . . . . . . . . . . . . . . . . 4

1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 MSBEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 REE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Scope of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 System overview 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Simulator block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Simulation initializer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Main simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

iii

Contents iv

2.5 Output analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Simulation tester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Module details 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Simulation Initializer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Reading geometry file . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Creating Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Reading Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.4 Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.5 Position Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.6 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.7 Space Charge Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Main Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Charge Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Field Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Trajectory Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Trap Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.2 Multipole Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.3 Poincare Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Results and Verification 34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Geometries used for verification . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Velocity and position distribution . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Electric potential distribution . . . . . . . . . . . . . . . . . . . . . . . . . 36

Contents v

4.5 Electric field distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.6 Trajectory and micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.7 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A Derivations used for LIT and RIT potential calculation 51

A.1 Potential due to a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A.2 Image charge calculation used in LIT and RIT potential calculation . . . . 55

A.3 Potential due to an infinite narrow strip inside cylinder . . . . . . . . . . . 58

A.4 Equipotential surface for two infinite line charges . . . . . . . . . . . . . . 60

List of Figures

1.1 Cross section of (a) Paul Trap (b) Cylindrical Ion Trap (CIT). . . . . . . . 3

1.2 Schematic diagram of a linear ion trap. . . . . . . . . . . . . . . . . . . . . . 4

1.3 Cross section of a linear ion trap. . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Schematic Diagram of Rectilinear Ion Trap (RIT). . . . . . . . . . . . . . . . . 5

1.5 Cross section of the RIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Mathieu stability plots for (a) the LIT and (b) the 3D Paul trap. . . . . . . . . 7

1.7 Timing diagram showing different stages of the experiment. Dead time (0-

t1), Ionization time(t1-t2),Cooling time (t2-t3), Ramp time(t3-t4), Ionization

voltage (V3), Cooling voltage (V2), Ramp start voltage (V1), Ramp end

voltage (V4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 Process of mass selective boundary ejection for Paul trap. The ring elec-

trode has been excited by rf only. . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Block diagram showing modules and their submodules. . . . . . . . . . . . 13

2.2 Flow chart for (a) Reading input ion distribution (b) Generating velocity

distribution (c) Generating position distribution. . . . . . . . . . . . . . . . 15

2.3 Flow chart for trajectory integration. . . . . . . . . . . . . . . . . . . . . . 17

2.4 Flow chart for spectrum generation by MSBEE. . . . . . . . . . . . . . . . 18

3.1 Hirarchical structure for geometry creation. . . . . . . . . . . . . . . . . . . 21

3.2 Graphical interface for (a) creating geometry (b) editing the part of the geometry. 22

3.3 Potential due to ith ring at a point on jth ring . . . . . . . . . . . . . . . 26

vi

List of Figures vii

3.4 Ramping of voltage in mass selective boundary ejection experiment . . . . 31

3.5 Potential due to the ring at a point on its axis . . . . . . . . . . . . . . . . 33

4.1 Cross section of (a) Paul Trap (b) Cylindrical Ion Trap (CIT). . . . . . . . 35

4.2 Analytical and numerical results for generation of normal distribution. . . . . . 36

4.3 Analytical and numerical results for Paul trap with no hole. (a) axial poten-

tial distribution (b) radial potential distribution. The continuous line shows

analytical output and dots represent numerical output. . . . . . . . . . . . . . 37

4.4 Numerical results for Paul trap with hole. (a) axial potential distribution (b)

radial potential distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 Numerical results for CIT with hole. (a) axial potential distribution (b) radial

potential distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.6 Numerical and analytical results for Paul trap with no hole. (a) axial field

distribution (b) radial field distribution . . . . . . . . . . . . . . . . . . . . . 39

4.7 Numerical and analytical results for Paul trap with hole. (a) axial field distri-

bution (b) radial field distribution. . . . . . . . . . . . . . . . . . . . . . . . . 40

4.8 Numerical and analytical results for CIT with hole. (a) axial field distribution

(b) radial field distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.9 Trajectory of an ion of mass 78amu in the Paul Trap at rf amplitude of

350V . The initial position of the ion is (0.0001m,0.0001m,0.0001m) and

the initial velocity has been set to zero. . . . . . . . . . . . . . . . . . . . . 41

4.10 Trajectory of an ion of mass 78amu in the Paul Trap at rf amplitude of

375V . The initial position of the ion is (0.0001m,0.0001m,0.0001m) and

the initial velocity has been set to zero. . . . . . . . . . . . . . . . . . . . . 42

4.11 Ion micromotion along (a) axial direction (b) radial direction. . . . . . . . . . . 44

4.12 FFT of ion micromotion along (a) axial direction (b) radial direction. . . . . . . 45

4.13 Spectrum of air obtained by Paul trap simulation without damping . . . . . . 47

4.14 Spectrum of benzene obtained by Paul trap simulation without damping . . . . 48

4.15 Spectrum of benzene obtained by Paul trap simulation with damping . . . . . 49

4.16 Spectrum of benzene obtained by CIT simulation with damping . . . . . . . . 50

List of Figures viii

A.1 Potential calculation for Sector with uniform charge distribution. . . . . . . . . 52

A.2 Calculating potential due to a triangular section. . . . . . . . . . . . . . . . . 53

A.3 Location of the image charge due to charge at S(ρs, 0). . . . . . . . . . . . . . 56

A.4 Cross section of an infinite line charge inside a grounded cylinder. . . . . . . . . 58

A.5 Cross section of an infinite strip. . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.6 Equipotential surface due to two line charges . . . . . . . . . . . . . . . . . 61

List of Tables

2.1 Description of the elements of electrode data structure. . . . . . . . . . . . . . 14

2.2 Description of the elements of ion data structure. . . . . . . . . . . . . . . . . 14

2.3 Description of the elements of trajectory data structure. . . . . . . . . . . . . 16

4.1 Geometry parameters of the traps studied. All dimensions are in mm. . . . . . 35

4.2 Mass and charge distribution of air sample. . . . . . . . . . . . . . . . . . . . 46

4.3 Mass and charge distribution of benzene sample. . . . . . . . . . . . . . . . . 46

A.1 Symmetry classes of common traps. . . . . . . . . . . . . . . . . . . . . . . . 51

ix

Chapter 1

Introduction

1.1 Introduction

Ion trap is a device used for trapping the ions and then facilitating their detection based

on mass to charge (m/e) ratio (March and Hughes, 1989). It is also used as an ion

storage device. Its trapping volume is formed by shaped electrodes across which rf/dc

is applied to generate the oscillatory electric field required for trapping of the ions. Ions

can be trapped using magnetic fields also. The penning trap uses this concept. But

there are certain limitations which restrict its suitability for mass spectroscopy. First,

to generate electric field either magnet or electromagnet is required. It makes the whole

device heavier. Also it is comparatively difficult to maintain the uniformity of the field.

Mass spectrometers are analytical instruments used for compositional and structural

analysis of a chemical sample. Ion trap mass spectrometers uses ion traps as its mass

analyzer. In these spectrometers chemical sample is ionized by electron bombardment to

form a characteristic mixture of ions of different mass to charge (m/e) ratio. The ions

of the mixture is first allowed to get concentrated near the center of the trap and then

they are selectively ejected out of the trap by ramping rf amplitude. The ions ejected out

of the trap are detected by electron multiplier which sends signal to the display unit for

generating the mass spectrum.

Based on electrode shapes, there are different types of ion traps. The most widely used

ion trap is Paul trap named after its inventor (Paul and Steinwedel, 1953). It has one ring

electrode and two endcap electodes. The hyperbolic shape of the electrodes define a linear

electric field inside the trap. But due to manufacturing difficulties involved in getting

these electrodes other geometries have been used. Cylindrical Ion Trap (CIT) proposed

by Cooks and coworkers uses cylindrical ring electrode and flat endcap electrodes. Both

Paul trap and CIT have rotational symmetry about its axis hence they are referred as

axially symmetric traps. There is another family of traps where only shaped rods and

plates have been used as electrodes. Linear Ion Trap (LIT) (Bier and Syka, 1995) is the

most widely used member of this family. LIT can be thought of the two dimensional

counterpart of the Paul trap. Its electrodes are in form of four rods with hyperbolic

surface. Another member Rectilinear Ion Trap (RIT) uses two pairs of flat plates as

electrode (Langmuir et al., 1962). Many variants of these geometries have also been used

in theoretical and practical studies.

1

Chapter 1. Introduction 2

In ion trap based mass spectrometry, spectrum can be generated by Mass Selective

Boundary Ejection Experiment (MSBEE) (Stafford et al., 1984) or Resonance Ejection

Experiment (REE). In MSBEE endcaps are grounded and rf is applied across the ring

electrode. The rf amplitude is ramped up linearly to make the ions unstable. In REE

an auxiliary excitation is applied to endcaps. The rf amplitude is ramped up so that ion

frequency matches the auxiliary excitation frequency and ion becomes unstable due to

resonance.

Need of a simulation study of ion traps becomes obvious from the fact that ideal Paul

trap is the only trap which can be analyzed fully by analytical means due to its linear

nature. All other traps offer a nonlinear field distribution. Factors like electrode trun-

cation, surface roughness, misalignment and feed holes also introduces nonlinearities in

practical traps. These nonlinearities can not be easily captured by analytical expressions.

Further, the ongoing efforts of trap size miniaturization require exploration of operational

characteristics of different trap geometries. This can be achieved by making hardware for

different geometries and then performing the experiments. But it will require lots of time

and money. Simulation allows this exploration as well as avoids the complexities of the

hardware path.

In this chapter different types of traps used as mass analyzer in spectrometers have

been introduced and then their operational parameters have been discussed. Later in this

chapter two ways of getting a mass spectrum MSBEE and REE have been elaborated.

This chapter end with the discussion of the scope of the project.

1.2 Trap Geometries

Different types of trap geometries have been proposed over the period. First among

these are Paul trap invented by Wolfgang Paul. Other geometries include CIT, LIT and

RIT. The proposal of trap geometries have been dictated mainly by factors like ease of

fabrication, scope of miniaturization and nonlinearity of the field distribution inside the

trap. Ideal Paul trap offers linear field distribution with in the trap volume. But getting

exactly hyperboloid electrode geometry for miniaturized traps is not easy. Any deviation

from the hyperboloid geometry introduces nonlinearity. CIT, LIT, and RIT are nonlinear

traps but they are relatively easier to fabricate. Following subsections gives the details of

geometry parameters of different traps.


1.2.1 Axially Symmetric Traps

These rotationally symmetric trap structures consists of a ring electrode and two endcap

electrodes. Endcaps are generally grounded and rf/dc is applied across ring electrode.

Feed holes are provided in the endcaps.

In the Paul trap, ring electrode is hyperboloid of one sheet and endcaps are hyperboloid

of two sheets. The CIT is a simplified version of Paul trap, where hyperboloid ring and

endcap electrodes have been replaced by cylinder and flat plates respectively. It retains

the axial symmetry of the Paul trap, but the field distribution becomes nonlinear. Fig.

1.1 shows the cross section of the Paul Trap and the CIT.

6?ds

- r0

- rh

6

?

z0

- r0

- rh

6

?

z0

Figure 1.1: Cross section of (a) Paul Trap (b) Cylindrical Ion Trap (CIT).

1.2.2 Linear Ion Traps

The LIT is a two-dimensional counterpart of the Paul trap. It consists of four rods

having hyperbolic surface. Fig. 1.2 shows the three-dimensional view of LIT electrode

arrangement. x- direction electrode is maintained at potential Φ0 whereas y- direction

electrode is at potential −Φ0.

Fig. 1.3 shows the cross sectional view of the LIT where four electrodes appear as

hyperbola. Ideally these hyperbola should extend to infinity, but in practice it is truncated

to a finite length. The width, height and slit size have been shown by w, h and hs

respectively.

The RIT is a modified form of the LIT, where curved electrodes have been replaced by

flat electrodes. Fig. 1.4 shows the perspective view of the RIT. The x-direction electrodes

have a slit along its length. These slits are used for the inlet of electrons and the outlet


2wx

zy

−φ0

−φ0

φ0φ0

Figure 1.2: Schematic diagram of a linear ion trap.

-

6

w

h

Figure 1.3: Cross section of a linear ion trap.

of destabilized ion. The endcap electrodes (not shown in the figure) are placed along

the z-axis. These electrodes don’t have any aperture. x- and y-direction electrodes are

maintained at potential Φ0 and −Φ0 respectively.

Fig. 1.5 shows the cross sectional view of the RIT. The half width and the half height

has been denoted by w and h respectively. The vertical and the horizontal electrodes are

not allowed to touch each other. They are separated by xs in the x- direction and by ys

in the y- direction. Electrode plate length p and slit width hs has also been shown in the

figure.


Figure 1.4: Schematic Diagram of Rectilinear Ion Trap (RIT).

-p

6

?

p 6?hs -

x

6y

-xs

6

?ys

6

?

h

- w

Figure 1.5: Cross section of the RIT

1.3 Equation of Motion and Stability Plot

In an ideal Paul trap, equation of motion of an ion is given by linear Mathieu’s equation.

The canonical form of the linear Mathieu equation can be written as (McLachlan, 1947)


d2u

dξ2+ (au + 2qucos(2ξ)u = 0. (1.1)

where u can be radial (r) or axial (z) position of the ion and ζ is dimensionless

parameter given by

ζ =Ωt

2(1.2)

au and qu are Mathieu parameters which are functions of experimental variables. For

radial and axial directions these parameters can be expressed as

az = −2ar =4eU

mr20Ω

2(1.3)

and

qz = −2qr =8eV

mr20Ω

2. (1.4)

where m is the mass of the ion, e is the charge on the ion, U is the DC excitation given

to the ring electrode, V is the rf excitation given to the ring electrode, r0 is the radius of

ring electrode at center level and Ω is the angular frequency of rf excitation.

For ideal LIT also Eq. 1.1 is applicable. In this case u can be x or y position of the

ion and Mathieu parameters can be written as

ax = −ay =2eU

mr20Ω

2(1.5)

and

qx = −qy =4eV

mr20Ω

2. (1.6)

For the CIT and the RIT ion motion can be described by nonlinear Mathieu equation

which becomes linear in the the vicinity of the trap center.

The nature of the solution of Mathieu equation governs the ion dynamics inside the

trap. A stable solution means ion amplitude will be bounded while unstable solution

means unbounded growth. So, the ions can be contained with in the trap only if its

motion is stable in all directions. For axially symmetric trap stability zone will be an

intersection of r- and z- stability zones. For LIT and RIT it will be intersection of x- and

y- stability zones.

Stability of the solution of the Mathieu equation is controlled by parameters au and

qu. For linear Mathieu equation, parameter βu , a function of au and qu, is used to define


stability boundary. The parameter βu can be expressed as (March and Hughes, 1989)

β2u =au +

q2u

(βu + 2)2 − au −q2u

(βu + 4)2 − au −q2u

(βu + 6)2 − au − . . .

+q2u

(βu − 2)2 − au −q2u

(βu − 4)2 − au −q2u

(βu − 6)2 − au − . . .

(1.7)

The stability boundary is marked by βu = 0 and βu = 1 line. The subfigures of the Fig.

1.6 show stability plot for the LIT and the Paul trap respectively. The isobeta lines have

also been shown for both directions. In both plots, stability boundary crosses q-axis at

0.908. The stability plot of LIT is symmetric about qx axis. Both βx = 1 and βy = 1

lines cross q axis simultaneously. It means ions will get out of the trap in either x- or y-

direction with equal probability. While in case of Paul trap, it is βz = 1 line which crosses

q axis first. So, the ions preferably eject out in z-direction.

1.4 Experiments

Mass spectrum can be generated by mass selective boundary ejection or resonance ejection

experiment. The timing diagram shown in Fig. 1.7 gives the voltage levels maintained

during different stages of the experiment. In the begining of the experiment rf amplitude

is kept zero for a period known as dead period (≈ 1 ms). After that voltage is ramped

for ionization to take place. It is called ionization period. It is followed by cooling period

(≈ 1 ms) in which ions are allowed to concentrate near the center. The end of cooling

period marks the begining of ramping period where voltage is first brought down and

then it is ramped up. It lasts for around 25-35 ms.

MSBEE and REE deploy different strategy to eject the ion out of the trap. These two

experiments have been detailed in the following subsections.

1.4.1 MSBEE

In MSBEE, endcap electrodes are grounded and only rf or rf/dc potential is applied to

the ring electrode. Generally pure rf is used for ring electrode excitation. So at the start

of the experiment all ions sit along qu axis of the Mathieu stability plot as per their m/e

ratio. The value of qu can be calculated from Eq. 1.4 or 1.6 depending on trap type.

Ions having higher m/e sit closer to the origin. In ramping period as the rf amplitude is


0 0.2 0.4 0.6 0.8 1 1.2 1.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.20.2

0.20.2

0.4

0.4

0.6

0.6

0.8

0.8

0.8

0.8

0.6

0.6

0.4

0.4 q = 0.908cut-off

βz= 1

βr= 0

βz= 0

βr= 1

βy= 0

βx= 0

βy= 1

βx= 1

q = 0.908cut-off

-0.7

xa za

xq z

q

(a) (b)

Figure 1.6: Mathieu stability plots for (a) the LIT and (b) the 3D Paul trap.

ramped up the qu value of the ions increase. When the qu value reaches the critical value

of 0.908, ions becomes unstable and leave the trap. The ramping process is equivalent of

pushing the ions along the qu-axis out of the stability zone. Lower and upper mass cutoff

is can be controlled by the voltage limits V1 and V2 shown in the Fig. 3.4.

1.4.2 REE

The motion of the ion within the trap has many frequency components. The most promi-

nent frequency among them is secular frequency. For an ideal Paul trap or LIT it is given

by

ωu =1

2βuΩ (1.8)

As shown in the Fig. 1.6, the boundary of the Mathieu stability plot is defined by βu = 1

line. So it is evident from Eq. 1.8 that maximum value of secular frequency will be the

half of the ring electrode excitation frequency.

The REE uses the principle of resonance to make the ions unstable. In contrast to the


V1

V2

V3

V4

t1 t2 t3 t4 time

volt

age

Figure 1.7: Timing diagram showing different stages of the experiment. Dead time (0-t1), Ionization time(t1-t2),Cooling time (t2-t3), Ramp time(t3-t4), Ionization voltage (V3),Cooling voltage (V2), Ramp start voltage (V1), Ramp end voltage (V4).

MSBEE, an auxiliary excitation is applied to the endcap electrodes. The rf amplitude is

ramped to the extent where the secular frequency of the ion matches auxiliary excitation

frequency. At this point ion comes into resonance and gets ejected out of the trap.

Auxiliary excitation frequency cannot be greater than the maximum possible value of

secular frequency, because in that case ion will get ejected out of the trap by boundary

ejection before coming into resonance. So in REE ions will become unstable much before

they cross boundary of the Mathieu stability plot. The working principle of the REE can

be exploited to increase the mass range of the trap.


Figure 1.8: Process of mass selective boundary ejection for Paul trap. The ring electrodehas been excited by rf only.

1.5 Scope of the Project

The project concentrates on developing a simulator which will generate the mass spectrum

from the given mass and charge distribution of the characteristic mixture formed by

ionization process. It will cover both axially symmetric traps and LITs. Along with

spectrum, the simulator can show the trajectory and micromotion of the ion within the

trap. Factors like space charge effect and damping have been considered during the process

of trajectory and spectrum generation. The simulator will also help in characterizing the

trap performance by generating stability plots and escape velocity plots. Functions have

been provided for calculation of trap capacitance, multipole coefficients and electrode

charges.

The simulator does not consider the chemistry involved in ionization process. Also

any chemical reaction taking place within the trap and hence changing the given mass

and charge distribution have not been accounted in this study.

Chapter 2

System overview

2.1 Introduction

The function of the ion trap simulator can be grouped into two categories namely ion dy-

namics simulation and trap characterization. Ion dynamics simulation helps in getting ion

trajectories, micromotion and ultimately the mass spectrum. While trap characterization

process is based on the computation of the multipole coefficient, capacitance, stability

plot, Poincare section and escape velocity plots. These two process are not independent

of each other. They are linked at various levels. Apart from these two major groups,

simulator gives many utility functions for conversion of simulation parameters from one

set to other and vice versa.

Both ion dynamics simulation and trap characterization pose a multivariable problem

with complex interdependencies. Also due to wide range of shapes and sizes of traps,

the set of possible inputs and desired output variables is also wide. The number of

variables and parameters involved is high enough to make it unsuitable for a monolithic

treatment. It gives rise to the need of dividing the bigger problem into smaller and

possibly independent subproblems. The solution to these subproblems has to be combined

appropriately to get the final result.

The implementation of divide and conquer approach for any problem has technical as

well as strategical issues associated with it. In case of the trap simulator choice of pro-

gramming language and the platform is the only technical issue. The process of dividing

the simulator into modules and submodules is more of a strategic issue. Both of these

require proper attention as they will have a great influence on efficiency, scalability and

usability of the whole simulation package.

This chapter gives an overview of the simulator architecture and layout. It starts

with the block diagram of the simulator followed by the discussion of main modules.

The discussion is concentrated on their functionality, algorithms, flowcharts and data

structures used.

11

Chapter 2. System overview 12

2.2 Simulator block diagram

The spectrum generation experiment is a multistage process. So naturally its simulation

also becomes multistage in nature. The simulator has been divided into modules to

address different stages. These modules forms a pipeline structure where output of the

previous module work as input to the next module. The main modules of the simulator

are

1. Simulation initializer

2. Main simulator

3. Output analyzer

4. Simulator tester.

Simulation initializer sets up the environment for the simulation to run. It reads simula-

tion inputs, creates geometry, generates ions and initializes the data structures to be used

in later stages. The simulation inputs include trap geometry, trap excitation, ion popula-

tion and experiment conditions like temprature, damping and space charge effect. Main

simulator solves the given trap geometry to get electric field and potential distribution.

It also handles ion trajectory integration process. There are many possible outputs of the

simulation. Along with mass spectrum it can generate multipole coeeficients, capacitance,

stability plots, escape velocity plots, single ion trajectory and micromotion details also.

The ouput analyzer module takes care of all possible outputs. The role of module simu-

lation tester is to validate the functioning of different submodules and provide feedback

for correction as well as improvement.

The modules have been further divided into submodules to take care of specific jobs.

Fig. 2.1 shows the block digrams of the modules and their submodules. Details of these

modules and submodules have been discussed in following sections.

2.3 Simulation initializer

Simulation starts running by calling the module simulation initializer. The module con-

tains many submodules responsible for different parts of simulation setup. This module

handles three main jobs

1. Creating geometry

2. Reading inputs


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Figure 2.1: Block diagram showing modules and their submodules.

3. Generating position and velocity distribution of ions

Geometry is created by successively adding electrodes. Both geometry and electrode

has been represented by a structure. The electrode structure contains entries required

to store the shape, size and applied potential distribution. The shape of the electrode is

described in form of a parametric equation. The form of electrode structure is given in

Table 2.1.

The simulation study uses the Boundary Electrode Method (BEM) for charge calcu-

lation. The BEM divides the electrodes in elementary charge strips and then sets up a

system of equation to be solved for knowing the charge distribution on the electrodes.

These strips are numbered for indexing purpose. The strip index for an electrode starts

from elemBeg and ends at elemEnd. The electrode structure has entry for number of

divisions (nDivs) also.

Geometry structure contains overall dimesions of the trap, ring and endcap potentials

and the entries required to address any elementary strip of an electrode.


Element DescriptiondescType Expression typerhoExpr Parametric equation for radial directionzExpr Parametric equation for axial directiontmin Minimum value of parameter ttmax Maximum value of parameter trhob Starting value of rhozb Starting value of zrhoe Ending value of rhoze Starting value of znDivs Number of divisionsu Potential applied to the electrodeelemBeg Index of the first charge strip of the electrodeelemEnd Index of the end charge strip of the electrode

Table 2.1: Description of the elements of electrode data structure.

Initial mass and charge distribution of the ion is read from an input file, where the

first entry gives the total number of ions to be simulated. Ions have been stored in an

array of structure the element of which has been shown in Table. 2.2.

Element Descriptionpos Position of the ionvel Velocity of the ionmamu Mass of the ion in amucharge Charge on the ion

Table 2.2: Description of the elements of ion data structure.

Space required to hold all ions is allocated dynamically. The mass and charge of each

ion is read from the file and is stored in the structure. Fig. 2.2(a) shows the flow chart

of the process.

The position and velocity of the ions is sampled from a probability distribution. For

velocity, Maxwell’s distribution is the default choice while for position uniform distribution

has been used. The mean and variance of the distribution is computed from the input

experiment parameters. Fig. 2.2(b) and 2.2(c) show the flow chart for sampling and

storing the ions velocity and position respectively.


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Figure 2.2: Flow chart for (a) Reading input ion distribution (b) Generating velocitydistribution (c) Generating position distribution.

2.4 Main simulator

This module is the core of the simulator. It is called after the simulation initializer has

created the geometry and initialized all data structures. The main simulator performs

the following three jobs.

1. Charge calculation

2. Field and potential calculation

3. Trajectory integration

Charge calculation has been done by the Boundary Element Method (BEM) discussed

in detail in next chapter. The computed charge value is stored in a charge array which

is a member of geometry structure. The charge on any elementary charge strip can be

indexing the charge array. The index of the strip is stored in the structure which represents

the electrode containing that strip.


Electric field and potential calculation uses principle of superposition. This calculation

requires coordinates of the elementary strips as well as the charge on them. All these values

are stored in geometry structure.

Trajectory integration is the process of tracing the path of the ion starting from its

initial position. The data required to update the position and velocity of the ion after a

time step is stored in an structure shown in Table reftrajdata.

Element Descriptiongeom Structure containing geometry detailsm amu Mass of the ion in amuq e Charge of the ion in amuVFn Function returning voltage valuevData Array containing waveform dataspace charge Array containing space charge datacoeffDamp Coefficient of damping

Table 2.3: Description of the elements of trajectory data structure.

The process monitors the ion for a prespecified time. It terminates prematurely if the

ion exits the trap. The postion of the ion in intermediate steps is stored against time to

give a graphical display of the trajectory. Fig. 2.3 shows the flow chart for this process.

2.5 Output analyzer

This module takes the service of previous two modules and generates the output of the

simulator. Apart from data generation it also does some processing. The function of this

module can be put under two categories namely

1. Spectrum generation

2. Trap characterization

The flow chart for spectrum generation has been shown in Fig. 2.4. As depicted in

the flow chart, after reading the inputs trajectory evolution of the ions are traced using

fourth order Runge-Kutta method. The trajectory evolution period is divided into two

parts namely cooling period and ramping period. In cooling period the evolution takes

place at constant voltage (init vol) while during ramping period voltage increases with

time from init vol to max vol. The voltage at which ions eject out of the trap is stored


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Figure 2.3: Flow chart for trajectory integration.

in a file which is used to generate the spectrum. The data structures used in this process

have been discussed earlier.

The trap characterization processes do not use any specific algorithms or data struc-

ture. The most of the data required for these outputs are already computed and stored

in different structures in previous steps.

2.6 Simulation tester

This module has been used to verify the output generated by the simulation. It involves

running the simulation under conditions for which analytical results are known. It forms

a platform for checking the correctness and accuracy of the numerical computation. The

property of axial symmetry also provides some check points. This module does not use

any specific algorithm and data structure.


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Figure 2.4: Flow chart for spectrum generation by MSBEE.

Chapter 3

Module details

3.1 Introduction

This chapter gives an detailed description of different modules of the simulator as dis-

cussed in Chapter Two. For each module, computational method as well as implementa-

tion approach has been discussed. To avoid unnecessary distractions some mathematical

derivation has been put in appendix. These modules are written in C, so they make use

of the standard C library. As the simulator also needs data structure augmentation for

some of its modules, glib library has also been used. Graphical interface for Poincare

section has been implemented using openGL.

3.2 Simulation Initializer

In this section different submodules of the module simulation initializer has been discussed.

The main function of the module is to create the environment for the main simulation to

run. So it handles jobs like geometry creation, reading input parametrs and experiment

conditions and generation of ions for a given sample.

To simulate the dynamics of ions inside the trap their initial velocity and position

must be known. As simulation will deal with a collection of ions having different masses

and charges, assuming same initial position and velocity for all ions will be impractical.

In this simulation these parameters have been specified using appropriate distribution

functions. These default distribution may not be the best in all cases so simulation gives

an option to use user defined distributions.

The details of the various submodule is given in following subsections.

3.2.1 Reading geometry file

The simulator can be used to create and solve any trap geometry for which electrode can

be expressed in form of a parametric equation. But there are some standard geometry

which are more common than other geometries. For such cases a geometry file has been

created where geometry parameters of the trap is stored. The format of the geometry file

is shown below.

19

Chapter 3. Module details 20

Geometry file format

Trapname Radial width(ρ0) Axial width(z0) Hole size Spacing Misc

These geometry can be loaded into the simulator by the function LoadGeom(trapname).

Currently the geometry file support the Paul trap and the CIT.

3.2.2 Creating Geometry

The simulator handles the geometry creation process in a hierarchical way. It divides the

geometry into two groups namely ideal and actual. The ideal group includes hyperbolic

and cylinderical geometry. The ideal hyperbolic geometry consists of infinitely long hy-

perboloid electrodes with no feed holes. The cylinderical geometry refers to a cylinder

with endcaps having no hole. The spacing between the ring electrode and the endcap elec-

trodes is also set to zero. The actual group include any practical trap gepmetries. These

geometries consists of electodes which are made up of segments of curves or straight lines.

The curves are specified by a set of parametric equations. The simulator does not put

any restriction on number of electrodes the geometry can possess. Also, the number of

curves or straight lines per electrode is not restricted. These geometries are created by

successively adding the elctrodes. Fig.3.1 shows a tree diagram of the geometry creation

process.

The ideal group geometries are represented by standard parameters like radial and ax-

ial width. For these geometries numerical field computation is not required and therefore,

they are handled by analytical approach. It makes the simulation run comparitively faster.

These geometry are useful in many theoretical studies. They also provide checkpoints for

correctness and accuracy of the simulator.

The actual group geometries come from practical traps. The simulator provides an

interface for creating theses geometries by adding the electrodes. Fig.3.2(a) shows a

snapshot of the interface. The interface gives the facility of specifying the parametric

equations for the electrode and the number of divisions. It also allows to deletion of the

electrode. These entries can be edited by clicking on the electrode and then selecting the

edit option to get an interface shown in Fig.3.2(b).

3.2.3 Reading Input File

The simulator reads initial mass and charge distribution from an input file. This file has

the following format


Figure 3.1: Hirarchical structure for geometry creation.

Total number of ionsmass1 Number of ions of mass1

mass2 Number of ions of mass2...

...massn Number of ions of massn

Function read input() scans the file line by line and dynamically allocates required

memory for the data structures. The memory is freed at the end of the simulation.

3.2.4 Velocity Distribution

By default the Maxwell-Boltzmann distribution based on kinetic theory of gas has been

used to specify initial velocities of ions. Maxwell-Boltzmann distribution asserts that

three velocity components are mutually independent and each of them is normally dis-

tributed with mean zero. So the probability density function for the Maxwell-Boltzmann

distribution is given by three dimensional Gaussian function as follows.

fv(vx, vy, vz) =( m

2πkT

)3

2

exp

(

−m(v2

x + v2y + v2

z)

2kT

)

(3.1)

where vx, vy and vz are velocity components in x-, y- and z-direction respectively, m is the

mass of the gas molecules, k is Boltzmann constant and T is absolute temperature. This


(a)

(b)

Figure 3.2: Graphical interface for (a) creating geometry (b) editing the part of the geometry.


distribution can be realized by sampling velocity components from independent normal

distributions.

The standard C library provides the function drand48 to generate uniform random

variate in the interval (0,1). This uniform random variate (U(0, 1)) is converted to mean

zero and unit standard deviation normal random variate (Z(0, 1)) by Box-Muller transfor-

mation. Given two independent uniform random variate U1(0, 1) and U2(0, 1) Box-Muller

transformation returns two normal random variate Z1(0, 1) and Z2(0, 1). The transforma-

tion equations are give by

Z1 =√

−2 ln U1 cos 2πU2 (3.2)

Z2 =√

−2 ln U1 sin 2πU2 (3.3)

A mean zero and unit standard deviation normal variate (Z(0, 1)) is transformed to

mean µ and standard deviation σ normal variate (N(µ, σ2)) by a linear transformation

given by

N = σZ + µ (3.4)

For Maxwell-Boltzmann distribution each velocity component is generated by putting

µ = 0 and σ = kTm

.

3.2.5 Position Distribution

Position distribution has been assumed to be zero mean uniform random variate dis-

tributed over prespecified interval.

3.2.6 Damping

Ion traps uses buffer gas, which acts as a damping agent for the ion motion. This effect

can be simulated in two ways.

• Viscous damping

• Collisional damping

In this study only viscous damping has been implemented.


Viscous Damping

Effect of viscous damping has been captured by using a damping coefficient calculated as

c =mn

m + mn

p

kTb

q

2ǫ0

√

αm + mn

mmn

(3.5)

where mn is the mass of bath gas(generally helium), α = 0.22 × 10−40Fm2 is the polar-

izability of bath gas, ǫ0 = 8.854 × 10−12Fm−1 is the permittivity of the free space, Tb is

absolute temperature, p is bath gas pressure, k is Boltzmann’s constant, m is the mass of

the ion and q is the charge of the ion. Damping force on the ion is calculated as

fd =c

mvu (3.6)

where Ω is frequency of applied rf potential and vu is velocity component in u-direction.

3.2.7 Space Charge Effect

Presence of large number of ions inside the trap forms a charged cloud which modifies

the original existing field distribution set up by charged electrodes. So, in calculation of

electrostatic force experienced by an ion inside the trap force due to charged cloud should

also be considered.

Force on ith ion due to charged cloud is given by

~f isc =

Nq∑

j=1j 6=i

~fi,j (3.7)

where ~fi,j is the electrostatic force on ith ion due to jth ion and Nq is the total number

of ion inside the trap. By Coulomb’s law expression for fi,j can be written as

~fi,j =1

4πǫ0

qiqj

|~ri − ~rj|3(~ri − ~rj) (3.8)

where qi is the charge on ith the ion and ri is the position vector of ith ion. From Eq.3.8 it

is obvious that magnitude of fi,j is same as that of fj,i but their directions are opposite to

each other. Components of fj,i can be easily obtained by sign reversal of the components

of fi,j. So once fi,j is calculated fj,i need not to be calculated separately.

3.3 Main Simulator

This module forms the core of the simulator. It solves the geometry to get electrode

charge distribution which is later used for potential and field calculation. It also handle

trajectory integration with or without ramping.


The different submodules of this module has been discussed in the following subsec-

tions.

3.3.1 Charge Calculator

The main aim of the charge calculator is to compute electrode surface charge density for

an electrode of given shape and size along with known applied potential. To achieve the

goal, it uses the Boundary Element Method (BEM), details of which is given below.

BEM

The BEM is a technique of solving the boundary value problems. It devides the boundary

into number of elementry strips and sets up equations for describing the behavior of the

system in terms of unknowns. Then those are solved to get the final solution.

In this case, the BEM divides the elctrode surface into number of elementry charge

strips. It asserts that potential on any strip is the resultant of the potential due to itself

and the potentials due to all other charge strips in the system. This potential balance

gives gives a linear system of equation with charge on the strips as unknown variable. The

system of equation is solved to get the charge distribution on electrode surface. But this

process requires an analytical expression for calculation of potential due to a charge strip

at any given point. This expression acts as a Green’s function for the BEM calculation.

The use of the BEM requires Green’s function evaluation. For axially symmetric

geometry expression for potential due to a ring a any point serves as the Green’s function.

Potential due to a uniformly charged ring at a point not on the ring

Applying cosine rule for ARB

cos α =r2i + r2

j − s2

2rirj

⇒ s2 = r2j + r2

i − 2rirj cos α (3.9)

Potential at point P on jth ring due to infinitesimal charge element around B on ith

ring is given by

dui,j =1

4πǫ0

dqi√z2 + s2

(3.10)

where dq is the charge on the element. If q is the total charge on the ring and the charge

element subtends an angle dα at the center of the ring then assuming uniform distribution

dqi =dα

2πqi (3.11)


O SR

A

B

P

rj

z = zj − zi

rj

s

riα

ith ring

jth ring

(zi, 0, 0) (zj, 0, 0)

Figure 3.3: Potential due to ith ring at a point on jth ring

Substituting Eq.3.11 in Eq.3.10 and integrating over 0 to 2π

ui,j =1

4πǫ0

qi

2π

∫ 2π

0

dα√z2 + s2

=1

4πǫ0

qi

2π

∫ 2π

0

dα√

z2 + r2i + r2

j − 2rirj cos α

=1

4πǫ0

qi

2π

∫ 2π

0

dα√

z2 + (ri + rj)2 − 2rirj(1 + cos α)

=1

4πǫ0

qi

2π

∫ 2π

0

dα√

z2 + (ri + rj)2 − 4rirj cos2 α2

=1

4πǫ0

qi

2π

1√

z2 + (ri + rj)2

∫ 2π

0

dα√

1 − 4rirj cos2 α2

z2+(ri+rj)2

=1

4πǫ0

qi

2π

1√

z2 + (ri + rj)2

∫ 2π

0

dα√

1 − k2 cos2 α2

(3.12)

Where

k2 =4rirj

z2 + (ri + rj)2(3.13)


By substituting α2

= π2− β in integral in Eq.3.12, we get

ui,j =qi

4πǫ0

2

π√

z2 + (ri + rj)2

∫ π2

0

dβ√

1 − k2 sin2 β(3.14)

Putting z = zj − zi, we get

ui,j =qi

4πǫ0

2

π√

(zi − zj)2 + (ri + rj)2K(k) = g(i, j)qi (3.15)

Where g(i, j) is the Green’s function value and

k2 =4rirj

(zi − zj)2 + (ri + rj)2(3.16)

Potential due to a uniformly charged ring at a point on the ring itself

Potential due to a uniformly charged ring at a point on the ring itself can be found by

ui,i =qi

4πǫ0

1

πri

(

1 + ln

(

16ri

∆wi

))

= g(i, i) (3.17)

If electrodes are divided in n charge strips, potential on ith ring will be equal to the

sum of potential due to all elementary charge rings. So the potential on ith ring will be

given byn

∑

j=1

ui,j =n

∑

j=1

gi,jqj = vi (3.18)

Eq.3.18 is valid for i = 1 . . . n. This system of equations can be written in matrix form as

Gn×nQn×1 = Vn×1 (3.19)

where Gij = gi,j,Qi = qi and Vi = vi.

Equation Solver

The purpose of equation solver is to solve Eq.3.19 numerically. Following properties of

matrix G associated with the system of equation helps in numerical solution.

• It is symmetric square matrix where, number of rows(and hence columns) is equal

to total number of elementary charge strips.


• It is a positive definite matrix. This property can be easily established by pre-

multiplying Eq.3.19 by qT to give

qT Gq = qT V (3.20)

where right hand side of the equation is nothing but the electrostatic energy stored

in the structure. As stored energy cannot be negative, qT Gq must be positive,

implying that G is a positive definite matrix.

• In numerical computations, improper choice of elementary charge strip width can

lead to unacceptable charge distribution making the matrix G non-positive definite

or marginally positive definite( near to machine precision). In this case Cholesky

factorization will fail.

Since matrix G is positive definite, it can be split into two triangular systems using

Cholesky factorization which can be solved by forward or backward substitution.

Cholesky factorization

Cholesky factorization theorem states that if G is real, symmetric, positive definite matrix

then, there exist a unique real, lower triangular matrix L with positive diagonal entries

such that G = LLT .

Putting G = LLT , Eq.3.19 converts to

LLT q = V (3.21)

where q can be determined by solving the two triangular systems Ly = V and LT q = y.

Cholesky factorization can be implemented either in Gaxpy version or outer product

version. In this package Gaxpy version has been implemented. Algorithm for Gaxpy

Cholesky for Gn×n can be written as

for j = 1 : n

if j > 1

G(j : n, j) = G(j : n, j) − G(j : n, 1 : j − 1)G(j, 1 : j − 1)T

end

G(j : n, j) = G(j : n, j)/√

(G(j, j)

end

Complexity of the algorithm is O(n3/3).


Triangular solve

Triangular systems can be either upper or lower triangular. Upper triangular systems are

solved by backward substitution algorithm. Given a upper triangular system Un×nxn×1 =

bn×1 following algorithm overwrites x with the solution vector.

b(n) = b(n)/U(n, n)

for i = n − 1 : −1 : 1

b(i) = (b(i) − U(i, i + 1 : n)b(i + 1 : n))/U(i, i)

end

Lower triangular systems are solved by forward substitution algorithm. Given a lower

triangular system Ln×nxn×1 = bn×1 following algorithm overwrites x with the solution

vector.

b(1) = b(1)/L(1, 1)

for i = 2 : n

b(i) = (b(i) − L(i, 1 : i − 1)b(1 : i − 1)/L(i, i)

end

Complexity of both forward and backward substitution algorithm is O(n2).

Function posDefSolve() implements Cholesky factorization and also gives the solution

of the two triangular systems. The overall complexity of the function is O(n3). It should

be noted that the element of the matrix has been accessed by a single pointer so the

indexing used in the function posDefSolve() is different from that mentioned in above

algorithms.

3.3.2 Field Calculator

Field calculator returns electric field vector value at any point inside the trap. Electric

field vector at any point is the resultant of electric field due to elementary charge strips. So

this subroutine accumulates contribution from each strip component wise and computes

net electric field vector.

3.3.3 Trajectory Calculation

Trajectory calculation is an initial value problem where the position and the velocity of

the ion are known at t = 0. The equation of motion is integrated numerically to determine


postion and velocity at subsequent time steps. Equation of motion of ions within trap

volume is given byd2u

dt2=

1

m(qEu + fsc + fd) (3.22)

where u can be x,y or z coordinate of the ion, Eu is the component of electric field along

u, fsc is the force due to space charge effect, fd is the force due to damping, q is charge

and m is mass of the ion. The second order differential equation is converted to a system

of two first order equation.

du

dt= vu (3.23)

dvu

dt=

1

m(qEu + fsc + fd) (3.24)

The system of first order equations has been solved numerically by Runge-Kutta fourth

order method.

Runge Kutta Fourth Order Method

Given a system of differential equations of the form dy

dt= f(y, t) where along with initial

condition y(t0) = y0, Runge Kutta fourth order method calculates y1 = y(t0 +h) by using

following equations.

k1 = hf(t0, y0) (3.25)

k2 = hf(t0 +h

2, y0 +

k1

2) (3.26)

k3 = hf(t0 +h

2, y0 +

k2

2) (3.27)

k4 = hf(t0 + h, y0 + k3) (3.28)

y1 = y0 +1

6(k1 + 2k2 + 2k3 + k4) (3.29)

This algorithm has been implemented in function rk4step(). It should be noted that if y

is a vector than f will also be a vector function and all steps of Runge Kutta algorithm

will involve a vector operation. In trajectory calculation y =

(

uvu

)

is a two dimensional

vector.

Ramp

In mass selective boundary ejection experiment rf amplitude is kept constant during cool-

ing period while in ramping period rf amplitude is increased at a prespecified rate.


V2

V1

t1 t2time

volt

age

Figure 3.4: Ramping of voltage in mass selective boundary ejection experiment

In Fig.3.4 V1 is the initial rf amplitude, V2 is the maximum rf amplitude,t1 is the cooling

period and t2 − t1 is the ramping period.As Eu directly proportional to rf amplitude,

righthand side of Eq.3.22 has to be calculated differently for cooling and ramping period.

Function vramp() implements this calculation assuming a linear ramping.

3.4 Trap Characteristics

3.4.1 Escape Velocity

Escape velocity is the minimum velocity required for an ion sitting at the center of the

trap to get out of the trap. It is a characteristic of field distribution inside the trap. It is

particularly important in resonance ejection experiment for deciding the point of ejection.

Escape velocity is a function of trap geometry parameters,rf amplitude, rf phase and

angle of projection. Numerically escape velocity for a given rf amplitude is approximated

as the minimum of the velocities required to escape corresponding to different rf phase

and angle of projection. Number of rf phases and angle of projections to be considered in

escape velocity calculation is user defined.

Function bracket escape() return limits of the interval containing velocity required to

escape corresponding to a given rf amplitude, rf phase and angle of projection. It starts

with an initial lower limit(0) and upper limit(100) and keeps shifting both limits by a fixed

amount(100 by default) till upper velocity limit causes ion to fly out of the trap while lower

limit velocity keeps the ion in trap. Function escape() calculates the velocity required to


escape by performing prespecified number of bisections starting with the limits returned

by bracket escape(). The function escapeChart() calls function escape() for different rf

amplitude, rf phase and angle of projections (as specified by user) and displays returned

velocities along with escape directions.

3.4.2 Multipole Coefficient

The potential at a point u(ρ, θ, φ) in spherical coordinates in an axially symmetric trap

can be expressed as

u(ρ, θ, φ) = Φ∞

∑

n=0

An

(

ρ

LN

)n

Pn(cos θ) (3.30)

where Φ is applied potential, An nth order multipole coefficient, Pn is Legendre polynomial

of nth degree and LN is normalizing length. Potential at a point on z-axis can be found

by putting θ = 0 and ρ = z in Eq.3.30 to get

u(z) = Φ∞

∑

n=0

An

(

z

LN

)n

(3.31)

On the other hand, potential at any point on z-axis can also be found by accumulating

the potential contribution of all elementary charge strips. Potential due to a ith ring

(elementary charge strip) at a point on its axis can be expressed as

ui(z) =qi

4πǫ0ρi

∞∑

n=0

(

z

ρi

)n

Pn(cos θi) (3.32)

As total number of elementary charge strips is N potentian at point on z-axis of the

trap will be given by

u(z) =N

∑

i=1

qi

4πǫ0ρi

∞∑

n=0

(

z

ρi

)n

Pn(cos θi) (3.33)

By comparing Eq.3.31 and Eq.3.33, An can found out to be

An =1

Φ

N∑

i=1

qi

4πǫ0ρi

(

LN

ρi

)n

Pn(cos θi) (3.34)


z

rizi

(0, 0, 0)

ρi

θi

Figure 3.5: Potential due to the ring at a point on its axis

3.4.3 Poincare Section

The equation of undamped motion of ion along z-axis inside the trap is given by following

second order differential equation

d2z

dt2+ 2qz cos(2t)

∞∑

n=1

A2n

A2

nz2n−1 = 0 (3.35)

Poincare section for ion motion is the graphical representation of periodically sampled

value of z and dzdt

evolving as per Eq.3.35.

For numerical computation infinite summation is generally truncated at some pre-

specified (normally 6 to 8) number of terms. To get Poincare section Eq.3.35 has been

solved numerically by Runge-Kutta fourth order method and value of z and dzdt

has been

sampled periodically. As the equation of motion involves cos(2t) term sampling has been

done at intervals of π. Curves in Poincare section are sensitive to initial condition. But

it is not computationally feasible to scan all possible initial condition in two-dimensional

space. Hence the plot has been made interactive by adding openGL based graphical

interface where every mouse click on the plot space gives an initial condition. The solution

corresponding to that initial condition is traced on plot window.

Chapter 4

Results and Verification

4.1 Introduction

Validation is the last but one of the most important part of any simulation study. It

checks for the correctness and accuracy as well as provides the feedback for the further

improvement. The simulation of the ion trap is no exception. At different stages of the

simulation many numerical and analytical computations have been performed. Also in this

study some assumptions have been used to simplify the implementation. All these requires

a through verification before the output of the simulator can be used with reliability.

The core of the simulator uses the BEM for electrode charge calculation which has been

later used for computing the field and the potential distribution in the trap. The BEM

divides the electrodes in numbers of elementary charge strips. The number of divisions

made in an electrode has a direct influence on the accuracy of the result. As the number

of divisions increases the accuracy increases, but simulation takes more time to run. So,

the choice of number of divisions per electrode is a trade off between accuracy and run

time. Keeping this point in view, the BEM implementation requires proper verification.

This chapter shows the results obtained by this simulator along with the its validation,

wherever possible. It starts with a description of trap geometries used for the verification.

After that, verification of position and velocity distribution generation is given. There-

after, results and verification of multipole coefficient, electrode charge, field and potential

distribution have been discussed. The next section shows the results of trajectory inte-

gration and micromotion studies. Some of sample stable and unstable trajectories along

with micromotion plots have also been shown. The chapter ends by showing the mass

spectrum of benzene and air obtained by the simulator.

4.2 Geometries used for verification

Two axis symmetric traps namely, the Paul trap and the cylindrical ion trap (CIT) have

used to validate the simulator output. Theses geometries have been chosen because results

for them are widely available in literature. Also property of axial symmetry provides some

check points for the results. Fig. 4.1 shows the cross section of these two traps. The radial

width has been marked by r0 while the axial width has been shown by z0 respectively.

34

Chapter 4. Results and Verification 35

The endcap hole size and the spacing between ring and endcap electrode is represented

by ds and rh respectively. Table 4.2 shows the geometry parameters values used in this

study for these traps.

For comparison purpose many times simulation has also been run for traps with no

hole. In those cases rh has been set to zero, keeping other geometry parameters same.

6?ds

- r0

- rh

6

?

z0

- r0

- rh

6

?

z0

Figure 4.1: Cross section of (a) Paul Trap (b) Cylindrical Ion Trap (CIT).

r0 z0 ds rh

Paul 7.1 5.0204 0.0 0.3CIT 5.0 6.1743 1.6 0.5

Table 4.1: Geometry parameters of the traps studied. All dimensions are in mm.

4.3 Velocity and position distribution

The initial velocity of the ions have been sampled from the Maxwell’s distribution. The

Maxwell’s distribution has been implemented by sampling the velocity components inde-

pendently from a normal distribution with mean zero and variance given by

σ2 =kT

m. (4.1)

where k is the Boltzman’s constant, T is the absolute temperature of the gas and m is

the mass of the gas molecule.

So the probability density function for each velocity component becomes

f(v) =1√2πσ

e−v2

2σ2 . (4.2)


0

0.0005

0.001

0.0015

0.002

0.0025

-600 -400 -200 0 200 400 600

Figure 4.2: Analytical and numerical results for generation of normal distribution.

Eq. 4.2 gives the analytical formula for the probability density function which should

be followed by velocity components of the ions. In the simulation, normal distribution has

been generated numerically by using Box Muller transformation. The uniform random

variables used by the this transformation has been generated by the inbuilt C library

function drand48(). Fig. 4.2 shows the comparison of analytically and numerically gen-

erated probability density functions. This graph has been generated for m = 78amu and

T = 298.15K (which corresponds to the room temperature of 25C). In the figure the

continuous line shows analytical values and dots show numerical values. From the figure

it is clear that numerical values follow the analytical expressions closely.

The initial position of the ions have been considered to be uniformly distributed around

the center of the trap. The position have been generated by directly using C library

function drand48(). As the function is a part of standard library, a separate verification

of the uniformity of the ion position around the trap center has not been discussed.

4.4 Electric potential distribution

An ideal Paul trap offers a quadratic potential distribution inside the trap. When a

potential of Φ0 is applied to the ring electrode and the endcaps are kept grounded, the


0

0.1

0.2

0.3

0.4

0.5

0.6

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Pot

entia

l

Axial distance

(a)

0.5

0.6

0.7

0.8

0.9

1

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Pot

entia

l

Radial distance

(b)

Figure 4.3: Analytical and numerical results for Paul trap with no hole. (a) axial potentialdistribution (b) radial potential distribution. The continuous line shows analytical output anddots represent numerical output.

potential can be expressed as

Φ(r, z) =Φ0

2r20

(r2 − 2z2) +Φ0

2(4.3)

Where r0 and z0 are radial and axial width of the trap. r and z are radial and axial

coordinate respectively.

The axial potential distribution can be found by putting r = 0 in Eq. 4.3. It will be

of form

Φ(0, z) = −Φ0

r20

z2 +Φ0

2(4.4)

Similarly the radial potential distribution can be found by putting z = 0 in Eq. 4.3.

The Radial potential distribution can be expressed as

Φ(0, z) =Φ0

2r20

r2 +Φ0

2(4.5)

Eq. 4.4 and 4.5 gives the theoretical potential distribution for an ideal Paul trap.

This has been used to check the correctness of potential computation performed in this

study. Fig. 4.3 shows the plot of axial and radial potential distribution. The continuous

line shows the potential distribution for an ideal Paul trap obtained analytically and dots

represent the numerical result for the Paul trap considered for the investigation. This

comparison is based on no hole geometry, because an ideal Paul trap does not account

for hole effect. The graph shows a good match between numerical and analytical values.

Fig. 4.4 shows the axial and radial potential distribution for the Paul trap with hole.

In potential distribution along the axis, it can be seen that the potential curve does not


0

0.1

0.2

0.3

0.4

0.5

-0.004 -0.002 0 0.002 0.004

Pot

entia

l

Axial distance

(a)

0.5

0.6

0.7

0.8

0.9

1

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Pot

entia

l

Radial distance

(b)

Figure 4.4: Numerical results for Paul trap with hole. (a) axial potential distribution (b) radialpotential distribution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

Pot

entia

l

Axial distance

(a)

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Pot

entia

l

Radial distance

(b)

Figure 4.5: Numerical results for CIT with hole. (a) axial potential distribution (b) radialpotential distribution.

touch zero line at endcap level. This is because of holes in the endcaps. The similar

observation can be made for the CIT also. Fig. 4.5 depicts the potential distribution for

the CIT.

4.5 Electric field distribution

For an ideal Paul trap, field distribution can be found by taking the negative gradient of

the potential expressed in Eq. 4.3. So, the field distribution can be computed to be

~E(r, z) = −Φ0

r20

~r +Φ0

r20

2~z (4.6)

The axial field distribution can be found by taking the ~z component of the field vector.


-200

-150

-100

-50

0

50

100

150

200

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Fie

ld

Axial distance

(a)

-150

-100

-50

0

50

100

150

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

Fie

ld

Radial distance

(b)

Figure 4.6: Numerical and analytical results for Paul trap with no hole. (a) axial field distri-bution (b) radial field distribution

It will be of form~E(0, z) =

2Φ0

r20

~z (4.7)

Similarly the radial field distribution can be found by taking the ~r component of the

field vector. It can be expressed as

~E(r, 0) = −Φ0

r20

~r (4.8)

Eq. 4.7 and 4.8 gives the theoretical field distribution for an ideal Paul trap. This

has been used to check the correctness of field computation performed in this study.

Fig. 4.6 shows the plot of axial and radial field distribution. The continuous line shows

the field distribution for an ideal Paul trap obtained analytically and dots represent the

numerical result for the Paul trap considered for the investigation. This comparison is

based on no hole geometry, because an ideal Paul trap does not account for hole effect.

The graph shows a good match between numerical and analytical values. The numerical

field calculation becomes unstable in region very close to electrode surface. All the graphs

showing the field distribution considers only 95% of the inner trap width. For numerical

computation of field very close to electrode surface indirect method like extrapolation

should be used.

Fig. 4.7 shows the axial and radial field distribution for the Paul trap with hole. The

nonlinear effect of the hole is visible in axial field distribution. But near the center of the

trap field is still linear. The radial field distribution is linear in this case also. Fig. 4.8

depicts the field distribution for the CIT. The graph reveals the nonlinear behaviour of

the CIT. Both axial and radial field distributions are nonlinear. The holes in the endcaps

also adds to the nonlinearity.


-200

-150

-100

-50

0

50

100

150

200

-0.004 -0.002 0 0.002 0.004

Fie

ld

Axial distance

(a)

-200

-150

-100

-50

0

50

100

150

200

-0.006 -0.004 -0.002 0 0.002 0.004 0.006

Fie

ld

Radial distance

(b)

Figure 4.7: Numerical and analytical results for Paul trap with hole. (a) axial field distribution(b) radial field distribution.

-250

-200

-150

-100

-50

0

50

100

150

200

250

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

Fie

ld

Axial distance

(a)

-25

-20

-15

-10

-5

0

5

10

15

20

25

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005

Fie

ld

Radial distance

(b)

Figure 4.8: Numerical and analytical results for CIT with hole. (a) axial field distribution (b)radial field distribution.


-0.002-0.0015

-0.001-0.0005

0 0.0005

0.001 0.0015

0.002-0.002-0.0015

-0.001-0.0005

0 0.0005

0.001 0.0015

0.002

-0.0025-0.002

-0.0015-0.001

-0.0005 0

0.0005 0.001

0.0015 0.002

0.0025

Figure 4.9: Trajectory of an ion of mass 78amu in the Paul Trap at rf amplitude of 350V .The initial position of the ion is (0.0001m,0.0001m,0.0001m) and the initial velocity hasbeen set to zero.

4.6 Trajectory and micromotion

The correctness of trajectory integration process can be checked by monitoring the rf

amplitude at which the ion sitting close to the center of the trap becomes unstable. In

the case where pure rf excitation is applied to the ring electrode, stability limit is marked

by the Mathieu parameter value q = 0.908. This is a theoretical result for an ideal Paul

trap. For the practical Paul traps, ions might get ejected at slightly lower or higher value.

The phenomenon is referred as early and delayed ejection respectively.

Fig. 4.9 shows trajectory of an ion of mass 78amu inside the Paul trap used in this

study. The initial position of the ion is taken as as (0.0001m,0.0001m,0.0001m). The

initial velocity has been set to zero while the rf amplitude has been maintained at 350V .

The ion has been allowed to evolve for 0.5ms. The stability limit of q = 0.908 for this

trap corresponds to the rf amplitude of 366V for ions of 78amu. The ion trajectory keeps

looping back and forth in a region close to the center. It is a stable trajectory. When

the rf amplitude is increased to 375V , the ion becomes unstable and leaves the trap in

z-direction. Fig. 4.10 shows ion trajectory for this case.

Another check for the trajectory integration process can be the frequency analysis of


-0.00025-0.0002

-0.00015-0.0001

-5e-05 0

5e-05 1e-04 0.00015

0.0002-0.00025-0.0002

-0.00015-0.0001

-5e-05 0

5e-05 1e-04

0.00015 0.0002

-0.005-0.004-0.003-0.002-0.001

0 0.001 0.002 0.003 0.004 0.005

Figure 4.10: Trajectory of an ion of mass 78amu in the Paul Trap at rf amplitude of 375V .The initial position of the ion is (0.0001m,0.0001m,0.0001m) and the initial velocity hasbeen set to zero.


the ion micromotion. It is known that ion motion inside the ideal Paul trap consists of

some discrete frequencies, which can be calculated analytically. The main frequency is

referred as secular frequency. For an ideal Paul trap of the same dimension as that of the

Paul trap considered in this study secular frequency for z-direction motion of an ion of

mass 78amu at 350V can be computed to be 0.41MHz. Fig. 4.11(a) shows the graph

of z-direction motion with time and 4.11(b) shows its frequency spectrum obtained by

FFT. The frequency spectrum shows the first peak at 0.41MHz. The second peak shows

the first harmonics. For r-direction motion of the same ion, the analytical value of the

main frequency is 0.16MHz. The FFT of numerically computed trajectory also shows

the main frequency of 0.16MHz. The graph of radial displacement with time and the

FFT of the motion has been shown in Fig. 4.12.

4.7 Spectrum

The final check of the correctness of the simulation has been done by generating the

spectrum of some well known chemical compounds/mixtures. Air and benzene are the

two examples discussed in this section. The mass and charge distribution of the input

air sample is shown in Table 4.7. Table 4.7 shows the details of the benzene sample.

The first row gives the mass of the ions, while the second row gives the charge on the

ions of different masses. The third row gives the number of ions of a given mass and

charge present in the sample. The spectrum has been plotted with voltage on x-axis and

intensity on y-axis. The intensity has been calculated by the following equation.

I =n

N100 (4.9)

where n is the number of ions coming out in a given voltage interval and N is the total

number of ions present in the sample.

The favourable property of the air mixture is that there is a wide separation between

the masses present in the sample. So it is expected to get a clean mass spectrum with

widely separated peaks. Fig. 4.13 shows the air spectrum obtained by the Paul trap

simulation. The peaks have been marked for the masses they represent. As expected, the

peaks are well separated and are located close to voltage level corresponding to Mathieu

parameter value q = 0.908 for the mass they represent. The spectrum has been generated

without taking damping effect into account.

The benzene mixture has contiguous masses (e.g. 26-27, 36-37, 50-52, 73-79) present

in the sample. SO the peaks representing these masses are expected to be close yet

distinct from each other. The spectrum of benzene obtained by Paul trap simulation has

been shown in Fig. 4.14. The peaks have been marked for the masses they represent.


-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 4.5e-05 5e-05

line 1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 106

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−5

(b)


-0.00025

-0.0002

-0.00015

-0.0001

-5e-05

0

5e-05

1e-04

0.00015

0.0002

0.00025

0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 4.5e-05 5e-05

line 1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 106

0

1

2

3

4

5

6

7

8

9x 10

−6


Mass 18 28 32 44 45Charge 1 1 1 1 1Number 5 75 19 2 1

Table 4.2: Mass and charge distribution of air sample.

Mass 26 27 36 37 39 50 51 52 63 73 74 75 76 77 78 79Charge 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Number 2 1 1 2 6 8 9 10 2 1 2 1 3 7 50 3

Table 4.3: Mass and charge distribution of benzene sample.

The peaks of different masses are distinctly visible. And also the peaks occur close to

voltage corresponding to q = 0.908 value. The spectrum has been generated without

taking damping effect into account. To improve the spectrum quality simulation was run

with damping. Fig. 4.15 shows the benzene spectrum for this case. By comparing these

to spectrums, it can be seen that the peaks are better separated in the case of simulation

with damping. Also ions are are coming out at higher voltage as compared to without

damping case. So, it is the case delayed ejection.

For comparison purpose, the spectrum of benzene obtained by the CIT simulation has

been shown in Fig. 4.16. The peaks representing different masses are not well separated.

The resolution is seen to be poor compared to that of the Paul trap of the similar dimen-

sions. The poor quality can be assigned to the nonlinear field distribution offered by the

CIT geometry. Although the CIT is easier to manufacture than the Paul trap, the quality

of the spectrum is worse for the CIT. Designing traps which have electrodes containing

easy to manufacture cylindrical sections and at the same time have spectrum as good as

that of the Paul trap is a topic of current research.

Chapter4.ResultsandVerificatio

n47

80 100 120 140 160 180 200 2200

5

10

15

20

25

30

35

40

Voltage

Inte

nsity

28

32

18 4445

Figure 4.13: Spectrum of air obtained by Paul trap simulation without damping


n48

100 150 200 250 300 350 4000

5

10

15

Voltage

Inte

nsity

79

78

26

27 36

37

39

50

51 52

63 73

77

Figure 4.14: Spectrum of benzene obtained by Paul trap simulation without damping


n49

100 150 200 250 300 350 4000

5

10

15

20

25

Voltage

Inte

nsity

26 27 36 3739

5051

52

6373

77

78

79

Figure 4.15: Spectrum of benzene obtained by Paul trap simulation with damping


n50

0

1

2

3

4

5

6

7

100 150 200 250 300 350 400

Inte

nsity

Voltage

Figure 4.16: Spectrum of benzene obtained by CIT simulation with damping

Appendix A

Derivations used for LIT and RIT potential calculation

In the boundary element method the electrodes are divided into elements on which the

surface charge density is nearly constant. Depending on the symmetry of the problem

these elements may be small polygons, thin circular rings, or narrow infinite strips. Table

A.1 provides the symmetry classification of some common trap geometries.

Table A.1: Symmetry classes of common traps.Symmetry Example Trap Geometries BEM Element Shape

Three dimensionalPaul trap with feed holes on the ring elec-trode, and/or on the endcaps off the centre

Small polygons, usu-ally triangles or rect-angles

AxialPaul trap with feed hole at the centre ofthe endcaps, Paul trap with no feed hole

Thin circular rings(to be considered asthin slices of the sur-face of a cone)

Two dimensional LIT, and RIT of infinite lengthThin infinite straightline strips

In BEM, the potential at any point in space due to an element of charge is needed.

In particular the self-potential or potential due to a uniformly charged element at the

location of the element is needed. Detailed derivations of the potentials due to some

important charge distributions are given here.

A.1 Potential due to a polygon

In this case we consider a planar polygon on which the charge is uniformly distributed

with the charge per unit area being σ. Far away from the polygon, the charge looks like

a point charge σA, where A is the area of this polygon. The potential at a point P far

from the polygon is

Φ =Aσ

4πǫor(A.1)

where is r is the distance between the centroid of the polygon and P . Although the term

far here means that the distance r should be several times the diameter of the polygon, in

practice it is possible to use this formula even for computing the potential at the centroid

51

Appendix A. Derivations used for LIT and RIT potential calculation 52

α

ρ

dρa

a

s

h

Figure A.1: Potential calculation for Sector with uniform charge distribution.

of an adjacent polygon without incurring significant error. Now we consider the problem

of determining the potential on a point on a polygon with m sides. Joining the vertices

of the polygon to the given point results in m triangles which have a vertex at the given

point. So the problem of determination of potential on a polygon can be reduced to the

determination of potential at the vertex of a triangle.

Potential due to a sector with uniform charge distribution: In figure A.1,

P is point of observation. It is the point where potential needs to be computed.

OAB is a charged sector with uniform surface distribution (σ)

α is the angle of the sector

Area of the shaded region is given by αρdρ

All the points on the shaded region are approximately at distance of√

h2 + ρ2 from P .

Charge in shaded region: σ × (Area) = σαρdρ

Potential at P , ΦP due to shaded region is derived to be,

ΦP =σαρdρ

4πǫ0

√

h2 + ρ2(A.2)


o

a

b

cy

dy

p

α

αd

x

Distance

Figure A.2: Calculating potential due to a triangular section.

Result A : Potential due to entire sector:

ΦP =σα

4πǫ0

∫ a

ρ=0

ρdρ√

h2 + ρ2(A.3)

=σα

4πǫ0

∫ a2

0

12dρ2

√

h2 + ρ2(A.4)

=σα

4πǫ0

[

√

h2 + ρ2

]ρ=a

ρ=0

(A.5)

=σα

4πǫ0

(

√

h2 + ρ2 − h)

(A.6)

=σα

4πǫ0

(s − h) (A.7)

Result B : Potential at O due to entire sector can be computed by letting h → 0 in the

previous formula. As h → 0, s → a and the potential at O, φO due to entire sector is

given by

ΦO =σ

4πǫ0

× (Angle) × (Distance) (A.8)

=σαa

4πǫ0

(A.9)

Potential at point O due to a triangle:

y = p tan α (A.10)

dy = p sec2 αdα (A.11)

dα =dy

p sec2 α(A.12)


Thus, for the shaded region,

(Angle) × (Distance) =dy

p sec2 αp sec α (A.13)

= cos αdy =p

√

p2 + y2dy (A.14)

Potential at O due to triangle:

ΦO =σ

4πǫ0

∫

√a2−p2

−√

b2−p2

p√

p2 + y2dy (A.15)

=σp

4πǫ0

∫

√a2−p2

−√

b2−p2

dy√

p2 + y2(A.16)

Let y = p sinh t

dy = p cosh tdt (A.17)√

p2 + y2 = p cosh t (A.18)

dy√

p2 + y2= dt (A.19)

when y =√

a2 − p2,

p sinh t =√

a2 − p2 (A.20)

a2 − p2 = p2 sinh2 t (A.21)

a2 = p2 cosh2 t (A.22)

t = cosh−1 a

p(A.23)

t = ln

(

a

p+

√

a2

p2− 1

)

(A.24)

t = lna +

√

a2 − p2

p(A.25)

Similarly when y = −√

b2 − p2,

t = lnb −

√

b2 − p2

p(A.26)

So potential at O due to triangle can be given as,

ΦO =σp

4πǫ0

(

lna +

√

a2 − p2

p− ln

b −√

b2 − p2

p

)

(A.27)

=σp

4πǫ0

lna +

√

a2 − p2

b −√

b2 − p2(A.28)


Now since (a +√

a2 − p2)(a −√

a2 − p2) = p2 = (b +√

b2 − p2)(b −√

b2 − p2), we have

a +√

a2 − p2

b −√

b2 − p2=

b +√

b2 − p2

a −√

a2 − p2=

a + b +√

(a2 − p2) +√

b2 − p2

a + b −√

a2 − p2 −√

b2 − p2=

a + b + c

a + b − c,

since√

a2 − p2 +√

b2 − p2 = c.

So,

ΦO =σp

4πǫ0

lna + b + c

a + b − c(A.29)

The potential of a point O on a uniformly charged polygon P1P2 . . . Pm can be found by

dividing the polygon into triangles using lines joining O to the vertices of the polygon. If

the total charge on the polygon is q, then σ = q/A and the potential at O is,

Φ =q

4πǫoA

m∑

k=1

pk lnak + bk + ck

ak + bk − ck

. (A.30)

Here ck is the length of the k-th side of the polygon, pk is the length of the perpendicular

from O to the k-th side of the polygon, and ak, bk are lengths of sides joining O to the

k-th side of the polygon.

A.2 Image charge calculation used in LIT and RIT potential ca lculation

Ion traps like the linear ion trap (LIT), and rectilinear ion trap (RIT), have electrodes

which are very long compared to the distance between the electrodes. For these traps the

analysis is simplified considerably if the electrodes are considered to be infinite in length.

Then a two dimensional treatment is possible. In the BEM potential calculation, the trap

is considered to be surrounded by a large radius cylinder which acts as a referance having

zero potential. In the actual calculation the cylinder has been electrostatically replaced

by the images of the elementry charge strips across the cylinder.

In the Fig. A.3 , the cross section of the cylinder has been shown. An elementry

charge strip having charge strip λ passes through point S(ρs, 0). The image of this strip

across the cylinder has been shown by point S ′(ρs′ , 0). The charge density on image of

charge strip has been shown by λ′. Es and Es′ show the electric field at point A due to

charge strips at S and S ′ respectively. Er and Et represents radial and axial component

of the field at point A.

Applying cosine rule in AOS

r2s = R2 + ρ2

s − 2Rρs cos φ (A.31)

Applying sine rule in AOS

ρs

sin α=

R

sin(φ + α)⇒ sin(φ + α)

sin α=

R

ρs

(A.32)


λ′λS ′(ρs′ , 0)S(ρs, 0)O(0, 0)

rs′

rs

R

A

Es

Er

Et

Es′

φ

α

Figure A.3: Location of the image charge due to charge at S(ρs, 0).

Eq.A.32 can be simplified to get

sin α =ρs sin φ

√

(R2 + ρ2s − 2Rρs cos φ)

(A.33)

Using Gauss law the electric field at point A due to infinite line charge at point S can

be computed to be,

Es =λ

2πǫ0rs

(A.34)

The tangential component of the electric field at point A due to charge strip at S can

be expressed as.

Est =λ sin α

2πǫ0rs

(A.35)

Substituting value of sin α from Eq.A.33, we get

Est =λ

2πǫ0rs

ρs sin φ√

(R2 + ρ2s − 2Rρs cos φ)

(A.36)

Substituting value of for rs from Eq.A.31 we find


Est =λ

2πǫ0

ρs sin φ

R2 + ρ2s − 2Rρs cos φ

(A.37)

Similarly tangential component of electric field at a point A due to line charge at S ′

is given by

Es′t =λ′

2πǫ0

ρs′ sin φ

R2 + ρ2s′ − 2Rρs′ cos φ

(A.38)

As the cylinder is an equipotential surface, we assert that the tangential component

of the resultant electric field must be zero

Est + Es′t = 0 (A.39)

Putting the value of Est and Es′t from Eq.A.37 and A.38, we get the following equation.

λ

2πǫ0

ρs sin φ


+λ′

2πǫ0

ρs′ sin φ


= 0 (A.40)

Making λ = −λ′, we get

ρs sin φ


=ρs′ sin φ


(A.41)

⇒ sin φR2

ρs+ ρs − 2R cos φ

=sin φ

R2

ρs′+ ρs′ − 2R cos φ

⇒ R2

ρs

+ ρs = R2

ρs′+ ρs′

⇒ R2(1

ρs

− 1

ρs′) = ρs′ − ρs

⇒ R2

ρsρs′= 1

So, we finally get

ρs′ =R2

ρs

(A.42)

The Eq.A.42 gives the image charge location.


x

S

S’

P

O

Reference circle of radius R

Figure A.4: Cross section of an infinite line charge inside a grounded cylinder.

A.3 Potential due to an infinite narrow strip inside cylinder

Let the cross section of the electrodes be on the x-y plane. Then any electrode can

represented by a curve on the x-y plane. The division of the actual electrode into narrow

infinite strips is seen on the cross section as the division of the curve into short line

segments. Due to the logarithmic nature of the dependence of the potential with distance

from the strip, the reference for zero potential cannot be chosen at a point infinitely far

away from the strip. In this work a cylinder of large radius R is chosen as the surface of

zero potential. The objective here is to find the potential due to a narrow infinite strip

with uniform surface charge density in the presence of the reference cylinder. The charge

per unit length of the strip is ql, and its width is w. At points whose distance from the

strip is large compared to the width, w, the strip may be regarded as an infinite line

charge. Figure A.3 shows the cross section of an infinite line charge at point S, with polar

coordinates (ρs, φs). The circle of radius R represents the reference cylinder. It is required

to the potential at point P , with coordinates (ρ, φ). The method of images (Weber, 1950)

can be used to express the potential in the presence of the reference cylinder. The location

of the image charge is at S ′, (ρs′ , φs), where

ρs′ =R2

ρs

(A.43)

The potential at point P with coordinates (ρ, φ) is given by

Φ(ρ, φ; ρs, φs) =ql

2πǫo

lnrs′

rs

+ Φ0 (A.44)

Where Φ0 is to be chosen so that the potential on the reference circle is 0. Here rs is

distance from (ρs, φs) to (ρ, φ), and rs′ is distance from (ρs′ , φs) to (ρ, φ). Setting (ρ, φ)


Figure A.5: Cross section of an infinite strip.

to (R, φs) makes Φ(ρ, φ) = 0, rs = R − ρs, rs′ = R2/ρs − R. So,

Φ0 = − ql

2πǫo

lnR2/ρs − R

R − ρs

= − ql

2πǫo

lnR

ρs

(A.45)

and consequently


2πǫo

lnρsrs′

Rrs

=ql

4πǫo

lnρ2

sr2s′

R2r2s

(A.46)

Substituting

r2s = ρ2

s + ρ2 − 2ρsρ cos(φ − φs) (A.47)

and

r2s′ = (R2/ρs)

2 + ρ2 − 2(R2/ρs)ρ cos(φ − φs) (A.48)

in Equation A.46 and simplifying we get


4πǫo

ln(ρsρ/R)2 + R2 − 2ρsρ cos(φ − φs)

ρ2s + ρ2 − 2ρsρ cos(φ − φs)

(A.49)

Equation A.49 is valid when rs is large compared to w. It also shows that interchanging the

source and observation points leaves the potential unchanged. When the point P coincides

with S, the concentrated line charge approximation cannot be used. The potential is to

be found by integration of contributions from the width of the strip. Figure A.3 shows a

charged strip. The points on the strip can be expressed in terms of t, the displacement

from the centre of the strip. On the strip t changes from −w/2 to w/2. The line charge

corresponding to the differential element dt is (ql/w)dt. We now proceed to find the

contribution to the potential at the centre of the strip due to this differential element

using Equation A.49. The denominator of the fraction inside the logarithm in Equation

A.49, ρ2s + ρ2 − 2ρsρ cos(φ − φs), is the distance between the source and the observation

points and equals t2. The numerator (ρsρ/R)2 + R2 − 2ρsρ cos(φ − φs) is approximately


(ρ2s/R)2+R2−2ρ2

s = (R−ρ2s/R)2 since ρs ≈ ρ and φs = φ on the strip. So the contribution

of the differential element is

dΦ =ql/w

4πǫo

ln(R − ρ2

s/R)2

t2dt (A.50)

The potential at (ρs, φs), found by the integration of the differential contribution from

t = −w/2 to t = w/2, is

Φ(ρs, φs; ρs, φs) =ql

4πǫo

(

2 + ln(R − ρ2

s/R)2

(w/2)2

)

=ql

4πǫo

(

2 + 2 lnR − ρ2

s/R

w/2

)

(A.51)

A.4 Equipotential surface for two infinite line charges

Fig. A.4 the cross sectional view of two infinite line charges with charge density −λ and

λ passing through point (−c, 0) and (c, 0) respectively. Let (a, 0 be the center and b be

the radius of equipotential surface in which we are intrested. Let (x, y) be an arbitrary

point on the equipotential surface. The point (x, y) is at a distance r and r′ from the

point (c, 0) and (c′, 0) respectively.

The resultant potential at point (x, y) due to two line charges has to be equal to V0.

So,−λ

2πǫ0

lnR∞

r+

λ

2πǫ0

lnR∞

r′= V0 (A.52)

where R∞ represents the distance between line charge and the surface of zero potential.

Eq.A.52 can be simplified to get

r2 = k2r′2

(A.53)

with value of k given by

k = e2πǫ0V0

λ (A.54)

Putting distance formula in Eq.A.53

(x + c)2 + y2 = k2[(x − c)2 + y2]

⇒ x2 + y2 + 2cx1 + k2

1 − k2+ c2 = 0 (A.55)

which is the equation of a circle centered at(

−c1+k2

1−k2 , 0)

But the assumed center location is (a, 0). Hence

−c1 + k2

1 − k2= a


(a, 0)(c, 0)O

r

λ−λ

r′

(−c, 0)

b

V0

(x, y)

Figure A.6: Equipotential surface due to two line charges

⇒ 1 + k2

1 − k2=

−a

c(A.56)

The assumed radius of the circle b can be expressed as

c2 (1 + k2)2

(1 − k2)2− c2 = b2 (A.57)

Using Eq. A.56 we get,

c =√

(a2 − b2) (A.58)

Applying Componendo-dividendo rule to Eq. A.56 we get,

k2 =a + c

a − c(A.59)

But, from Eq.A.54

e2πǫ0V0

λ = k

Squaring both side

e4πǫ0V0

λ = k2


Putting the value of k2 from Eq.A.59 and taking logarithm on both side, we get

⇒ 4πǫ0V0

λ= ln

a + c

a − c

Therefore

λ

4πǫ0

=V0

ln a+ca−c

(A.60)

subtituting for c from Eq. A.58 we get,

λ

4πǫ0

=V0

lna+√

(a2−b2)

a−√

(a2−b2

(A.61)

The Potential at any point (x, y) can be written as

V (x, y) =λ

2πǫ0

ln(x +

√

(a2 − b2))2 + y2

(x −√

(a2 − b2))2 + y2(A.62)

Substituting Eq.A.61 in the previous equation, we get

V (x, y) =V0

lna+√

(a2−b2)

a−√

(a2−b2

ln(x +

√

(a2 − b2))2 + y2

(x −√

(a2 − b2))2 + y2(A.63)

Bibliography

Bier, M. E. and Syka, J. E. P. (1995), U. S. Patent 5,420,425.

Langmuir, D. B., Langmuir, R. V., H., S., and F., W. R. (1962), U.S. Patent No. 3065640.

March, R. E. and Hughes, R. J. (1989), Quadrupole Storage Mass Spectrometry, Wiley-

Interscience, New York.

McLachlan, N. W. (1947), Theory and Applications of Mathieu Functions, Clarendon,

Oxford.

Paul, W. and Steinwedel, H. (1953), Ein neues Massenspektrometer ohne Magnetfeld,

Zeitschrift fur Naturforschung A, 8, 448-450.

Stafford, G. C., Kelley, P. E., Syka, J. E. P., Reynolds, W. E., and Todd, J. F. J. (1984),

Recent improvements in and analytical applications of advanced ion trap technology,

International Journal of Mass Spectrometry, 60, 85-98.

Weber, E. (1950), Electromagnetic Fields: Theory and Applications. Vol 1 - Mapping of

Fields, John Wiley & Sons, New York.

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