Plasma Dynamics in Paul Traps

Plasma Dynamics in Paul Traps

A Thesis

submitted by

Kushal Kumar Shah

for the award of the degree

of

Doctor of Philosophy

Department of Electrical Engineering

Indian Institute of Technology Madras, India

March 2010

c© Copyright Kushal Kumar Shah, March 2010

All rights reserved.

So close, no matter how far

Couldn’t be much more from the heart

Forever trusting who we are

And nothing else matters

I never opened myself this way

Life is ours, we live it our way

All these words I don’t just say


Yeah, trust I seek and I find in you

Everyday for us something new

Open mind for a different view


— James Hetfield

This thesis is dedicated to

The Pursuit of HappIness

THESIS CERTIFICATE

This is to certify that the thesis entitled “Plasma dynamics in Paul traps” sub-

mitted by Kushal Kumar Shah to the Indian Institute of Technology Madras

for the award of the degree of Doctor of Philosophy by research is a bonafide

record of research work carried out by him under my supervision. The con-

tents of this thesis, in full or in parts, have not been submitted to any other

Institute or University for the award of any degree or diploma.

Place: Chennai

Date:

Prof. Harishankar Ramachandran

(Research Guide)

i

ACKNOWLEDGEMENTS

One looks back with appreciation to the brilliant teachers, but with gratitude

to those who touched our human feelings. The curriculum is so much

necessary raw material, but warmth is the vital element for the growing plant

and for the soul of the child.

— Carl Jung

I really feel very honored for having come across brilliant teachers who not

only taught me the academic subjects but also touched the deepest chords

of my heart. I would like to convey my genuine appreciation and heart-felt

gratitude to all of them.

PhD is not just another degree, but it marks the culmination of very long

years of academic education. I would like to take this opportunity to thank

one and all who has contributed to my learning process since I was a child in

school.

Out of all subjects in my school, I loved mathematics the most. I would

like to sincerely thank my teacher, Mrs. Chapala Mohanty, for having sensed

my interest early on and for having provided me excellent guidance to get a

strong foundation in the beautiful subject. She is the first teacher to have had

a great positive impact on my life.

My first interaction with my PhD guide, Dr. Harishankar Ramachandran,

was during my undergraduate days at IIT Madras. I did my final year B.Tech.

project in plasma physics (Landau damping) under him and then joined back

ii

iii

for a PhD in January 2007. I have had a great time working with him. I am

extremely glad that I had a chance to do my PhD under someone who truly

practices and appreciates deep thinking. I will be forever indebted to him for

all the time he has spent with me in discussions. Thank you very much, Sir.

During my PhD, I have also had the wonderful opportunity to discuss ideas

and concepts with a few faculty members apart from my guide. I would like to

thank Dr. Arul Lakshminarayan, Dr. V. Balakrishnan and Dr. Suresh Govin-

darajan of IIT Madras, Dr. Sudip Sengupta, Dr. Prabal Chattopadhyay, Dr.

Amita Das and Dr. R. Ganesh of Institute for Plasma Research (Gandhina-

gar), and Dr. M. Lakshmanan of Bharathidasan University (Trichy). I would

also like to thank Dr. Sayan Kar of IIT Kharagpur, with whom I interacted

over a few weeks before joining for PhD at IIT Madras.

I would also like to thank the members of my Doctoral Committee, Dr. V.

Jagadeesh Kumar (Head and Chairman), Dr. Anil Prabhakar, Dr. Bijoy Kr-

ishna Das, Dr. Suresh Govindarajan and Dr. Neelima M. Gupte, for annually

reviewing my work and for giving me positive feedback. Special thanks to Dr.

M. S. Ananth (Director of IITM) and Dr. K. Krishnaiah (Dean, Academic

Research) for accepting my submission of this thesis.

During my days of preparation for the IIT-JEE examination, I used to go

to Mr. Snehanshu Dwibedi to clear my doubts in Chemistry. It was only after

coming in contact with him that I truly started appreciating this subject. And

apart from the subject, he also taught me many things that helped me broaden

my perspective in life. He also gave me a lot of confidence. My sincere thanks

to him.

I have learnt a lot of my life’s lessons from my parents, specially my father.

One of the most important ingredients that goes into making a good researcher

is curiosity. My father has contributed a lot towards increasing my curiosity

and the spirit to seek a deep understanding of the world around me. He

iv

always encouraged me to ask as many questions as possible and tried his best

to answer them. Along with my parents, I would also like to sincerely thank

all my aunts and uncles for having showered so much love on me since my

childhood.

I would also like to thank all my friends for the great time I have always

had with them. Apart from the fun, some of my friends have also been a great

source of learning and inspiration. Thanks a lot to one and all and special

thanks to Sanjay Mitra and Naga Pranay. Sanjay has been a dear friend since

my school days and Naga since my undergraduate days at IITM. There are,

of course, many others whom I would like to mention, but the list is too long

and the space too short. I will always cherish the moments I have shared with

all my friends.

I would like to thank Mrs. Usha Rani for helping me with the day-to-day

office requirements and Samkiruba for help with technical problems related

to my PC. I would also like to convey my gratitude to the staff members of

the office of the Dept. of Electrical Engineering, specially Mrs. Suganthi and

Mrs. Tamilselvi, and also the administrative staff of IIT Madras, specially

Mrs. Saraswati, for their help and support all throughout my PhD.

I came to know about Linux in 2001, after I joined IITM for undergraduate

studies. And since then I have been using Linux and only Linux. Thanks a lot

to the open source community for the wonderful work they have been doing

over the past many years.

To be able to work efficiently, a human being also requires a source of en-

tertainment. My heartfelt thanks to the Open Air Theater (OAT) and LAN

at IITM and to YouTube and TED for being such great sources of entertain-

ment. Apart from the entertainment value, I have also learnt a lot from all

the movies I have watched and all the interesting talks I have heard.

Many more people have touched my life in many wonderful ways. I would

v

like to convey my sincere appreciation and heart-felt gratitude to one and all.

— Kushal Shah

Chennai

July 2009

ABSTRACT

Paul traps are of immense importance in confining plasmas of particles of a

single species. Though lot of progress has been made in this field experimen-

tally, a good theoretical understanding of the distribution function and density

of plasmas in Paul traps is sill lacking.

This thesis theoretically investigates the dynamics of a non-neutral plasma

in a Paul trap. To begin with, space charge effects are neglected and an

analytic, nonlinearly exact expression for the distribution function and density

is obtained. It is found that the plasma is a Maxwellian at all times and at

all spatial locations. However, the spatially constant plasma temperature is

not temporally constant and keeps oscillating in time at the same frequency as

that of the applied RF field. Since the distribution function is a Maxwellian,

the solutions obtained remain valid even in the presence of collisions. The

linear limit of weak RF fields is recovered. These exact solutions have also

been used to validate some of the approximate results present in the literature

and at the same time some of the well accepted results have been shown to be

incorrect.

It is shown in this thesis that the plasma distribution function remains

periodic with time only for a particular specification of initial distribution

function. The case of a kicked plasma is also considered where the plasma is

initially confined by a DC field and an RF field is switched on at t = 0. It is

shown that if the RF field is abruptly switched on, then the plasma has slow

vi

vii

breathing apart from oscillating at the RF frequency. However, if the RF field

is increased sufficiently slowly, it is numerically shown that it is possible to

recover a time periodic distribution function asymptotically.

The expression for the time varying density is then used to find the field in-

duced by the plasma. The complete induced field is very complicated. To make

further progress, the nonlinear term of the lowest order has been retained and

the modified force equation is thus obtained. This modified equation is linear

in RF but has a nonlinear DC term. This equation is then solved by using

the well known Lindstedt-Poincare method to obtain the expressions for the

particle orbits. A surprising finding has been that the expressions for the stro-

boscopic orbit corresponding to these particle trajectories can be transformed

(scaling+rotation) into the expressions for the time averaged orbit. This has

been used in constructing the time-dependent invariant corresponding to these

orbit expressions. The distribution function of the plasma is then simply an

exponential of this invariant. Like the linear DC field case, the distribution

function has again been found to be a Maxwellian at all times and all spatial

locations. However, in this case, the plasma is a Maxwellian only up to the

lowest order in the nonlinearity.

At the end, a nonlinear RF field is considered and the force equation has

been solved to obtain the expressions for the particle orbit. The expression

for time averaged orbit and the stroboscopic orbit has been obtained. Unlike

the linear RF field case, in the nonlinear RF field case, there does not seem to

be any way to transform the stroboscopic expressions into those for the time

averaged orbits. Thus, the complete time dependent invariant corresponding

to the stroboscopic orbit is constructed perturbatively. This corresponding

distribution function has been found to be non-Maxwellian even to the lowest

order in the RF field magnitude.

Contents

1 Introduction to Paul Traps 1

1.1 Paul’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Mathieu Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Conventional Ponderomotive Theory . . . . . . . . . . . . . . . 11

1.4 Ponderomotive Density . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Paul trap experiments . . . . . . . . . . . . . . . . . . . . . . . 14

2 Introduction to Perturbation Methods 19

2.1 Solving Nonlinear Equations . . . . . . . . . . . . . . . . . . . 21

2.1.1 Multiple Scale Methods . . . . . . . . . . . . . . . . . . 23

2.1.2 Lindstedt-Poincare (LP) Method . . . . . . . . . . . . . 25

2.1.3 Modified Lindstedt-Poincare (MLP) Method . . . . . . . 27

2.1.4 Small Denominator Problem . . . . . . . . . . . . . . . . 29

2.2 Krapchev’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . . 34

3 Spatially linear DC and RF field — I 36

3.1 Time evolution of the distribution function and density . . . . . 37

3.1.1 General distribution function . . . . . . . . . . . . . . . 39

3.1.2 Maxwellian distribution function . . . . . . . . . . . . . 41

3.1.3 Time periodic distribution function . . . . . . . . . . . . 42

3.2 Time averaged density . . . . . . . . . . . . . . . . . . . . . . . 49

viii

CONTENTS ix

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 The Ponderomotive Potential . . . . . . . . . . . . . . . 51

3.3.2 Comparison with Krapchev’s Theory . . . . . . . . . . . 55

3.3.3 Extent of non-Maxwellianity . . . . . . . . . . . . . . . . 59

3.3.4 Relation to BGK Theory . . . . . . . . . . . . . . . . . 60

3.3.5 Collisional Effects . . . . . . . . . . . . . . . . . . . . . . 66

3.3.6 Multi-Species Plasmas . . . . . . . . . . . . . . . . . . . 67

3.3.7 Conservation of particle number and energy . . . . . . . 69

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Spatially linear DC and linear RF field — II 72

4.1 Plasma distribution function . . . . . . . . . . . . . . . . . . . . 73

4.2 Validation of Linear Response Theory . . . . . . . . . . . . . . . 76

4.2.1 Linear estimate from exact theory . . . . . . . . . . . . . 76

4.2.2 Linear Vlasov Equation . . . . . . . . . . . . . . . . . . 78

4.3 Quiet Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Spatially nonlinear DC and linear RF field 86

5.1 Stroboscopic Map . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 The force equation . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Solving the force equation . . . . . . . . . . . . . . . . . . . . . 92

5.4 Time averaged motion . . . . . . . . . . . . . . . . . . . . . . . 95


5.6 Stroboscopic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.7.1 Space charge effects in RF traps . . . . . . . . . . . . . . 109

5.7.2 Distribution function . . . . . . . . . . . . . . . . . . . . 110

5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

CONTENTS x

6 Spatially nonlinear RF Fields 116

6.1 The particle trajectory . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Time averaged motion . . . . . . . . . . . . . . . . . . . . . . . 118

6.3 Stroboscopic Map . . . . . . . . . . . . . . . . . . . . . . . . . . 120


6.4.1 O (q) Invariant . . . . . . . . . . . . . . . . . . . . . . . 124

6.4.2 O (q2) Invariant . . . . . . . . . . . . . . . . . . . . . . 132

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 Discussions and Conclusions 138

7.1 Invariants of motion . . . . . . . . . . . . . . . . . . . . . . . . 138

7.2 Canonical Adiabatic Theory . . . . . . . . . . . . . . . . . . . . 144

7.3 Gyro-Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4 Three Dimensional Distribution Function . . . . . . . . . . . . . 149

7.5 Time averaging and Poincare Map . . . . . . . . . . . . . . . . . 151

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A Trajectory of particle in nonlinear RF field 162

List of Figures

1.1 This figure shows how the coefficients, c2r decay as r → ±∞.

As can be seen, the coefficients decay exponentially fast. . . . . 7

1.2 The solid line shows the plot of the path of the particle as

per numerical integration of Eq. (3.1). The dashed line shows

the path as predicted by the mathematical expression given in

Eq. (1.10), considering only the first three terms. This was

done for q = 0.16, p = −0.01 and the initial conditions are

x0 = 0.91918, v0 = 0.We can see that there is close agreement

between mathematics and simulation and the little discrepancy

can be reduced further by considering higher order terms in the

mathematical expression. . . . . . . . . . . . . . . . . . . . . . . 9

1.3 This is the phase space plot of the trajectory shown in Fig.

1.2. We can see that near x = 1 and x = −1, the particle

is undergoing high frequency oscillations. This region of phase

space near the turning point is where ponderomotive theory is

applicable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

xi

LIST OF FIGURES xii

3.1 Density plots with q = 0.16, p = −0.01 at two different times

τ = 0 and τ = kπ, where k ∈ N is such that kπ is close

to π/2ν. The three plots are for different values of γ0. (a)

γ0 = 0.5p+0.25q2 (b) γ0 = γ, where γ is given in Eq. (3.21) (c)

γ0 = 20p+ 10q2 γ. In (a), it can be seen that the two curves

are quite close to each other. This shows that, approximately,

γ = 0.5p+0.25q2 is the value of γ0 at which the density function

does not have a large ν dependence. The curve in (b) clearly

shows that Eq. (3.22) gives a much more accurate expression

for the γ which leads to a more accurate expression for the

ponderomotive energy of a particle under the linearly varying

oscillatory electric field. The overlap of the curves at τ = 0 and

τ = kπ is so good that it is not visible in this graph. In (c) it

can be clearly seen that for this value of γ0, the density function

has a strong dependence on ν. . . . . . . . . . . . . . . . . . . 47

3.2 This is the contour plot of the distribution function of the

plasma for the case γ0 = 20p + 10q2 with q = 0.16, p = −0.01.

The two superimposed contour plots correspond to the two

times of the curves shown in Fig. 3.1c. This clearly shows that

the drastic change in the distribution function is the reason for

the huge change in the density function over the ν time scale. . 49

LIST OF FIGURES xiii

3.3 This plot compares the time averaged density of the plasma,

Eq. (3.25) obtained from exact expressions with that predicted

by the conventional ponderomotive theory. (a) The difference

in the plasma density at x = 0 is due to the factor 1 − 0.5q

present in Eq. (3.25). In conventional ponderomotive theory, it

is assumed that the plasma density at the origin is not effected

by the RF field. (b) The plot has been normalized so that

the plasma densities are same in both cases at x = 0. It can

be clearly seen that there is a difference in the spatial density

profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 This is a plot of the time averaged distribution function at differ-

ent spatial locations. The solid line is obtained from a numerical

integration of Eq. (3.10) and the dashed lines are the ones given

in the work of Krapchev [9] and are normalized and scaled. The

labels are scaled by x. As can be seen, the curves cease to be

monotonic after a certain threshold in x. . . . . . . . . . . . . . 56

3.5 This shows the spatial variation of the magnitude of various fre-

quency components present in the Fourier transform of Ei(x, t)

as given by Eq. (3.33). This is for the case when γ is given by

Eq. (3.21) and q = 0.16, p = −0.01. As can be clearly seen,

leaving the DC component, the component at ω is dominating.

And we also have small contributions from components at fre-

quencies 2ω , 3ω (ω = 2 in our normalization). The remaining

harmonics are lower in magnitude than the the ones shown. The

component at 2ω is two orders of magnitude lower than that at

ω. So, the field given by Eq. (3.33) is essentially a nonuniform

monochromatic electric field for all practical purposes. . . . . . . 63

LIST OF FIGURES xiv

3.6 This shows the relative spatial variation of the fields corre-

sponding to the RF solution considered in this paper and the

fields in static equilibrium. If the time-averaged density of the

plasma goes like exp (−β0γ0x2), then the “Effective static Field”

corresponds to the Electric field for which the potential goes

like β0γ0x2. The “Induced Field” is the field induced by the

exp (−β0γ0x2) electron density in the absence of ions. The “Ap-

plied static Field” is the sum of these two fields, and is the exter-

nal Electric field which has to be applied to get this particular

density profile. Now, if the same time averaged profile has to be

achieved by using an RF field, then the total field seen by the

plasma has a much steeper slope and is shown by the straight

line labeled “Effective RF Field”. The curves labeled “Induced

Field” and “Applied static Field” are not straight lines. These

curves are purely qualitative and are not to scale. . . . . . . . . 64

4.1 Plot of the magnitude of the coefficients of cos 2ντ and cos 2τ

in the expression for A as given in Eq. (4.8), normalized by q.

It can be clearly seen that magnitude of the response at w lies

below the response at 2ν. The solid thick line labeled p = 0.5q2

serves to divide two prominent region of plasma response. When

p = 0.5q2, the RF response is as large as the DC response and

as we approach this region, plasma behavior is highly nonlinear.

There is another solid thick line labeled p = 0.25q. This also

demarcates two regions in the p−q space. As q crosses this line,

there is a visible change in the slope of the curves. The plots

clearly show that as q becomes large compared to p, the first

nonlinearity to set in is of the order of q2/4p. Also, the curves

are well in agreement with the expressions derived in Eq. (4.11). 75

LIST OF FIGURES xv

4.2 This figure shows the solution of Eq. (7.2) for initial condition

ρ(0) = 1/

4√p and q(τ) = q1

(1 + tanh

[(τ − τ0)

/w])

with q1 =

0.01 and p = 0.1. The two curves correspond to τ0 = 20π

and two different values of w = 0.5π, 5π. As can be seen, for

w = 0.5π, ρ(τ) is not asymptotically periodic. But when w is

increased to w = 5π, ρ(τ) tends to a periodic function after a

small region of non-periodicity. . . . . . . . . . . . . . . . . . . 82


ρ(0) = 1/

4√p and q(τ) = q1 exp

[−w

(1/τ − 1

/τ0)2]

with q1 =

0.01 and p = 0.1. The two curves correspond to τ0 = 50π and

two different values of w = π, 100π. As can be seen, for w = π,

ρ(τ) is not asymptotically periodic. But when w is increased to

w = 100π, ρ(τ) tends to a periodic function after a small region

of non-periodicity. . . . . . . . . . . . . . . . . . . . . . . . . . 83


ρ(0) = 1/

4√p and q(τ) = q1τ

/w with q1 = 0.01 and p = 0.1. The

two curves correspond to two different values of w = 5π, 50π.

As can be seen, for w = 5π, ρ(τ) is not asymptotically periodic.

But when w is increased to w = 50π, ρ(τ) tends to a periodic

function after a small region of non-periodicity. . . . . . . . . . 84

5.1 This figure shows the stroboscopic plot of the phase space tra-

jectory of particles with different initial conditions under the

force equation given by Eq. (5.4) keeping terms up to O (q2). It

can be seen that some orbits are simple closed curves and some

orbits form islands in phase space. The numerical values of the

various parameters were pe = −0.01, qe = 0.16 ≈ q, ωp = 0.5ν0

and x = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

LIST OF FIGURES xvi

5.2 This is a plot of the error function. Curve (1) shows the exact

error function. Curve (2) is the linear approximation and curve

(3) is the plot with the cubic nonlinearity taken in. This shows

that 2(x− x3

/3) /√

π is a good approximation to the exact

error function, erf(x), up to x = 1. . . . . . . . . . . . . . . . . 91

5.3 This plot shows the stroboscopic map of the particle orbits in

phase space. Curve (1) is the full orbit as obtained by integra-

tion of Eq. (5.4). Curve (2) is the stroboscopic plot obtained

by a sampling of the particle orbit at a fixed time step of 2π/ω

starting from τ = 0. The big crosses correspond to a few points

on this level curve. These points form a countable set that is

dense on the level curve given by Eq. (5.28). Curve (3) is

the time averaged orbit and corresponds to the time averaged

ponderomotive Hamiltonian, Eq. (5.11). Curve (4) is also a

stroboscopic plot like curve (2), but, in this case, the sampling

begins at τ 6= 0. The arrow labeled v shows the direction of

sampled velocity, vs, at one particular instant of time. It can

be clearly seen that vs is not along the stroboscopic level curve,

which explains the reason why vs 6= dxs

/dt in Eq. (5.15) and

Eq. (5.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Comparison of stroboscopic map of the particle orbit obtained

by numerical integration of Eq. (5.4) and the analytical expres-

sions given by Eq. (5.6). As can be seen, as the initial condition

of the particle moves away from the origin in phase space, the

analytical expression become less and less accurate. . . . . . . . 114

LIST OF FIGURES xvii

6.1 This plot compares the invariant given by Eq. (6.30) and the

results of numerical integration of the force equation, Eq. (6.2).

Curve (1) is the stroboscopic plot of the orbits obtained by nu-

merical integration of Eq. (A.1). Curve (2) is the stroboscopic

plot of the analytic solutions, up to O (q), to Eq. (6.2). Curve

(3) is the level curve corresponding to the invariant, Eq. (6.30),

corresponding τ = 0. The parameters used were p = 0.011,

q = 0.1, α = 0.1 and A = 2.5. . . . . . . . . . . . . . . . . . . . 129

6.2 This plot shows the same set of curves as in Fig. (6.1) but for

a particular τ 6= 0. As can be seen in the figure, Eq. (6.30)

gives good estimate of the invariant for the rest of the curve,

but near the turning points terms of O (q2) become important

and cannot be ignored. . . . . . . . . . . . . . . . . . . . . . . 130

6.3 This figure compares the time average of the distribution func-

tion given by Eq. (6.31) to a Maxwellian. As can be clearly

seen, the time averaged distribution function is very close to a

Maxwellian in the plasma bulk but deviates as v increases. . . 131

6.4 The two unlabeled curves in the plot compare the stroboscopic

plot of the particle orbit in phase space obtained by numerical

integration and the level curve of the invariant predicted by Eq.

(6.33) and Eq. (6.35). This corresponds to the case when the

stroboscopic sampling is begun at a particular τ > 0. As can

be seen, the two curves match very well. The curve labeled “q”

is the level curve corresponding to the O (q) invariant in Eq.

(6.30). Thus, we can clearly see an improvement as we go from

O (q) to O (q2) for the invariant expression. The parameters

used in this figure are the same as in Fig. (6.2). . . . . . . . . . 135

LIST OF FIGURES xviii

7.1 The curve labeled (1) is the stroboscopic plot of the numerical

solutions of Eq. (5.4) and the curve labeled (2) is the level curve

corresponding to the invariant 0.5v2 + 0.5ν2x2 with ν given by

Eq. (5.5). As can be seen, it is very hard to say from a numerical

plot that the invariant 0.5v2 + 0.5ν2x2 is incorrect. . . . . . . . 153

Chapter 1

Introduction to Paul Traps

Plasma confinement has been an active area of research since the beginning

of plasma studies. Though confining a plasma is difficult, the complexity is

greatly reduced if the plasma consists of charged particles of only one species,

i.e. for non-neutral plasmas. The most important mechanisms for trapping

non-neutral plasmas are Paul trap [1], linear RF trap [2], [3] and Penning trap

[4]. Paul traps and linear RF traps trap charged particles by using oscillating

electric fields and Penning trap uses a combination of static electric and mag-

netic fields. There is no magnetic field involved in a Paul trap or a linear RF

trap.

From Earnshaw’s theorem [5], we know that a time independent electric

field cannot have have a local minima in 3 dimensional space and hence, cannot

confine a charged particle. However, Paul showed in 1953 that charged particles

of a single species can, however, be confined in 3d using an oscillatory electric

field. There is no magnetic field required. These traps have come to be known

as Paul traps [1] and Paul was awarded the Nobel Prize in 1989 for this novel

idea.

1

CHAPTER 1. INTRODUCTION TO PAUL TRAPS 2

1.1 Paul’s solution

In a Paul trap, the externally applied potential is of the form [6]

φ(x, y, z, t) =U0 + V0 cosωt

r20 + 2z2

0

(x2 + y2 − 2z2

)(1.1)

where U0 is the DC potential, V0 is the RF potential, r0 is the radial extent

of the trap, z0 is the axial extent and ω is the RF frequency. It can be easily

checked that this potential, φ, satisfies the Laplace equation, ∇2φ = 0. The

equations of motion along the three directions of motion are

x = [−pe + 2qe cos 2τ ]x

y = [−pe + 2qe cos 2τ ] y

z = [− (−2pe) + 2 (−2qe) cos 2τ ] z (1.2)

where the frequency has been normalized to ω = 2 and pe, qe are normalized

constants. The subscript e stands for externally applied field. The normaliza-

tion has been done so as to make the three equations in Eq. (1.2) to have the

same form as the well known Mathieu’s equations [7].

Depending on the values of pe, qe, the solutions of Eq. (1.2) are either

bounded or unbounded. Thus, for the Paul trap to be able to confine particles,

it is important to choose the values of pe, qe appropriately. The details of the

Mathieu equation are discussed in the next section.

1.2 Mathieu Equation

As shown in Eq. (1.2), the equations of motion of particles in a Paul trap get

decoupled along the three coordinate axes and each of these three equations

can be written in the form of the standard Mathieu equation. The Mathieu


equation in its standard form is

d2x

dτ 2+ [p− 2q cos 2τ ]x = 0 (1.3)

where τ = ωt/2 is the scaled time coordinate.

Equation (1.3) does not have bounded solutions for all p, q. The bounded

solutions solutions of Eq. (1.3) are, in general, aperiodic. However, for any

given q, there is a countable set of values of p denoted by a0 (q) , a1 (q) , a2 (q) , ...

for which Eq. (1.3) has even periodic solutions and there is another countable

set of values of p denoted by b1 (q) , b2 (q) , b3 (q) , ... for which the equation has

odd periodic solutions. These two sets of values (a0, a1, a2, ...) and (b1, b2, b3, ...)

form two countably infinite sets and are called the characteristic values. If q

is real, the characteristic values ar and br are also real and are distinct. For

q > 0, we have, a0 < b1 < a1 < b2 < .... For values of p ∈ (ar, br+1), where

r = 0, 1, 2, .., the solutions to Eq. (1.3) are aperiodic and bounded. However

for other values of p, i.e. p ∈ (br, ar+1) for r = 1, 2, ..., the solutions grow

exponentially to infinity. For p = 0, the solutions are stable if 0 ≤ |q| ≤ qc

where qc ≈ 0.9. This corresponds to the case when the particles see only the

RF field. If p < 0, since the DC field is trying to throw away the electrons

to infinity, any arbitrarily small positive value of q will not be sufficient to

confine the plasma. Thus, when p < 0, there is both a non-zero positive lower

bound and an upper bound on the value of q for the solutions of Eq. (1.3)

to be stable. This corresponds to the case when the total DC field is trying

to destabilize the plasma but the RF field comes in and makes the particle

orbits bounded and hence stable. For p > 0, both the DC and the RF field

contribute towards confining the plasma.

Equation (1.3) belongs to the class of linear ordinary differential equations

with periodic co-efficients. According to Floquet theorem [8], the solutions to


such equations will be of the form

x(τ) = eiντP (τ) (1.4)

where P (τ) is a periodic function of τ , with the same period as the coefficients

in the equation, and ν is called the characteristic exponent and is, in general,

complex. When ar < p < br+1, ν is purely real and, hence, leads to bounded

solutions which are aperiodic, in general. When br < p < ar+1, ν is purely

imaginary and, hence, leads to unbounded solutions that grow exponentially

to infinity.

Since Eq. (1.3) is a second order linear differential equation, it has two

linearly independent solutions, in general, and all other solutions can be written

as a linear superimposition of these two solutions. Using Eq. (1.4), we can

write these two linearly independent solutions as

φ(τ) =∞∑

r=−∞

c2r cos (ν + 2r) τ

ψ(τ) =∞∑

r=−∞

s2r sin (ν + 2r) τ (1.5)

where c2r and s2r are real constants to be determined. The reason for which

we are able to write the linearly independent solutions as a pure cosine series

and a pure sine series is that the product of a cosine with the term, cos 2τ ,

gives a sum of two cosines and the product of a sine with the term, cos 2τ ,

gives a sum of two sines. Substituting φ(τ) from Eq. (1.5) into Eq. (1.3), and

equating coefficients of the same frequency, we get

[p− (ν + 2r)2] c2r = q [c2r+2 + c2r−2] (1.6)

which is a recurrence relation for the coefficients c2r. For s2r also we get the


same recurrence relation as Eq. (1.6) and, thus, s2r = c2r.

For a one sided infinite sequence of coefficients (w0, w1, w2, ...), what we do

to solve the corresponding recurrence relation is start from w0 and keep solving

for the coefficients of higher index one by one. However, the coefficients, c2r,

form a two sided infinite sequence and, hence, Eq. (1.6) cannot be solved

in the same way as we solve recurrence relations for one sided sequence of

coefficients. This is because if we start from c0 or any other index, and proceed

in the two directions, the sequence will blow up as we move further. To solve

such recurrence relations, what is done is to truncate the sequence at some

r = ±R and move towards r = 0 from the two opposite directions. These two

directions yield two different values for c0. On reaching r = 0, the values of

the two sequences, for negative and positive r, are normalized so as to yield

c0 = 1. The value of r = ±R where the series should be truncated depends on

the desired accuracy. For low enough values of q ∈ (0, qc), R = 8 yields good

results.

For p, ν 1 and very large values of r, Eq. (1.6) becomes

−4r2c2r = q [c2r+2 + c2r−2] (1.7)

⇒ qc2r+2 + 4r2c2r + qc2r−2 = 0

Taking r ≈ constant, we have c2r ≈ αr, which implies

α2 +4r2

qα+ 1 = 0


⇒ α = −2r2

q±

√4r4

q2− 1

= −2r2

q± 2r2

q

(1− q2

4r2

)1/2

≈ −2r2

q± 2r2

q

(1− q2

8r2

)since

q2

r2 1

= −q4,−4r2

q+q

4(1.8)

Thus, we always have one unstable root, |α| > 1. Our goal is to construct a

series that converges, i.e. |c2r| → 0 as r → ±∞. So, we start at r = ±N with

arbitrary cN , cN+1 and work backwards. From Eq. (1.8), we have

c2r = Aαr1 +Bαr

2

where |α1| < 1 and |α2| > 1. The solution at r = N is

AαN1 +BαN

2 = c2N = 1

AαN+11 +BαN+1

2 = c2N+2 = 0

Since |α2| > 1, B → 0 is essential as N → ∞. Actually, we should have

B = O[(c2N , c2N+2)

/αN

2

]. Hence, only the decaying solution is seen as shown

in Fig. (1.1). These two branches must agree at r = 0:

1. c0 predicted by both should be same. This fixes cN and c−N in terms of

c0.

2. The recurrence relation involving c−2, c0, c2, i.e. [p− ν2] c0 = q [c2 + c−2]

must hold.

But c2, c−2 are fixed for a given c0. So, this condition on the coefficients

c2, c0, c−2 is independent of c0. This implies that this is a consistency condition

on ν.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -3 -2 -1 0 1 2 3 4

c 2r

r

Figure 1.1: This figure shows how the coefficients, c2r decay as r → ±∞. Ascan be seen, the coefficients decay exponentially fast.


Thus, to be able to solve Eq. (1.6), we also need to know the value of ν.

There are two ways of doing this. One is to use a method of continued fractions

[8]. This method is, however, very cumbersome. For low enough values of

p, q, it is preferable to use perturbation techniques like the Lindstedt-Poincare

method [7], which gives,

p = ν2 +q2

2 (ν2 − 1)+

(5ν2 + 7) q4

32 (ν2 − 1)3 (ν2 − 4)+O

(q6)

When p, q 1, the above equation can be simplified to give

ν =

√p+

q2

2(1.9)

which is the same as derived in Eq. (1.16).

As an example, consider the case of q = 0.16 and p = −0.01 . This case

corresponds to a field given by

E(x, τ) = −(m/e) [−0.01 + 0.32 cos(2τ)]x

The high frequency response should be roughly δx ≈ −0.08x0 cos(2τ). Thus,

this is a regime where |δx| |x0|, as required for conventional Ponderomotive

theory to be valid. For this case, the expressions for φ and ψ are,

φ(τ) ≈ cos(νt)− 0.03789 cos(ν + 2)τ − 0.04211 cos(ν − 2)τ (1.10)

+0.0003688 cos(ν + 4)τ + 0.0004322 cos(ν − 4)τ

ψ(τ) ≈ sin(νt)− 0.03789 sin(ν + 2)τ − 0.04211 sin(ν − 2)τ

+0.0003688 sin(ν + 4)τ + 0.0004322 sin(ν − 4)τ

where ν = 0.0529.


-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200

x

t

Figure 1.2: The solid line shows the plot of the path of the particle as pernumerical integration of Eq. (3.1). The dashed line shows the path as predictedby the mathematical expression given in Eq. (1.10), considering only the firstthree terms. This was done for q = 0.16, p = −0.01 and the initial conditionsare x0 = 0.91918, v0 = 0.We can see that there is close agreement betweenmathematics and simulation and the little discrepancy can be reduced furtherby considering higher order terms in the mathematical expression.

The solid line in Fig. (1.2) displays the plot of numerical solution of Eq.

(1.3) for q = 0.16, p = −0.01 and initial conditions x0 = 0.91918, v0 = 0.

The dashed line is the analytic solution as given in Eq. (1.10) considering the

first three terms. This reveals the simple structure lying behind these complex

looking orbits. There is a small mismatch between the two plots which is due to

the fact that the full analytic solution contains infinite number of terms and

accuracy can be improved by including more terms in the expression. This

regime is clearly one where the δx due to the high frequency wave is small

compared to the distance between turning points. The phase space plot for

this case is shown in Fig. (1.3). When q becomes large, |δx| ∼ |x0| and the


assumptions leading to the derivation of the ponderomotive force expression

become invalid.

-0.2

-0.1

0

0.1

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

v

x

Figure 1.3: This is the phase space plot of the trajectory shown in Fig. 1.2.We can see that near x = 1 and x = −1, the particle is undergoing highfrequency oscillations. This region of phase space near the turning point iswhere ponderomotive theory is applicable.

The general solution to Eq. (1.3) is

x(τ) = Dφ(τ) + Eψ(τ) (1.11)

where D and E are constants that depend on the initial conditions, namely

the particle position and velocity at τ = 0 .

As can be seen in Eq. (1.10), the co-efficients in the expression for φ

and ψ that really matter are c0, c2 and c−2 . From Eq. (1.6), for r 0,

|c2r/c0| ≈ O(q|r|/4r2). Thus, c0 dominates over the other coefficients since

q 1. The particle trajectory is a low frequency sinusoid (both cosine and


sine components) with frequency equal to ν, which is irrationally related to w

in general (in our normalization w = 2), and two low-amplitude high frequency

components, of frequency w + ν and w − ν, superimposed on that and other

higher frequencies as can be seen from the expressions in Eq. (1.5). The low

frequency path is the one predicted by the ponderomotive force expression.

1.3 Conventional Ponderomotive Theory

Equation (1.2) belongs to a more general class of force equations where the

particle is under the effect of an arbitrary DC field and an oscillatory field,

x = h(x) + g(x) cosωt (1.12)

where h(x) and g(x) are smooth but otherwise arbitrary functions of the spatial

coordinate, x. If the spatial gradient of the oscillatory component, g(x), is

small enough, the particle trajectory, x(t), can be written as a superposition

of small amplitude high frequency oscillations, x1, on a slow time averaged

component, xa [9]. Substituting x = xa + x1 into Eq. (1.12) and Taylor

expanding h(x) and g(x) about xa, we get,

xa + x1 = h (xa) + x1h′ (xa) + g (xa) cosωt+ x1g

′ (xa) cosωt (1.13)

Equating the high frequency components of the above equation, we get,

x1 = x1h′ (xa) + g (xa) cosωt


Assuming that h′ (xa) and g (xa) change on a time scale large compared to

2π/ω, the above equation can be solved to give,

x1 = −g (xa) cosωt

ω2 + h′ (xa)

Substituting the above expression for x1 in Eq. (1.13) and time averaging the

resulting equation, we get,

xa = h (xa)−1

2

g (xa) g′ (xa)

ω2 + h′ (xa)

Assuming ω2 h′ (xa), we get,

xa = h(x)− 1

4ω2

dg2(x)

dx

∣∣∣x=xa

(1.14)

The time averaged ponderomotive Hamiltonian corresponding to Eq. (1.14) is

v2

2+ φ0(x) +

g2(x)

4ω2= Hp (1.15)

where φ0 is the potential corresponding to the force h(x), i.e. h(x) = −dφ0

/dx.

Thus, ponderomotive theory gives an equation for the time averaged motion

of a particle under the effect of spatially non-uniform oscillatory forces.

Applying Eq. (1.14) to Eq. (1.2), we get the following time averaged

equations

xa =

[−pe −

q2e

2

]xa (1.16)

ya =

[−pe −

q2e

2

]ya

za =

[− (−2pe)−

(−2qe)2

2

]za

Thus, to trap the plasma, all we need to do is to choose the trap parameters


in such a way that pe + 0.5q2e > 0 and −2pe + 2q2

e > 0.

The special case when g(x) and h(x) are proportional to x has applications

in 2d linear RF traps [2], [3], 3d ion traps [1] and combined RF traps [10],

[11], [12]. The linear RF trap and ion trap are mechanisms for confining single

species plasma and are widely used in mass spectroscopy [13] and also have

applications in areas like quantum information [14] where a single or very few

charged particles have to be trapped. The combined RF trap is used to trap two

different charged species of widely varying masses in the same region of space

and is mainly used in anti-hydrogen production. These traps take advantage

of the fact that the boundedness of particle orbits in a linearly varying RF

field is independent of the initial conditions. It only depends on the applied

field strength.

1.4 Ponderomotive Density

Depending on the particular application, the number of ions in a Paul trap can

range from just a few to about 103 or more. If there are only a few ions in the

trap, then its enough to solve the force equations, Eq. (1.2), and keep track

of individual particles. But if the number of ions is larger, then it becomes

important to know the distribution function and density of the plasma in the

trap.

It is well known in equilibrium statistical mechanics that if a system of

particles is under the effect of time independent forces, x = −dφ(x)/dx, then

the density of particles follows a Boltzmann distribution [15] given by

n(x) = n0 exp (−β0φ) (1.17)

where β0 is a measure of the temperature. Equation (1.14) describes the time

averaged motion of the particle under the effect of a force equation, Eq. (1.12).


This equation can be written as

xa = −dφp

dx

∣∣∣x=xa

(1.18)

where φp = φ0 + g2/4ω2 is the effective ponderomotive potential. Equation

(1.18) along with the Boltzmann density, Eq. (1.17), has led to the conjecture

[9] that the time averaged distribution function of the plasma, under the force

equation Eq. (1.12), will be given by

n(x) = n0 exp (−β0φp) (1.19)

Conventionally, equation (1.19) is believed to follow from the conjecture that

the time averaged distribution function of the plasma is constant on curves

of time averaged motion of the individual particles [16]. However, Eq. (1.19)

was formally derived by Krapchev [17] for the case when the DC field is not

present, i.e., h(x) = 0 in Eq. (1.12), by proceeding directly from the Vlasov

equation. This result is discussed in detail in Sec. (2.2).

1.5 Paul trap experiments

Though the theoretical understanding of plasma in Paul traps has a long way

to go, extensive experiments have been done on Paul traps and the results

have been reported in the literature. From among all the published data, the

observables that matter most are the following:

1. Confinement time

2. Maximum number of particles that can be confined

3. Spatial density profile of the plasma

4. Plasma temperature


In an experiment on Ba+ ions, it was reported that the confinement times in

an ideal Paul trap exceeded one hour [18] when the background gas consisted of

He with a pressure below 10−6 mbar, while the partial pressure of the heavier

elements was below 10−9 mbar. A higher background gas pressure, however,

triggers a diffusion process and this leads to a loss of most of the confined

ions. Confinement times were also found to reduce to about 10-15 minutes

when a horizontal and vertical slit was cut into the ring electrode to allow

laser scanning.

It has been observed in RF trap experiments that the plasma temperature is

higher than the background gas temperature [19]. To explain this observation,

it was conjectured that RF field causes heating in the plasma and collisions

with the background gas cause the plasma to cool down. The measured plasma

temperature is thought to be an equilibrium between these two opposing mech-

anisms. In agreement with this conjecture, the plasma temperature has been

found to decrease when the background gas pressure is increased[18]. However,

the background gas pressure can be increased only up to about 5× 10−6 mbar

since beyond this, the plasma is observed to be lost rapidly. Though it is widely

accepted that an RF field in a Paul trap leads to heating and there also have

attempts to explain this effect [20] [21], there is no convincing proof available

in the literature. For the case of highly localized RF fields, a mechanism for

RF heating was proposed in [22]. This will be discussed in later chapters.

The ubiquitous RF heating has also been proposed as a mechanism for

achieving fusion [23]. The Phaedrus tandem mirror experiment had a central

cell and two end plugs. RF fields were sent into the chamber using one antenna

at the central cell and two more antennas for the two end plugs respectively.

The static magnetic field in the central cell was so high that no resonance heat-

ing was caused by the RF fields. However, when an ion reached the end plugs,

the RF field caused resonant heating leading to an increase in the ion energy in


a direction perpendicular to the axis of the apparatus. This led to the particles

being turned back into the central cell, hence leading to plasma confinement.

In 1988, it was proposed [24] that future experiments on the Phaedrus will

concentrate on scaling of the RF stabilization caused due to ponderomotive

effects. These ponderomotive effects were thought to result from the electro-

static waves excited in the bulk plasma due to the electromagnetic RF fields.

However, these effects were not observed in the Phaedrus-B experiment [25].

Thus, there are two main differences between the Phaedrus experiment and

the Paul trap. Firstly, in the Phaedrus experiment, the RF field was of an

electromagnetic wave unlike the electrostatic field in a Paul trap. Secondly, in

the Phaedrus experiment, the RF field was not used for canonical confinement

of the plasma. Due to these differences between Phaedrus and the Paul trap,

we do not consider the Phaedrus experiment in this thesis.

Another issue of importance in an actual RF trap experiment is the maxi-

mum amount of plasma that can be trapped. As it turns out, this depends on

the parameters of the applied field. From the point of view of ponderomotive

theory, it is easy to show that the plasma frequency, ωp =√

4πn0e2/m, must

be less than the slow frequency, ν0, that characterizes the drift motion of the

particles. Consider a plasma trapped by RF fields. The time averaged plasma

density contributes to a repulsive force on particles given by

e

mEx(x) =

x

0

ω2pdx

′ ' ω2px

for a nearly uniform density profile. Given a trap characterized by

e

mEext,x(x) = (−pe + 2qe cos 2t)x


the space charge adds to the DC field to yield the following orbit equation

x =(−pe + ω2

p + 2qe cos 2t)x

From Eq. (1.14), we know that ponderomotive theory turns this into the

following equation in slow time

xa =

(−pe + ω2

p −q2e

2

)xa (1.20)

Thus, for stable orbits, the plasma frequency must be bounded above by

ν0 =√pe + 0.5q2

e . This is exactly what is observed in experiments where

the maximum plasma frequency is found to be close to ν0. It has also been

reported that the plasma density in the trap is highest for trap parameters,

pz = −0.03 and qz = 0.55 [26]. The maximum number of protons that can

be trapped has been observed to be of the order of 102 [27]. This corresponds

to the density of around 105 cm−3, which in turn corresponds to a plasma

frequency of ωp = 0.17 MHz. The slow frequency corresponding to the values

pz = −0.03 and qz = 0.55 is approximately ν0 = 0.5 MHz. Thus, these experi-

mental parameters satisfy the condition ωP < ν0. This condition, however, has

not been explicitly mentioned in the literature. These ideas that are based on

naive ponderomotive theory are given a more rigorous treatment in this work.

Finding the spatial density profile of the plasma in a Paul trap experiment

directly is not an easy task. This is done indirectly by measuring the fluo-

rescence intensity along a particular direction. Using this technique, it was

observed in the same experiment that the plasma density is a Gaussian[18],

[28], which is what is expected according to conventional ponderomotive the-

ory, Eq. (1.19), in the presence of a linear RF field. As the particle number

goes up, the induced field in the trap also goes up. This induced field is not

exactly linear and, hence, distorts the plasma density away from a Gaussian


profile. However, there is an upper limit to the amount of distortion since, as

discussed above, there is only a finite amount of plasma that can be trapped by

this mechanism. Thus, one question of interest is the extent of this distortion

from Gaussianity when the particle number in the trap approaches its upper

limit.

Thus, from the experimental point of view, there are mainly three questions

that we would like to answer through our work:

1. What could be a possible mechanism for the RF heating observed in Paul

traps?

2. What is the maximum number of particles that can be confined in a Paul

trap with given parameters?

3. How much can the plasma density deviate from Gaussianity as the par-

ticle number approaches its upper limit?

Chapter 2

Introduction to Perturbation

Methods

As mentioned before, there are two aspects of studying plasma dynamics in

non-uniform RF fields. One is solving for the individual particle orbits. And

second is obtaining expressions for the distribution function and density of the

plasma. Since the plasma density is simply the integral of the distribution

function over velocity space, there are two equations involved. One equation

for the particle orbits, Eq. (1.12), and the other is for the distribution function,

i.e. Vlasov equation [9],

d

dtf (x, v, t) =

∂f

∂t+ v

∂f

∂x+E(x, t)

m

∂f

∂v= 0 (2.1)

The force equation for individual particles, Eq. (1.12) and the Vlasov

equation, Eq. (2.1), for the plasma as a whole are not independent equations.

The solution of Eq. (2.1) is constant over the particle orbits obtained by

solving by Eq. (1.12). Thus, the solution of Eq. (2.1) is an arbitrary function

of the fundamental invariants associated with Eq. (1.12), namely the initial

conditions.

There are two methods of solving Eq. (2.1),

19

CHAPTER 2. INTRODUCTION TO PERTURBATION METHODS 20

1. Solve Eq. (2.1) directly without making any reference to the solutions

of Eq. (1.12)

2. Solve Eq. (1.12) to obtain analytic expressions for the particle trajec-

tories, x(t) and v(t), and then combine the solutions in order to find a

suitable invariant. This is what we have done in the thesis.

When h(x) and g(x) are linear functions of the spatial coordinate, x, the force

equation, Eq. (1.12), becomes the well known Mathieu’s equations (Sec. (1.2))

. For this case, solutions for the expressions for particle orbit to all orders are

well known and as we will show in the next chapter, these solutions can be

combined to obtain an exact expression for the distribution function. However,

when h(x) or g(x) or both are nonlinear, though we can solutions to Eq. (1.12)

using certain perturbation techniques, these solutions are only asymptotically

valid because the solutions of nonlinear equations suffer from the problem of

small denominators [29]. And for nonlinear equations, it is also very difficult

to combine the resulting solutions to obtain invariant expressions.

Thus, it would be very nice if we could directly solve Eq. (2.1) without

having to bother about obtaining solutions to Eq. (1.12). If the electric field is

time independent, then the maximum entropy solution to Eq. (2.1) is simply

given by the Maxwell-Boltzmann distribution[15]

f(x, v) = n0

√β0

2πexp

[−β0

(1

2v2 + φ(x)

)](2.2)

where φ(x) is the electric potential. Equation Eq. (2.2) is, however, not a self-

consistent solution. Self-consistent solutions of a plasma in a time independent

electric potential are known as BGK Modes [30]. However, if the electric field,

E(x, t) varies with time, the distribution function cannot be time indepen-

dent. And as of now, there are no known methods of obtaining exact analytic

solutions to Eq. (2.1) when the electric field is time dependent.


If the electric field is periodic in time, it is reasonable to look for “station-

ary” solutions of the distribution function that are periodic in time with the

same time period. A solution of Eq. (2.1) for time periodic electric fields is

not plagued by the problem of small denominators (discussed in Sec. (2.1.4))

because there is no slow drift frequency involved in the distribution function.

The single particle orbits, however, do have a slow drift frequency which leads

to the small denominator problem for nonlinear fields. This problem of solv-

ing the distribution function directly was addressed by Krapchev [17] and is

discussed in this chapter.

In this chapter, we will first review the standard techniques used in solving

nonlinear equations of motion and then discuss Krapchev’s method to solve

the Vlasov equation.

2.1 Solving Nonlinear Equations

Solving nonlinear equations can be very tricky. Application of regular per-

turbation expansions leads to resonances and secular terms [31]. To see this,

consider the Duffing’s equation

d2y

dt2+ ω2y + εy3 = 0 (2.3)

Substituting a perturbative expansion of y

y =∞∑

n=0

εnyn

into Eq. (2.3), and equating terms with like powers of ε, we get,

y0 + ω2y0 = 0


⇒ y0 = A cos (ωt+ φ)

y1 + ω2y1 + y30 = 0

⇒ y1 + ω2y1 = −A3 cos3 (ωt+ φ)

= −A3

4[cos (3ωt+ 3φ) + 3 cos (ωt+ φ)]

As can be seen, the presence of cos (ωt+ φ) term on the right hand side of the

above equation is in resonance with the left hand side and leads to unbounded

solutions for y1. Such terms are known as secular terms. It is known that

application of regular perturbation methods to nonlinear equations leads to

a convergent series solution for y(t) in powers of ε [31]. Though each of the

terms of this series, yn, is unbounded, summing up each of these unbounded

terms to all orders in ε leads to a bounded solution as must be the case for

Eq. (2.3). However, the presence of these unbounded terms implies that this

is not a valid perturbation series. That is because, in a valid perturbation

series solution, one should be able to truncate at some point depending on the

amount of accuracy required. Such a truncation is, however, not possible with

the solutions obtained by this method. Also, although summing up of the all

the terms to all orders in the regular series expansion of y(t) leads to bounded

solutions, the procedure required is very lengthy.

There are three methods by which bounded solutions can be obtained with-

out requiring such lengthy calculations

1. Multiple Scale Method

2. Lindstedt-Poincare (LP) Technique

3. Modified Lindstedt-Poincare (MLP) Technique


2.1.1 Multiple Scale Methods

As shown in Sec. (2.1), an application of regular perturbation series to Eq.

(2.3) leads to secular terms. This is because, through this naive method, the

solution of Eq. (2.3) to zeroth order is y0 = A cos (ωt+ φ). Multiple scale

analysis aims at circumventing this problem by seeking solutions where A, φ

are also functions of time. However, to avoid confusion, a new time scale, τ , is

introduced which varies slowly compared to the normal time scale, t. Thus, in

multiple scale analysis, an artificial time τ = εt is introduced in the problem

and y is written as a function of both the actual time, t, and the new time, τ ,

i.e. y = y(t, τ). This new function, y, is then written as a power series in ε

y(t, τ) = Y0(t, τ) + εY1(t, τ) + ... (2.4)

and the total derivative d/dt is written in terms of partial derivatives, ∂

/∂t

and ∂/∂τ , by using the chain rule

d

dt=

∂

∂t+ ε

∂

∂τ

⇒ dy

dt=∂Y0

∂t+ ε

(∂Y0

∂τ+∂Y1

∂t

)+ ...

⇒ d2y

dt2=∂2Y0

∂2t+ ε

(2∂2Y0

∂t∂τ+∂2Y1

∂τ 2

)+ ... (2.5)

We now substitute Eq. (2.4) and Eq. (2.5) into Eq. (2.3) and equate terms of

the same order in ε. The behavior of y as a function of τ can now be chosen

at every order such as to eliminate secular terms. This is the whole idea of

introducing an artificial time, τ . After we have obtained solutions for y(t, τ),

we substitute τ = εt to get the final solution, y = y(t).

Up to O (ε0), we have

∂2Y0

∂t2+ ω2Y0 = 0


⇒ Y0 = A(τ)eiωt + A∗(τ)e−iωt

where A(τ) is an arbitrary function of τ . Up to O (ε1), we have,

∂2Y1

∂t2+ ω2Y1 = −Y 3

0 − 2∂2Y0

∂t∂τ

= −(A(τ)eiωt + A∗(τ)e−iωt

)3 − 2∂2

∂t∂τ

(A(τ)eiωt + A∗(τ)e−iωt

)= −

(A3e3iωt + A∗3e−3iωt + 3A2A∗eiωt + 3AA∗2e−iωt

)−2(iωA′eiωt − iωA∗′

e−iωt)

where the prime denotes differentiation with respect to τ . Thus, to eliminate

secular terms, we choose,

−3A2A∗ − 2iωA′ = 0

To solve the above equation, we write A in the polar form, A(τ) = R(τ)eiθ(τ) ,

−3R3eiθ − 2iω (R′ + iθ′R) eiθ = 0

Equating the real and imaginary parts, we get,

dR

dτ= 0

dθ

dτ=

3

2ωR2

⇒ θ =3

2ωR2(0)τ + θ(0)


Thus, we have

Y0 = Aeiωt + A∗e−iωt

= 2R(0) cos

(3

2ωR2(0)τ + θ(0) + ωt

)= 2R(0) cos

(ω

(1 +

3ε

2ω2R2(0)

)t+ θ(0)

)(2.6)

Similarly, we can obtain the higher order terms of y in Eq. (2.4) by choosing

the behavior of each Yn as a function of τ to eliminate secular terms.

Though the method of multiple scale is quite elegant and gives bounded

solutions for the Duffing’s equation, Eq. (2.3) at every order, it is still quite

lengthy. There is another method known as the Lindstedt-Poincare technique

that gives bounded solutions to Eq. (2.3) in a still simpler way.

2.1.2 Lindstedt-Poincare (LP) Method

As mentioned in the Sec. (2.1.1), the multiple scale analysis is effective but is

very lengthy. To make the procedure simpler, instead of assuming that A, φ

are functions of a new time scale, τ , we begin, in the LP method, by seeking

solutions of the type y0 = A cos (Ωt+ φ), where Ω is obtained order by order

as a power series in ε. This method works for the class of nonlinear equations

where the zeroth order force equation is that of simple harmonic motion. This

is because, for these types of problems, the new time scale, τ , in multiple

time analysis appears as an additive term to the usual time scale, t. Thus,

unlike the multiple scale analysis, where an additional artificial time scale is

introduced, in the Lindstedt-Poincare method [32] the time, t, is transformed

to a new time, τ = 2πt/T = Ωt, where T is the time period of the solution, y.

Substituting this in the Duffing’s equation, Eq. (2.3), we get,

Ω2 d2y

dτ 2+ ω2y = −εy3 (2.7)


Now, both the oscillation frequency, Ω, and y are written as a power series in

ε,

Ω2 = α0 + εα1 + εα2 + ...

y = y0 + εy1 + εy2 + ...

These expansions are then substituted in the scaled equation, Eq. (2.7), and

the terms, αn are chosen such as to eliminate secular terms.

Up to O (ε0), we have

α0d2y0

dτ 2+ ω2y0 = 0

⇒ y0 = A cos (τ + φ)

and α0 = ω2, since the time period of motion in the new time coordinate, τ ,

is 2π. To O (ε1), we have

α0d2y1

dτ 2+ α1

d2y0

dτ 2+ ω2y1 = −y3

0

⇒ ω2d2y1

dτ 2− α1A cos (τ + φ) + ω2y1 = −A

3

4[cos (3τ + 3φ) + 3 cos (τ + φ)]

To eliminate secular terms, we require,

α1 =3A2

4

This gives,

d2y1

dτ 2+ y1 = − A3

4ω2cos (3τ + 3φ)

⇒ y1 =A3

32ω2cos (3τ + 3φ)


Thus, the solution of Eq. (2.3), up to O (ε1) is given by

y(t) = A cos (Ωt+ φ) +εA3

32ω2cos (3Ωt+ 3φ) (2.8)

where

Ω =

√ω2 +

3εA2

4

Thus, it can be clearly seen that the Lindstedt-Poincare method is much easier

to implement than the multiple scale method.

One problem associated with perturbation methods for nonlinear equations

is the question of convergence. It is well known [31] that the perturbation

solutions obtained for nonlinear equations are only asymptotically valid. Thus,

it is of interest to come up with methods that give rise to solutions that are

valid for higher and higher orders in the expansion parameter.

2.1.3 Modified Lindstedt-Poincare (MLP) Method

A modified version of the Lindstedt-Poincare method is also known in the

literature and is widely used to solve the linear Mathieu equation and its

nonlinear modifications [33]. In the LP method described in Sec. (2.1.2), we

solve for the particle orbit frequency, Ω, for a given ω. However, in the MLP

method we assume a given particle frequency, ν, and solve for the ω that would

lead to this value of ν. Thus, in this method, instead of transforming to a new

time scale, the ω2 term in Eq. (2.3) is written as a power series in ε,

ω2 = ν2 + εβ1 + ε2β2 + ... (2.9)

along with

y = y0 + εy1 + ε2y2 + ...


where, ν, is the frequency of motion and the βns are chosen such as to eliminate

secular terms order by order. Substituting this in Eq. (2.3) and equating the

terms of O (ε0), we get,

d2y0

dt2+ ν2y0 = 0

⇒ y0 = A cos (νt+ φ)

Equating the terms of O (ε1), we get,

d2y1

dt2+ ν2y1 + β1y0 = −y3

0

⇒ d2y1

dt2+ ν2y1 + β1A cos (νt+ φ) = −A

3

4[cos (3νt+ 3φ) + 3 cos (νt+ φ)]

Thus, we choose,

β1 = −3A2

4

to get,

y1 =A3 cos (3νt+ 3φ)

32ν2

Thus, up to O (ε1), the solution to Eq. (2.3) obtained by the modified LP

method is

y(t) = A cos (νt+ φ) +εA3

32ν2cos (3νt+ 3φ) (2.10)

where

ν =

√ω2 +

3εA2

4(2.11)

Thus, though the oscillation frequency obtained by the LP method and the

modified LP method are the same, but the denominators in the second term

of the solutions, Eq. (2.8) and Eq. (2.10), are different. The solutions as those

in Eq. (2.10) can also be obtained by using a method proposed by Amore and

Aranda [32].

For solving periodically forced systems of equations, the MLP method has

immense advantages over the LP method. In the LP method one has to trans-


form to a new time scale, and hence the time dependence of the forcing term

also undergoes a transformation. This leads to unnecessary complications. On

the other hand, in the MLP method we do not need to transform to a new

time scale. We assume a certain breathing frequency for the particle orbits

and solve for the parameters in the equation that would lead to the assumed

temporal behavior.

2.1.4 Small Denominator Problem

As can be seen in Eq. (2.10), the second term has integral powers of ν2 in

their denominator. Now, in these problems, ν is a small number, and thus,

presence of higher powers of ν leads to what is known as the small denominator

problem [29]. In Eq. (2.11), if 0.75εA2 ω2, the second term in Eq. (2.10) is

of O (ε) and this leads to a series that is asymptotically convergent. However,

if 0.75εA2 ≈ ω2, then the second term is of O (ε0) which leads to convergence

problems when the series is summed up. Thus, the small denominator problem

severely limits the convergence of the series obtained by perturbation methods

described above.

However, solving the single force equation is only a step towards the main

problem of solving the Vlasov equation to obtain the distribution function of

the plasma. If the Vlasov equation could be solved directly by some perturba-

tion method, the solutions would be free of the problem of small denominators.

This is because, unlike the single particle trajectory, which is aperiodic, we seek

to solve for a distribution function that has the same time dependence as the

applied field. If the field is time independent, we seek a time independent

distribution function and if the applied field is periodic, we seek a distribution

function that is periodic in time with the same time period, 2π/ω, as the RF

field. For example, as mentioned above, the solutions to Eq. (2.3) given by Eq.

(2.10) suffer from the problem of small denominators. However, the invariant


corresponding to the force equation, Eq. (2.3), is

I (x, v) =v2

2+ω2x2

2+εy4

4(2.12)

and the distribution function is any arbitrary function of this invariant, Eq.

(2.12). As can be seen, Eq. (2.12) is exact and does not suffer from any

convergence issues.

2.2 Krapchev’s Method

To the best of our knowledge, V. B. Krapchev [17] is the only one before us to

have made an attempt at rigorously solving the Vlasov equation for the field

prescribed in Eq. (1.12). However, he considered the case when h(x) = 0.

For particles under a force, g(x) cosωt, the Vlasov equation for the plasma

distribution function, f(x, v, t), is given by

∂f

∂t+ v

∂f

∂x+ g(x) cosωt

∂f

∂v= 0 (2.13)

The individual particle orbits under a spatially non-uniform time periodic

force will in general have a slow drift motion as shown in Eq. (1.14). However,

it is reasonable to expect the distribution function, f(x, v, t), to be be time

periodic with the same period as the forcing function. Krapchev used this

basic concept and expanded the distribution function as a harmonic series,

f(x, v, t) = f0(x, v) +∞∑

n=1

fn(x, v)e−jnωt +∞∑

n=1

f ∗n(x, v)ejnωt (2.14)

where f0(x, v) is the time averaged distribution function. Substituting Eq.


(2.14) into Eq. (2.13), we get,

−∞∑

n=1

jnωfne−jnωt +

∞∑n=1

jnωf ∗nejnωt + v

∂f0

∂x+

∞∑n=1

v∂fn

∂xe−jnωt +

∞∑n=1

v∂f ∗n∂x

ejnωt

+g(x)ejωt + e−jωt

2

(∂f0

∂v+

∞∑n=1

∂fn

∂ve−jnωt +

∞∑n=1

∂f ∗n∂v

ejnωt

)= 0

From the above equation, using the orthogonal properties of the harmonic

functions, we get an infinite set of equations,

v∂f0

∂x= −g(x)

2

(∂f1

∂v+∂f ∗1∂v

)(−jnω + v

∂

∂x

)fn = −g(x)

2

(∂fn−1

∂v+∂fn+1

∂v

)(2.15)

Up to here, all the steps carried out are exact. However, to proceed further, one

needs to make certain assumptions. Krapchev made two crucial assumptions.

Firstly, he assumed that

v

nωfn

∂fn

∂x 1 (2.16)

which when applied to Eq. (2.15) led to

fn ≈1

jnω

(1 +

v

jnω

∂

∂x

)g(x)

2

(∂fn−1

∂v+∂fn+1

∂v

)(2.17)

Ignoring the higher derivatives of fn in Eq. (2.17) by using Eq. (2.16) is valid

while solving for the lowest order expressions for the various harmonics. But

when one goes to the higher orders, one has to successively introduce more and

more derivatives of fn in Eq. (2.17). Secondly, he made the assumption that

there is a ordering in the fns with fn+1 being one order higher, in the applied

field magnitude, than fn. The consequence of this assumption was that f0 can

be written as a series expansion in even powers of g(x). On summing to all


orders, this led him to a time averaged distribution function

f0(x, v) =n0

2πvT

exp

(− g2(x)

4ω2v2T

) ∞

−∞exp

(jpv

vT

)exp

(−p

2

4

)J0

(pg(x)

vT

)dp

(2.18)

where vT is the thermal velocity. Integrating Eq. (2.18) with respect to veloc-

ity, yields the ponderomotive density expression,

n = n0 exp

(− g2(x)

4ω2v2T

)(2.19)

Though the time averaged plasma density obtained by Krapchev is the

same as that predicted by conventional ponderomotive theory, there is a big

difference in the predictions regarding the underlying distribution functions.

Conventional ponderomotive theory assumes that the time averaged distribu-

tion function is constant on the curves of time averaged motion. This leads to

a distribution function that is Maxwellian at all points in space. However, the

time averaged distribution function obtained by Krapchev is not a Maxwellian

at any point in space. Also, Eq. (2.18) leads to a double humped distribution

function beyond a certain threshold in space.

Krapchev then used Eq. (2.18) to obtain an expression for the plasma

temperature,

T = T0

(1 +

g2(x)

ω2v2T

)(2.20)

which is spatially non-uniform.

The assumptions and results of Krapchev can be summarized as follows:

1. The plasma distribution function has small spatial gradients,(v/nωfn

)∂fn

/∂x 1

2. The time averaged distribution function can be written as a series ex-

pansion in the even powers of g(x)


3. The time averaged distribution function is given by Eq. (2.18) and is

double humped beyond a certain threshold in space

4. The time averaged plasma density is given by Eq. (2.19) and is the same

as that predicted by conventional ponderomotive theory

5. The plasma temperature is given by Eq. (2.20) and is spatially non-

uniform

In order to solve for the plasma distribution in Paul traps, the ideal thing to

do would be to proceed along the lines of Krapchev’s paper up to Eq. (2.15)

and then solve these recurrence relations in the correct way. However, it is

not clear what this correct way could be. Thus, it becomes necessary to adopt

other ways of solving for the distribution function. For the case of linear RF

fields, we have used the solutions of the single particle orbit to construct the

plasma distribution function. However, for the case of nonlinear RF fields,

there didn’t seem to be a way of constructing the distribution function from

the single particle orbit solutions. For this case, we have directly solved the

Vlasov equation perturbatively. Our method is free from all inconsistencies, in-

cluding those present in Krapchev’s results. Though this perturbation method

of solving the Vlasov equation could have been used for the case of linear RF

fields also, we have used the solutions for the single particle orbits mainly for

two reasons:

1. Constructing the distribution function from single particle orbits gives an

understanding of the role that each particle plays in the over all plasma

dynamics.

2. It helps us in comparing our solutions to those of conventional pon-

deromotive theory. This is because conventional ponderomotive theory

is basically a single particle picture which has been generalized to the

whole plasma.


2.3 Contribution of this thesis

Ponderomotive theory is a very useful concept and is widely used in theoretical

analysis of RF traps, laser wake field acceleration [34], [35], [36] and also in

the study of wave interactions in the context of Zakharov equations [37]. The

equation for time averaged motion and the expression for time averaged density

simplify the analysis of such periodically driven systems. Though this theory

gives a lot of information about the time averaged behavior of particle paths

and the system as a whole, it gives no information about the detailed time

evolution. And without knowing the detailed time evolution of the plasma

distribution function, it is not possible to study the statistical properties of

system under consideration.

Though Eq. (1.19) was formally derived by Krapchev [17], it is not clear

if the assumptions made by him and the subsequent results are correct. It is

important to take some special cases, construct the complete time varying dis-

tribution function and density for the plasma and then compare the exact time

averaged density with the prediction of conventional ponderomotive theory.

In this thesis, we have taken up one dimensional charged particle motion

under the influence of the linear oscillatory electric field in a Paul trap. An-

alytic, nonlinearly exact expressions for the distribution function and plasma

density are obtained. This is then time averaged and compared with Eq.

(1.19). We find that the time average of the exact density is different from

the prediction of conventional ponderomotive theory. The plasma distribution

function is, however, found to be exactly a Maxwellian to all orders. This

makes the distribution function immune to collisions.

Though the applied field in a Paul trap is linear, the plasma inside the trap

itself induces a field that is nonlinear. The lowest order contribution of this

induced field is a DC cubic nonlinearity. We take this into account and solve

the find analytic expressions for the distribution function and density. Again,


the plasma is found to be a Maxwellian, but this time only up to the lowest

order in the nonlinearity. As we go to higher orders, the plasma ceases to be a

Maxwellian. Like the linear field case, the temperature is found to be spatially

uniform but oscillates with time at the same frequency as the applied field.

Towards the end, we have also solved for the distribution function of a

plasma under a nonlinear RF field. Unlike the case of linear RF field, a

nonlinear RF fields yields a non-Maxwellian distribution function even to the

lowest order in the RF field magnitude. In a Paul trap experiment, this non-

Maxwellianity of the distribution function will lead to relaxation processes

being triggered. This could be a potential cause for the heating observed in

Paul traps.

One of the most important contributions of this thesis is that we have

constructed the detailed time varying distribution function and shown that it

is Maxwellian at all instants of time when the RF field is linear. The plasma

temperature is spatially uniform but oscillates in time at the RF frequency.

Though the oscillatory nature of the temperature was conjectured in the past

[21], but, to the best of our knowledge, we are the first to provide a proof. For

the case of a nonlinear RF field, we have shown that the distribution function

is non-Maxwellian even to the lowest order in the RF field magnitude.

Chapter 3

Spatially linear DC and RF field

— I

In this chapter, we consider charged particle motion under the effect of the

externally applied linear field in a Paul trap, Eq. (1.2). As shown in Eq. (1.2),

since the equations of motion along the three directions are decoupled, we can

solve only one of them and the methods used can then be easily extended to

the full 3d problem. Thus, in this chapter, we are effectively going to solve for

a 1d problem. The normalized equation of motion can be written as

d2x

dτ 2= [−p+ 2q cos(2τ)]x (3.1)

which is the well known Mathieu equation and whose solutions were discussed

in the previous chapter.

36

CHAPTER 3. SPATIALLY LINEAR DC AND RF FIELD — I 37

3.1 Time evolution of the distribution func-

tion and density

As mentioned before, to study the statistical properties of the plasma, we need

to find an expression for the distribution function. The distribution function,

f(x, v, τ) is a solution of the Vlasov equation

∂f

∂τ+ v

∂f

∂x+ [−p+ 2q cos(2τ)]x

∂f

∂v= 0

This is a very simple problem conceptually, since the electric field is not

linked back to the distribution function. The methods of the previous chap-

ter can be immediately applied. As the particle moves in phase space under

the action of the applied field, it possesses two fundamental invariants which

are the initial conditions, x0, v0. And since the distribution function is also a

constant over the particle orbits, it must be a function of these fundamental in-

variants. For nonlinear equations, it is not easy to solve the force equation and

then invert the solutions to obtain the initial conditions, x0 (x, v, t) , v0 (x, v, t)

as time dependent functions of the position and velocity at a later time. How-

ever, since the applied force in a Paul trap is linear, this inversion is easy to

do. Thus, in this case, we can easily combine these initial conditions in a way

such as form a meaningful distribution function.

As shown in Chapter 1 (Sec. 1.2), Eq. (3.1) has two linearly independent

solutions, φ(τ) and ψ(τ), given by Eq. (1.5)

φ(τ) =∞∑

r=−∞

c2r cos (ν + 2r) τ

ψ(τ) =∞∑

r=−∞

c2r sin (ν + 2r) τ


where c2r are constants and ν is the slow frequency given by Eq. (1.9),

ν ≈√p+

q2

2

And any other solution of Eq. (3.1) can be written as a linear combination of

these two solutions. Thus, the expressions for x and v are,

x = D1φ (τ) +D2ψ (τ)

v =dx

dτ= D1φ

′ (τ) +D2ψ′ (τ) (3.2)

where D1, D2 are arbitrary constants and ′ represents differentiation with re-

spect to τ . Putting τ = 0 in Eq. (3.2), we get,

x0 = D1φ0 +D2ψ0

v0 = D1φ′0 +D2ψ

′0 (3.3)

where the subscript 0 refers to the value of the corresponding functions at

τ = 0. From Eq. (1.5), it is clear that φ (τ) contains only cosine terms and

ψ (τ) contains only sine terms. This means that φ′ (τ) will contain only sine

terms and ψ′ (τ) will contain only cosine terms. Thus, φ′0 = 0 and ψ0 = 0.

Substituting this in Eq. (3.3), we solve for D1 and D2, to get

D1 = x0/φ0

D2 = v0/ψ′0 (3.4)

Substituting this in Eq. (3.2) and solving for x0 and v0, we obtain,

x0 =1

ψ′0(xψ′ (τ)− vψ (τ)) (3.5)

v0 =1

φ0

(vφ (τ)− xφ′ (τ))


It is important to note that Eq. (3.5) is linear in x and v since φ (τ) and

ψ (τ) are functions of time alone. Since φ0ψ′0 is the non-zero Wronskian for

Eq. (3.1), the denominators in Eq. (3.5) are strictly non-zero.

3.1.1 General distribution function

The distribution function of the plasma at any time τ can now be written as,

f(x, v, τ) = f0 (x0 (x, v, τ) , v0 (x, v, τ))

where, f0(x0, v0) is the distribution function of the plasma at τ = 0 and the

functions x0 (x, v, τ) , v0 (x, v, τ) are given by Eq. (3.5). It must be noted

that f0(x0, v0) can be any arbitrary function of x0 and v0. And since we

have analytic expressions for x0 (x, v, τ) , v0 (x, v, τ), the problem is completely

solved from the mathematical point of view. However, from the physical point

of view, we need to find a distribution function that satisfies certain statistical

properties and also makes contact with conventional ponderomotive theory.

With this in mind, we choose the distribution function of the plasma at τ = 0 to

be f0(x0, v0) = F (0.5v20 + γ0x

20), where F is a smooth but otherwise arbitrary

function of its argument and γ0 > 0 is an arbitrary constant and is a measure

of the scale length of the plasma. Since x0, v0 are linear functions of x, v (Eq.

(3.5)) and we have chosen f0 such that it is a function of a quantity that is

quadratic in x0, v0, the time dependent distribution function, f(x, v, τ) will also

be quadratic in x, v. Using Eq. (3.5), the time evolution of the distribution


function is given by,

f(x, v, τ) = F

(1

2

[1

φ0

(vφ (τ)− xφ′ (τ))

]2

+ γ0

[1

ψ′0(xψ′ (τ)− vψ (τ))

]2)

= F

(1

2φ20ψ

′20

[a (τ) v2 + b (τ)x2 − 2c (τ)xv

])= F

(a (τ)

2φ20ψ

′20

[(v − c (τ)x

a (τ)

)2]

+1

2φ20ψ

′20

[x2

a (τ)

(a (τ) b (τ)− c2 (τ)

)])

where, a (τ) , b (τ) , c (τ) are given by,

a (τ) = φ2 (τ)ψ′20 + 2γ0ψ2 (τ)φ2

0 (3.6)

b (τ) = φ′2 (τ)ψ′20 + 2γ0ψ′2 (τ)φ2

0

c (τ) = φ (τ)φ′ (τ)ψ′20 + 2γ0ψ (τ)ψ′ (τ)φ20

which yields

a (τ) b (τ)− c2 (τ) = 2γ0φ40ψ

′40

which is a time-independent constant. Thus, the plasma distribution function

becomes,

f(x, v, τ) = F

(η(τ)

2(v − ξ(τ)x)2 + γ0

x2

η(τ)

)(3.7)

where,

η(τ) =a (τ)

φ20ψ

′20

=φ2 (τ)ψ′20 + 2γ0ψ

2 (τ)φ20

φ20ψ

′20

ξ(τ) =c(τ)

a(τ)

=φ (τ)φ′ (τ)ψ′20 + 2γ0ψ (τ)ψ′ (τ)φ2

0

φ2 (τ)ψ′20 + 2γ0ψ2 (τ)φ20

(3.8)


Since f (x, v, 0) = F (0.5v2 + γ0x2), we require η (0) = 1 and ξ (0) = 0 which

can be verified from Eq. (3.8) using ψ0 = 0 = φ′0. The density of the plasma

at any arbitrary time is thus,

n(x, τ) =

∞

−∞f(x, v, τ)dv

=

∞

−∞F

(η(τ)

2(v − ξ(τ)x)2 + γ0

x2

η(τ)

)dv

=

√1

η(τ)

∞

−∞F

(u2

2+ γ0

x2

η(τ)

)du; where u =

√η(τ) (v − ξ(τ)x)

=

√1

η(τ)n

(x√η(τ)

, τ = 0

)(3.9)

These results are for the artificial problem where the total electric field is

Etotal = [−p+ 2q cos 2τ ]x. However, no assumption has been made about

the strength of the field. Equations (3.7) and (3.8) are nonlinearly correct.

3.1.2 Maxwellian distribution function

As can be seen in Eq. (3.7) and Eq. (3.9), the time evolution of the distribution

function and density of the plasma has a strong dependence on its expression

at τ = 0. Ideally, one would like the distribution function to be a Maxwellian

since this would make the system immune to Coulomb collisions. With this in

mind, we choose the function, F in Eq. (3.7) to be an exponential. Thus, we

get,

f(x, v, τ) = n0

√β0

2πexp

(−β0

2η(τ) (v − ξ(τ)x)2

)exp

(−β0γ0

x2

η(τ)

)(3.10)

where, γ0 > 0 is any non-negative real number which defines the scale length

of the plasma for τ < 0, n0 is the plasma density at x = 0, τ = 0 and β0

is a measure of the plasma temperature. Equation (3.10) has the form of

a Maxwellian with a time and space dependent drift velocity, xξ(τ), and a


time dependent temperature, 1/β0η(τ). Since η (τ) , ξ (τ) are oscillatory and

bounded functions of time, the plasma stays confined in the trap. The time

dependent density of the plasma is given by

n(x, τ) =n0√η(τ)

exp

(−β0γ0

x2

η(τ)

)(3.11)

As can be seen in Eq. (3.11), n(x, τ) remains a Gaussian in x whose width

fluctuates in time.

3.1.3 Time periodic distribution function

For arbitrary values of γ0, the functions, η and ξ, in Eq. (3.8) will depend both

on the slow frequency, ν, and the RF frequency, ω = 2. Thus, in general, the

distribution function and density have fluctuations both at the slow frequency

and the RF frequency.

For q 1, we have,

η(τ) =a

ψ′20 φ20

=2

ψ′20

(ψ′202φ2

0

φ2 + γ0ψ2

)=

2

ψ′20

(γφ2 + γ0ψ

2)

=2

ψ′20γ [cos(ντ) + c2 cos(ν + 2)τ + c−2 cos(ν − 2)τ + ...]2

+2

ψ′20γ0 [sin(ντ) + c2 sin(ν + 2)τ + c−2 sin(ν − 2)τ + ...]2

=1

ψ′20[γ + γ0]

(1 +

γ − γ0

γ + γ0

cos(2ντ) + 2 (c2 + c−2) cos(2τ) +O(q2))

=γ + γ0

ψ′20

(1 +

γ − γ0

γ + γ0

cos(2ντ)− q cos(2τ) +O(q2))

(3.12)

where γ = ψ′20/2φ2

0. Thus, for γ0 = γ, η(τ) becomes independent of ν. This

shows that although the density, in general, fluctuates over both the slow and

RF frequency, there is a particular choice of initial density profile, γ0 = γ, for

which there is no breathing at the slow frequency, ν. For γ0 = ψ′20/2φ2

0, Eq.


(3.12) reduces to

η(τ) =γ + γ0

ψ′20

(1 +

γ − γ0

γ + γ0


=1

φ20

(1− q cos(2τ) +O

(q2))

= 1 + q − q cos(2τ) +O(q2)

(3.13)

This result can be generalized to all ordering: for γ0 = ψ′20/2φ2

0, we have

η(τ) =2

ψ′20

(ψ′202φ2

0

φ2 + γ0ψ2

)=

1

φ20

(φ2 + ψ2

)=

1

φ20

[∞∑

r=−∞

c2r cos(ν + 2r)τ

]2

+1

φ20

[∞∑

r=−∞

c2r sin(ν + 2r)τ

]2

=1

φ20

∞∑r=−∞

∞∑s=−∞

c2rc2s cos (ν + 2r) τ cos (ν + 2s) τ

+1

φ20

∞∑r=−∞

∞∑s=−∞

c2rc2s sin (ν + 2r) τ sin (ν + 2s) τ

=1

φ20

∞∑r=−∞

∞∑s=−∞

c2rc2s cos 2 (r − s) τ

=∞∑

r=0

ηr cos (2rτ) (3.14)

where ηr are constants that depend on p, q. Since each term in Eq. (3.14) is

periodic in τ with period 2π/ω = π, η (τ) is also periodic in this time interval,

i.e. it depends only on the fast time and not on the slow time, 2π/ω. Similarly,


for γ0 = ψ′20/2φ2

0, the expression for ξ in Eq. (3.8) becomes,

ξ(τ) =φ (τ)φ′ (τ)ψ′20 + 2γ0ψ (τ)ψ′ (τ)φ2

0

φ2 (τ)ψ′20 + 2γ0ψ2 (τ)φ20

=φ (τ)φ′ (τ) + ψ (τ)ψ′ (τ)

φ2 (τ) + ψ2 (τ)

=η′(τ)

2η(τ)

=−∑∞

r=1 2rηr sin (2rτ)

2∑∞

r=0 ηr cos (2rτ)

=∞∑

r=0

ξr sin (2rτ) (3.15)

where the last step follows from the fact that the ratio of an odd function

and an even function, is odd. Equations (3.14) and (3.15) show that, for

γ0 = ψ′20/2φ2

0, η(τ) and ξ(τ) are periodic functions of time to all orders in q

with period, 2π/ω.

Substituting Eq. (3.14) and Eq. (3.15) in Eq. (3.10) and Eq. (3.11), the

distribution function and density become, for this case,

f(x, v, τ) = n0

√β0

2πexp

−1

2

∞∑r=0

ηr cos (2rτ)

(v − x

∞∑r=0

ξr sin (2rτ)

)2(3.16)

× exp

(− β2

0γ∑∞r=0 ηr cos (2rτ)

x2

)

and

n(x, τ) = n0

√β0∑∞

r=0 ηr cos (2rτ)exp

(− β2

0γ∑∞r=0 ηr cos (2rτ)

x2

)(3.17)

As shown in Eq. (3.12), for the ν dependence to vanish from the expression


for f(x, v, τ), we need,

γ0 = γ =ψ′202φ2

0

=1

2

(∑r(ν + 2r)c2r∑

r c2r

)2

=1

2

(ν +

2 (c2 − c−2)

c0 + c2 + c−2

+ ...

)2

; since c4, c−4 c0 (3.18)

From Eq. (1.6), we have

[p− (2r + ν)2

]c2r = q(c2r+2 + c2r−2)

Assuming c6 = c−6 = 0, we get,

c2 =p− (4 + ν)2

q

q2

[p− (2 + ν)2] [p− (4 + ν)2]− q2

c−2 =p− (−4 + ν)2

q

q2

[p− (−2 + ν)2] [p− (−4 + ν)2]− q2(3.19)

This gives,

c2 − c−2 = qp− (4 + ν)2

[p− (2 + ν)2] [p− (4 + ν)2]− q2

−q p− (4− ν)2

[p− (2− ν)2] [p− (4− ν)2]− q2(3.20)

≈ −q(

(16 + 8ν)

(4 + 4ν)(16 + 8ν)− (16− 8ν)

(4− 4ν)(16− 8ν)

)≈ 0.5qν

Substituting Eq. (3.20) in Eq. (3.18), we obtain,

γ =1

2

(ν +

2 (c2 − c−2)

c0 + c2 + c−2

+ ...

)2

; since c4, c−4 c0

=ν2

2

(1 + 2q +O

(q2))

(3.21)


Thus, for the value of γ given in Eq. (3.21), the time-varying terms in the

distribution function and density have no dependence on the slow frequency,

ν, and the fluctuations in f(x, v, τ) and n(x, τ) are a Fourier series with fun-

damental frequency ω = 2. This corresponds to a solution that is stationary in

“slow time”, one that oscillates at the RF frequency and its harmonics. This

solution corresponds to an initial distribution that is invariant on curves of

constant E = 12v2 + γx2.

For the case of p = 0 and q > 0, ν = q/√

2 +O (q3). Thus, for this case,

γ =q2

4

(1 + 2q +O

(q2))

(3.22)

which for q 1 confirms the “ponderomotive energy”, Ep = 0.5v2 + 0.25q2x2.

For the case of p 6= 0, this expression of γ can be viewed as a definition of the

“generalized ponderomotive energy” concept where the particles see both the

DC as well as the RF field [1].

So, the improved expression for ponderomotive energy for the linear case

when p = 0 is,

E =1

2v2 +

q2

4

(1 + 2q +O

(q2))x2 (3.23)


0

1

2

-30 0 30

n(x,

t)

(a)

0

1

2

-30 0 30

n(x,

t)

(a)

(b)

0

1

2

-30 0 30

n(x,

t)

x

(a)

(b)

(c)

Figure 3.1: Density plots with q = 0.16, p = −0.01 at two different timesτ = 0 and τ = kπ, where k ∈ N is such that kπ is close to π/2ν. The threeplots are for different values of γ0. (a) γ0 = 0.5p + 0.25q2 (b) γ0 = γ, whereγ is given in Eq. (3.21) (c) γ0 = 20p + 10q2 γ. In (a), it can be seen thatthe two curves are quite close to each other. This shows that, approximately,γ = 0.5p + 0.25q2 is the value of γ0 at which the density function does nothave a large ν dependence. The curve in (b) clearly shows that Eq. (3.22)gives a much more accurate expression for the γ which leads to a more accurateexpression for the ponderomotive energy of a particle under the linearly varyingoscillatory electric field. The overlap of the curves at τ = 0 and τ = kπ is sogood that it is not visible in this graph. In (c) it can be clearly seen that forthis value of γ0, the density function has a strong dependence on ν.

Each of the three plots in Fig. 3.1 show two curves for the spatial variation


of density corresponding to Eq. (3.11) at two different times τ = 0 and τ =

2kπ/ω. Here, k ∈ N and is chosen such that 2kπ

/ω is close to π

/(2ν). These

three plots are for three different values of γ0, when q = 0.16 and p = −0.01.

Since the times are integer number of 2π/ω apart, the response at RF frequency

is the same. Any difference between these curves is due to a dependence on

the slow frequency, ν. Figure 3.1(a) is for γ0 = 0.5p+0.25q2, Fig. 3.1(b) is for

γ0 = γ as given by Eq. (3.21) and Fig. 3.1(c) is for γ0 = 20p + 10q2. It can

be seen in Fig. 3.1(a) that for γ0 = 0.5p + 0.25q2, which corresponds to the

conventional Ponderomotive theory, the density is not absolutely invariant on

the slow time-scale. But if γ0 is chosen according to Eq. (3.21), the density

does not change on the slow time-scale corresponding to frequency ν, which can

be seen in Fig. 3.1(b). This confirms the accuracy of Eq. (3.21). Figure 3.1(c)

shows that if γ0 is arbitrarily chosen, then the density changes significantly on

the slow time-scale. This substantial change in density can be understood by

considering the evolution of the distribution function in phase space which can

be seen in Fig. 3.2. The density of particles at the turning points at τ = 0

is the density at x = 0 at time τ ≈ π/2ν. For arbitrary loading, this density

differs from the density at x = 0 at time τ = 0. This results in fluctuation of

f(x, v, τ) at each point x, resulting in density fluctuations.


-40 -20 0 20 40-3

0

3

0.20.4

0.60.8

x

v

Figure 3.2: This is the contour plot of the distribution function of the plasmafor the case γ0 = 20p+ 10q2 with q = 0.16, p = −0.01. The two superimposedcontour plots correspond to the two times of the curves shown in Fig. 3.1c.This clearly shows that the drastic change in the distribution function is thereason for the huge change in the density function over the ν time scale.

3.2 Time averaged density

The time evolution of the density of the plasma is given by Eq. (3.11). From

Eq. (3.11), we have,

n(x, τ) =n0√η(τ)

exp

(−β0γ0

x2

η(τ)

)


As mentioned before, the plasma density, in general, has time fluctuation both

at the slow frequency, ν, and the RF frequency, ω = 2. However, as shown

earlier, for the case, γ0 = γ = ψ′20/2φ2

0, there is no fluctuation at the slow fre-

quency and the time evolution of the density becomes periodic with frequency,

ω = 2.

To make connection to conventional Ponderomotive theory, we evaluate

the time averaged density for the case q 1 and p = 0. Substituting the Eq.

(3.13) in Eq. (3.11), we get,

n(x, τ) = n0

[1− q

2+q

2cos 2τ +O

(q2)]

exp

[−β0

2

q2

2(1 + q − q cos 2τ)x2

]= n0

[1− q

2+q

2cos 2τ − β0

2

q3x2

2− β0

2

q3x2

2cos 2τ +O

(q2)]

(3.24)

× exp

[−β0

2

q2x2

2

]

The time averaged density of the plasma is then given by,

n(x) = n0

[1− q

2− β0

2

q3x2

2+O

(q2)]

exp

[−β0

2

q2x2

2

](3.25)

and is shown in Fig. 3.3.


0

0.2

0.4

0.6

0.8

1

-30 -20 -10 0 10 20 30

Den

sity

x

(a)Time averagedPonderomotive

0

0.2

0.4

0.6

0.8

1

-30 -20 -10 0 10 20 30

Den

sity

x

(a)

(b)Time averagedPonderomotive

Figure 3.3: This plot compares the time averaged density of the plasma, Eq.(3.25) obtained from exact expressions with that predicted by the conventionalponderomotive theory. (a) The difference in the plasma density at x = 0 is dueto the factor 1 − 0.5q present in Eq. (3.25). In conventional ponderomotivetheory, it is assumed that the plasma density at the origin is not effected bythe RF field. (b) The plot has been normalized so that the plasma densitiesare same in both cases at x = 0. It can be clearly seen that there is a differencein the spatial density profile.

3.3 Discussion

3.3.1 The Ponderomotive Potential

The standard definition of the “ponderomotive potential” is based on first

order theory. As a result, the value of ν is in error for larger field gradients.

For Electric fields linearly varying in space, the theory of Mathieu functions

provides a better expression for the ponderomotive force, that includes the

stability boundary. The expressions obtained in the previous sections describe

exact solutions of the 1-Dimensional problem for the case of constant gradient

fields. This is important since it permits us to validate theories in the literature

against this exactly solvable case.

For the case of a linearly varying field, conventional ponderomotive force

expressions approximately predict the low frequency path or the path along


which the particle will drift. And based on this expression, the expression for

the spatial variation of the time-averaged plasma density is derived. In a paper

published in 1979, Krapchev [17] attempted at proving that under the action of

an electric field −(m/e)wv0(x) cos(wt), the time-averaged density of the pure

ion plasma will be n = n0 exp(v2

0(x)/2v2

T

), where vT is the thermal velocity

of the plasma ions. This is in agreement with the conventional ponderomotive

theory. Krapchev’s result, and indeed all of the conventional work, applies

only to the q 1 limit, as it is assumed that the gradients are weak.

When these expressions are applied to the field distribution studied in this

paper, they predict that the time-averaged density should vary as

n = n0e−β0q2x2

/4

From the expressions derived in this paper in Eq. (3.11), the actual instanta-

neous density is given by

n =n0√η(τ)

exp

[−β0γ0

x2

η(τ)

]

For γ0 = γ, the time-averaged expression for this density is given by, Eq.

(3.25),

n(x) = n0

[1− q

2− β0

2

q3x2

2+O

(q2)]

exp

[−β0

2

q2x2

2

](3.26)

While the exponential dependence is in agreement with Ponderomotive theory,

the additional multiplicative factors are not. These factors arise from the fact

that the ion response is adiabatic and hence temperature is not a constant.

In conventional Ponderomotive theory, an implicit assumption is made that

the RF field is connected to a heat bath held at β0. Hence this factor is not

seen. However, such an assumption is questionable at RF frequencies. When

the exact problem is solved, we find the additional factors. It is startling


that there is a correction factor of O (q), which is of lower order than the

Ponderomotive effect itself. While this result has only been obtained for a

very specific field profile, the source of the deviation from conventional theory

suggests that a similar correction term would appear in any bounded system.

Of course, in the case of a linear gradient, this term is particularly significant,

since the plasma response is exactly adiabatic at all collision frequencies, as

discussed in Sec. (3.3.5).

A question of interest is what changes are required in plasma fluid equations

to obtain Eq. (3.26). The Ponderomotive force is seen by all particles and as

seen in Eq. (1.5), the shape of the particle orbit is independent of the particle’s

initial conditions. Hence, on taking the moment, this force remains in its single

particle form. However, as seen in Eq. (3.10), the temperature oscillates in

“fast time” yielding a changed value for the average temperature as seen in

Eq. (3.26). The equation of state therefore needs to be changed to P = nκT ,

where fluid behavior is assumed to be isothermal. From Eq. (3.10) and Eq.

(3.13), the time averaged temperature is,

T = T0

(1− q +O

(q2))

(3.27)

where T0 = 1/κβ0 is the initial temperature of the plasma. Substituting this

into the fluid momentum equation

0 = n∂Φp

∂x+κT0

n0

∂n

∂x

the long term steady state density becomes,

n = n0 exp

[−φP

κT

](3.28)

where φP is the Ponderomotive potential energy of the each ion. The above


expression for density is the well known Boltzmann relation. Since RF in-

teraction preserves fluid, n0 in Eq. (3.28) can be obtained from the initial

conditions to yield,

n(x) = n0

√T0

Texp

[−φP

κT

]Substituting φP = 0.25q2x2 (1 + 2q) from Eq. (3.23) and T from Eq. (3.27),

we obtain to O (q2),

n(x) = n0

√1

1− q +O (q2)exp

[−β0

0.25q2x2 (1 + 2q +O (q2))

1− q +O (q2)

]= n0

(1 +

q

2+O

(q2))

exp

[−β0

2

q2x2

2

(1 + 3q +O

(q2))]

= n0

[1 +

q

2− β0

2

3q3x2

2+O

(q2)]

exp

[−β0

2

q2x2

2

](3.29)

which is different from Eq. (3.26). Thus, it is clear that a fluid dynamics

approach the RF driven plasmas is incorrect and it is important to solve the

full kinetic equations. One more difficulty with this modification of the fluid

equations is that the solutions obtained in this work exhibit adiabatic behavior

for all RF frequencies and all collision frequencies. However, we conjecture

that non-idealities would relax the slow time behavior to make the isothermal

equation of state, P = nκT valid.

In general, the exact plasma response also contains a term at the slow

frequency, 2ν. This corresponds to the nonlinear, transient response. Conven-

tional ponderomotive theory looks for steady state solutions, i.e., it assumes

that the distribution function of the plasma can be expanded in a Fourier series

which has the same fundamental frequency, w, as the applied field. However,

unless γ is given by Eq. (3.21), a ν dependent term appears, and modifies the

density perturbation.


3.3.2 Comparison with Krapchev’s Theory

The correct time-averaged distribution function is of the form

f(x, v) =ω

2π

π/ω

−π/ω

f(x, v, τ)dτ (3.30)

where f(x, v, τ) is given by Eq. (5.29). As discussed in the previous section, the

distribution function, f(x, v, τ) is constant on the detailed orbit and not on the

time averaged orbit. Figure 1.3 makes it clear that the excursions in velocity

are not small. Indeed, they are as large as the time-averaged velocity itself.

Thus, a time average of the detailed distribution function cannot be replaced

by a distribution function that is constant on the time averaged orbits.

The time averaged distribution function for such problems was obtained by

Krapchev [17]. In his analysis, Krapchev obtained the distribution function by

solving for the steady state solution of the Vlasov equation directly. The orbits

obtained in this paper are the characteristics of that equation. Krapchev used

the idea that the distribution function does not see the slow frequency at all,

and expanded f as a Fourier series in the RF frequency and also made certain

simplifying assumptions to obtain the time averaged solution.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

aver

age

f

v

x=0

x=0.2

x=0.4

From exact solution (scaled)Krapchev’s solution

Figure 3.4: This is a plot of the time averaged distribution function at differentspatial locations. The solid line is obtained from a numerical integration ofEq. (3.10) and the dashed lines are the ones given in the work of Krapchev [9]and are normalized and scaled. The labels are scaled by x. As can be seen,the curves cease to be monotonic after a certain threshold in x.

Equation (3.30) was numerically evaluated and compared with the analyt-

ical results obtained by Krapchev for the case of p = 0 and α = 0. Fig. 3.4

shows the time averaged distribution function of the plasma as a function of

velocity. Each curve in the figure corresponds to a different spatial location,

separated by 0.2x. The solid curves are the result (normalized and scaled) of

numerical integration in Eq. (3.30) and the dashed lines are the result obtained

by Krapchev. As can be seen, at about x = x, the time averaged distribution

function ceases to be monotonic in velocity and becomes double humped. This

suggests that the presence of instabilities in the plasma, but, by using Penrose’s

criteria, Krapchev concluded that the plasma is stable. The use of Penrose’s

criterion in the context of RF plasmas is debatable. The solution obtained


by Krapchev does not satisfy the time-averaged Vlasov equation, which would

yield a Maxwellian distribution confined by a ponderomotive potential. Thus,

there is no time-independent kinetic equation whose equilibrium yielded the

double humped distribution obtained for x > x. This being the case, it is not

clear how Penrose’s criterion is applicable.

In this analysis, we have shown that though the time averaged distribution

function is double humped for these spatial locations, the exact time varying

distribution function in Eq. (5.29) is a single humped global Maxwellian at all

instants of time. It is quite reasonable, therefore, to believe that the plasma

will be stable to small perturbations. Even this, however, is not a proof.

There is a source of power in the external RF field, and there might well be a

fluctuation that could tap into that field and grow.

Krapchev’s analysis began with the assumptions that v∇f/ωf 1 and

neglected higher spatial derivatives of f even in the higher order analysis.

He also assumed that the time averaged plasma distribution is a perturbed

Maxwellian. He assumed that response at the nth RF harmonic was of order

O (qn). A consequence of this assumption was that the time averaged distri-

bution function became an even function of q. He used this to obtain a power

series for the response and summed up the series for f to infinity by means of

a conjecture that the correction of order O (q2n) took the form of a Laguerre

polynomial of order n.

There are several problems with Krapchev’s approach. The response at

O(q) is usually much stronger than the response at DC, which is of O (q2) since

it is related to the ponderomotive potential. Also, as mentioned above, double

humped, strongly non-Maxwellian solutions are predicted by this theory, which

is not consistent with the perturbative approach. It is quite surprising that

these findings are validated by the orbit calculations carried out in this paper.

It is our conjecture that his summing up all the higher order corrections to


obtain an analytic result extended the validity of his analysis beyond that of

his perturbation analysis.

Krapchev’s expressions for f and n, if correct, should agree even with

the expressions for the response of the plasma to an exactly linear RF field.

However, in Eq. (3.26), the time-averaged density has been shown by us to be

n(x) = n0

(1− q

2− β0

2

q3x3

2+O

(q2))

exp

[−β0

2

q2x2

2

]

which is different from Krapchev’s result,

n = n0 exp

[−β0

2

q2x2

2

]

Clearly, our expression has odd powers of q whereas Krapchev’s expression

implicitly requires the density to be an even function of q. The differences

between the two expressions could be interpreted as the difference between the

steady state distribution in oscillation center coordinates and the distribution

resulting from an initial value problem.

Krapchev’s analysis showed the temperature to be spatially non-uniform.

However, an exact analysis in this work shows that though the temperature

oscillates in time, it is spatially uniform. Krapchev obtained a spatially non-

uniform temperature because he used the time averaged distribution function

to define temperature. However, this is not correct, since temperature can

only be defined for thermodynamic states. In the current problem, the system

does go through a sequence of thermodynamic states, but these are continu-

ously oscillating in time. Thus the system cannot be characterized by a single

temperature.


3.3.3 Extent of non-Maxwellianity

One interesting finding of both Krapchev and our work is the non-Maxwellian

nature of the time-averaged distribution function. A natural question to ask

is the extent of non-Maxwellianity for given parameters, p, q.

For q 1, neglecting terms of O (q2) in the distribution function, Eq.

(3.10), we get

f(x, v, τ) ≈ n0

√β0

2πexp

[−β0

(1− q) v2

2− β0

p (1 + q)x2

2

]× exp

[(−β0

qv2

2+ β0

pqx2

2

)cos 2τ + β0qxv sin 2τ

]

Under this approximation, the time averaged distribution is given by

f(x, v, τ) ≈ n0

√β0

2πexp

[−β0

(1− q) v2

2− β0

p (1 + q)x2

2

]× 1

π

π

0

dτ exp

[(−β0

qv2

2+ β0

pqx2

2

)cos 2τ + β0qxv sin 2τ

]= n0

√β0

2πexp

[−β0

(1− q) v2

2− β0

p (1 + q)x2

2

]

×I0

β0q

√(v2

2− px2

2

)2

+ x2v2

(3.31)

where I0 is the modified Bessel function of the 1st kind [8]. As can be clearly

seen, Eq. (3.31) is clearly non-Maxwellian. For large values of x, Eq. (3.31)

can be further approximated to give,

f(x, v, τ) ≈ n0

√β0

2πexp

[−β0

(1− q) v2

2− β0

p (1 + q)x2

2

]×I0

[β0pq

2x2 + β0

q

pv2(1− p

2

)](3.32)

This equation clearly shows that the extent of non-Maxwellianity goes like q/p.

It must be noted that Eq. (3.31) and, hence, (3.32) is not valid for very large


values of x, since for very large values of x, we cannot neglect the O (q2) terms

in the distribution function. However, though Eq. (3.32) is valid only for a

certain bounded range of values of x, it gives a qualitative understanding for

the nature of the non-Maxwellianity in the time averaged distribution function.

3.3.4 Relation to BGK Theory

As can be seen in Fig. 3.5, the externally applied field has a significant DC

component. This is the Electric field that is required to balance the self-field

induced by the plasma ions. If we choose a value of γ for a particular p, q

other than that given by Eq. (3.21), the applied field will have components

at frequency ν, in addition to a DC component and components at the RF

frequency w and its harmonics. Existing treatments of this problem deal only

with the field seen by particles, and thus neglect the need to balance this self

field.

The derivation in section 3.1 was for a ion distribution function whose

velocity variation was a Maxwellian. However, any arbitrary distribution could

have been chosen that depended on x and v through Ep = 0.5v2+γ0x2. Let the

distribution function of the plasma at τ = 0 be f0(x0, v0) = F (0.5v2 + γ0x2),

where F is a smooth but otherwise arbitrary function of its argument. This

distribution will evolve with time, and at an arbitrary time τ , it can be written

as,

f(x, v, τ) = F

(η(τ)

2(v − ξ(τ)x)2 + γ0

x2

η(τ)

)where, η(τ) and ξ(τ) were defined in Eq. (3.8). The density of the plasma at


any arbitrary time is thus,

n(x, τ) =

∞

−∞f(x, v, τ)dv

=

∞

−∞F

(η(τ)

2

(v − ξ(τ)x

)2

+ γ0x2

η(τ)

)dv

=

√1

η(τ)

∞

−∞F

(u2

2+ γ0

x2

η(τ)

)du; where u =

√η(τ)

(v − ξ(τ)x

)=

√1

η(τ)n

(√1

β(τ)x, τ = 0

)

Corresponding to this density expression we can now find the induced field

and, hence, the total external field required.

The same derivation shows that any such function of EP yields a distri-

bution that is time stationary with fluctuations at the RF frequency and its

harmonics when γ is given by Eq. (3.21). Each such distribution function

corresponds to a different applied Electric field. There is, thus, an interesting

correspondence to BGK theory.

In the case of a BGK Mode [30], if we specify the distribution function as

a function of energy, we can find the potential required to confine the plasma

according to that particular distribution. If the distribution of the electrons is

given by f(E), this leads to the nonlinear Poisson’s Equation,

d2φ(x)

dx2= 4πe

∞

−eφ

f(E)dE√2m(E + eφ(x))

In this paper, we started with an initial distribution f0(0.5v20 + γx2

0) at τ = 0.

We assumed that the total field seen by the plasma must be−(m/e) [−B + A cos(wt)]x.

For this field, the particle paths are given by the Mathieu’s functions. The dis-

tribution function and the density were obtained by the method given in Sec.


3.1. From this, we found the induced field,

Ei(x, t) = −4πe

x

0

n(x′, t)dx′ (3.33)

and hence, the total external field required to confine the plasma

Ee(x, τ) = Et(x, τ)− Ei(x, τ)

Ee(x, τ) and f(EP , τ) constitute a self-consistent solution of the plasma evo-

lution equations. There is, however, a lack of self-consistency in the fact that

∂Ee

/∂x 6= 0. It requires that ~E vary along y or z, yet the particles be con-

strained to move only along x, perhaps by a magnetic field. For any given

distribution of this form, we can determine the Electric field that is to be

imposed to make the problem self-consistent. The reverse problem, of de-

termining f (x, v, τ) given Ee (x, τ) such that Et = [−p+ 2q cos 2τ ]x, is not

solvable. In general, Ee(x, τ) has an infinite number of harmonics, each of

which has an independent spatial profile. Essentially what this means is that

Ee(x, τ) can be an arbitrary function of time at each point in space. At most,

f(EP , τ) can match to one spatially varying harmonic of Ee(x, τ). Thus, for an

arbitrary specification of Ee(x, τ), it is not possible to determine a distribution

function that is invariant under the application of that field. The analogy with

BGK theory is therefore limited.


1e-4

1e-2

1

0 5 10 15 20 25

mag

(E(x

,w))

x

DCw

2w

3w

Figure 3.5: This shows the spatial variation of the magnitude of various fre-quency components present in the Fourier transform of Ei(x, t) as given byEq. (3.33). This is for the case when γ is given by Eq. (3.21) and q = 0.16,p = −0.01. As can be clearly seen, leaving the DC component, the componentat ω is dominating. And we also have small contributions from componentsat frequencies 2ω , 3ω (ω = 2 in our normalization). The remaining harmon-ics are lower in magnitude than the the ones shown. The component at 2ωis two orders of magnitude lower than that at ω. So, the field given by Eq.(3.33) is essentially a nonuniform monochromatic electric field for all practicalpurposes.

The presence of plasma means that a DC field becomes present. This needs

interpretation. BGK modes also correspond to time-stationary distributions

that are confined by DC fields. If a DC field is also present here, what difference

is there between the two?

The two types of solutions are clarified in Fig. 3.6. The figure shows the

confining fields when a Maxwellian plasma is present. In order to compare,

the static and the RF solutions are assumed to have the same time-averaged

density profiles (Gaussian in shape). The static solution corresponding to


a Maxwellian has a linear confining Electric field (shown as the curve labeled

“Effective static Field” in the figure). Charges execute simple harmonic motion

in this field. The externally applied field is the difference of the desired linear

field and the field induced by the charge density of the plasma. This is the

curve labeled “Applied static Field” in Fig. 3.6. The self-consistent field itself

is shown as the line labeled “Induced Field”.

-10

0

10

20

30

40

50

0 5 10 15 20 25

E(x

)

n(x)

x

Applied RF Field

Applied static Filed

Effective static Field

Induced Field

Applied DC Field

n(x)

Figure 3.6: This shows the relative spatial variation of the fields correspondingto the RF solution considered in this paper and the fields in static equilibrium.If the time-averaged density of the plasma goes like exp (−β0γ0x

2), then the“Effective static Field” corresponds to the Electric field for which the potentialgoes like β0γ0x

2. The “Induced Field” is the field induced by the exp (−β0γ0x2)

electron density in the absence of ions. The “Applied static Field” is the sumof these two fields, and is the external Electric field which has to be applied toget this particular density profile. Now, if the same time averaged profile hasto be achieved by using an RF field, then the total field seen by the plasmahas a much steeper slope and is shown by the straight line labeled “EffectiveRF Field”. The curves labeled “Induced Field” and “Applied static Field” arenot straight lines. These curves are purely qualitative and are not to scale.

When RF is used to confine the plasma, the applied field is determined by


the two parameters p and q. When q is zero (no RF), the resulting density

corresponds exactly to the static solution above. When p is zero, the external

applied field is assumed to have a DC component that exactly cancels the

self-repulsion of the plasma. Then ions see only an RF field, which is the case

studied in Ponderomotive force studies in the literature. When both p and

q are present, we have some very interesting possibilities. Fig. (3.6) shows

the case of q = 0.16 and p = −0.01. The p value is assumed to be such that

it corresponds exactly the self-repulsion of the plasma at x = 0. At larger x

there is a mismatch, since the self-field of the plasma grows as the error function

while the total field is required to be linear. Thus an additional repelling field is

necessary and is externally applied. This is shown as the negative dot-dashed

line in Fig. (3.6). For p that is more negative than this value, the dot-dashed

line will acquire a linear slope as well. For less negative p, a positive (confining)

field develops that turns into a repelling field at greater distances.

The possibility of a repelling DC applied field is quite interesting. As is well

known, static fields cannot confine charges. Thus, if perpendicular confinement

is achieved electrostatically, parallel confinement must be achieved by other

means such as RF. Yet, we see that a pure RF solution still requires a confining

DC field which invalidates the solution. The solution to this problem is to use

an extracting field along x. This ensures that the static field is consistent with

the averaging theorem of Laplace’s Equation. To confine the plasma along x,

we apply an RF field that not only keeps the particles in via Ponderomotive

Force, but also overcomes the DC repelling field that is present. Since there

are small self-consistent contributions at the RF frequency and its harmonics,

the applied field is slightly modified from 2qx cos (2τ). The RF field amplitude

is shown as the nearly straight line labeled “Applied RF Field” in Fig. (3.6).

The figure is only intended to convey the qualitative differences between static

equilibrium and the RF solutions obtained in this paper, and is not to scale.


3.3.5 Collisional Effects

The self-consistent solution we have obtained for the distribution function of

the plasma has a Maxwellian velocity distribution at all spatial positions x and

at all times τ . As is well known, the Maxwellian annihilates the point collision

operator. Thus, the solution obtained in the previous section by solving the

collisionless Vlasov equation is, in fact, a global thermal equilibrium. This

shows that even in the presence of collisions, these solutions are still valid,

because the system is already in the maximum entropy state by virtue of being

described by a Maxwellian. This situation is quite different from that of a local

thermal equilibrium where the distribution function is Maxwellian only to the

lowest order. A distribution that is invariant to collisions can, obviously, not

be subject to stochastic heating. This solution is therefore a solution where

the plasma is confined by an RF field without undergoing heating.

If the plasma interaction is adiabatic, the temperature, T , and volume, V ,

of the plasma must satisfy the condition,

TV Cp/Cv−1 = constant

where for a one dimensional system Cp/Cv = 3. The time evolution of the

distribution function of the plasma is evaluated in Eq. (3.10) and found to be,

⇒ f(x, v, τ) = n0

√β0

2πexp

(−β0

2η(τ) (v − ξ(τ)x)2

)exp

(−β0γ0

x2

η(τ)

)

From the above expression, we can see that T ∝ 1/η(τ) and V ∝ η0.5(τ) .

Thus, T ∝ V −2. This gives,

TV Cp/Cv−1 ∝ V −2V 3−1 = constant

The distribution function obtained in Eq. (3.10) clearly satisfies the adiabatic-


ity condition.

Even more intriguing are the solutions for which γ is very different from

the value given in Eq. (3.21). These are time-varying, exact solutions in

which the plasma undergoes large-scale reorganization over the π/2ν timescale.

These solutions also have Maxwellian velocity distributions at all times and all

spatial locations! Thus, these time-varying solutions are also exact solutions

of the 1-Dimensional, Vlasov-Boltzmann equation. Such solutions behave very

similarly to the quasistatic compressions and rarefactions of ideal gases trapped

in a piston, except that the compressions are not quasistatic, but happen at

bounce times, and that the “walls” of the plasma piston are moving back and

forth at RF frequencies (with harmonics at rω±ν, r = 0, 1, . . .). The breathing

of the plasma at the bounce frequency ν is a self-consistent, exact response of

the collisional plasma.

3.3.6 Multi-Species Plasmas

In this work, we have considered a single species ion plasma. If we had elections

in addition to electrons the problem changes somewhat. Each plasma species

satisfies Eq. (3.1) with a different ms

/qs ratio. Hence, Eq. (3.1) is satisfied by

each species for p and q values that are scaled by ms

/qs. The low frequency ν

corresponding to the ions will therefore be different from that corresponding

to the electrons. Nonetheless, a self-consistent, collisionless solution can be

obtained, where

Ee(x, τ) = −me

[−B + A cos 2τ ]− Ei(x, τ)

where Ei(x, τ) is now the Electric Field induced by all the species present

in the plasma. This solution will not however be invariant at the slow time.

Each species has its own breathing frequency νs. By choosing q it is possible


to eliminate the slow time variation of any one species, but the other species

will continue to oscillate. However, for the special case of a two species plasma

consisting of two oppositely charged species, it is possible to find p and q such

that both species are time invariant at the slow time scale. Consider electrons

and singly charged ions of mass mi. Then,

pe

pi

=qeqi

= −mi

me

= −k (say)

where, k 1. Now, we can approximately write γ from Eq. (3.11) as

γ =pi

2+q2i

4=pe

2+q2e

4

⇒ pi +1

2q2i = −kpi +

1

2k2q2

i

⇒ pi(1 + k) =1

2(k2 − 1)q2

i

⇒ pi

q2i

=k − 1

2

For this choice of p and q, both the electron and ion distributions will be

stationary in slow time.

Whatever the solution found, the presence of multiple species makes the

solution susceptible to collisions. Clearly when different species breathe at

different frequencies, the distributions collisionally drag on each other. Even

in the case where low frequency oscillations are eliminated for both species, the

high frequency oscillations of ions and electrons are opposite in phase. This

is obvious since the forces are opposite in direction. Hence there is a high

frequency vi − ve present, which means that collisional drag operates on these

oscillations. Since these oscillations are driven by the applied RF field, the

collisional stability of such solutions and the possibility of collisionally driven

heating of the distributions are issues that need investigation.


The special case of infinitely massive ions is interesting. The ion profile can

be chosen to cancel the self-consistent field of the electrons. No DC confining

field is now necessary (but is always permitted). The breathing solutions

found are not collisionally valid, since the ions act as point scatterers that

conserve energy but isotropize momentum. The stationary solutions are also

collisionally invalid since the high frequency oscillations now see a momentum

drag.

3.3.7 Conservation of particle number and energy

An important question that arises in the analysis of statistical systems is the

issue of conservation of particle number and energy.

For a plasma in a Paul trap, the particle number can increase either if there

is an injection of fresh ions during the experiment or if there are ionization

processes triggered. In this thesis, we are not considering these effects. And

the number of particles can decrease if the individual particle trajectories are

unbounded or if the statistical relaxation processes lead to an evacuation of

particles. As mentioned in the introduction, the individual particle trajecto-

ries are bounded if the Paul trap parameters are within the stability region

of the Mathieu equation. Thus, the only question that remains is whether

statistical relaxation processes can lead to an evacuation of the plasma parti-

cles. As mentioned in Sec. 3.3.5, the plasma distribution function is an exact

Maxwellian and hence, annihiliates the point-collision operator. Thus, it is

reasonable to believe that the particle number is a conserved quantity in most

Paul trap experiments under suitable conditions.

The energy of a particle subject to conservative forces is a constant if the

associated Hamiltonian is autonomous. However, since the particles in a Paul

trap are periodically forced by the RF field, the particle energy keeps changing

with time. For single particles, the energy is given by 0.5v2 and is aperiodic


for almost all choices of the Paul trap parameters. However, the energy of the

plasma as a whole is given by the second moment of the plasma distribution

function and is proportional to the plasma temperature. And as shown in this

chapter, the plasma temperature can be periodic for a specific choice of the

initial plasma density profile. Thus, the plasma temperature is not a conserved

quantity and keeps oscillating with time.

3.4 Conclusions

We have obtained nonlinearly exact solutions for the 1-Dimensional RF con-

finement problem, for the case of spatially linear Electric fields. The electric

field considered is a combination of a DC field and an RF field, both of which

vary linearly in space. Exact expressions for the distribution function and the

particle density have been obtained in closed form. We find that single species

plasmas confined in this field do not experience any stochastic heating. The

solutions are valid even in the presence of strong collisions, where a Brownian

Motion or Fokker-Planck approach might be expected to yield more accurate

results.

The nature of the ponderomotive force has been clarified to a certain extent

in this study. Unless the plasma is already in a distribution of the form f(EP )

when the RF field is turned on, the plasma response is not well approximated

by the ponderomotive force equation. Instead, the plasma executes complex

breathing cycles at a new frequency. The reason for this discrepancy in behav-

ior lies in the implicit assumption of harmonic plasma ponderomotive response

to an RF field. The plasma response to an RF field is not in general harmonic.

Instead a bounce frequency appears, which is, in general, irrationally related

to the RF frequency.

An implicit assumption made in the conventional theory of Ponderomotive


effect [16] is that the nonuniform RF field does not effect the velocity space

distribution of the plasma. But for the case of a linear field profile, we have

found that this is not true. The temperature is found to oscillate on the RF

time scales adiabatically and the time average of the temperature is found to

depend on the RF field strength (Eq. (3.26)). This is very significant and it is

expected that deviations from conventional theory will be present for the case

of spatially nonlinear RF fields also.

Among the surprising findings is a class of non-stationary solutions that do

not relax in the presence of collisions. These are maximum entropy states that

breathe at the ion bounce frequency. The plasma response shows spectral lines

at rω ± kν for r ∈ Z and k = 0, 1, 2 and is not quasistatic by any definition

of that term. Yet the solutions behave adiabatically in that they move from

thermodynamic state to thermodynamic state along the iso-entropic curves, in

finite time.

The main reason why the linear field problem yielded closed form solutions

is also its weakness: all the particles in the plasma respond with the same

frequency regardless of their initial position. Their orbits are not the same.

However, fast and slow particles both respond identically. This is rather like

a complex variant of simple harmonic motion.

Chapter 4

Spatially linear DC and linear

RF field — II

As shown in Chapter 3, for a particular choice of γ0 in Eq. (3.10), the distri-

bution function becomes ω-invariant, i.e. periodic with the same time period,

2π/ω, as the applied RF field. This condition on γ0 translates to a very specific

initial condition for the distribution function. If γ0 is different from this special

value, then the distribution is still given by Eq. (3.10) but it now breathes at

the slow frequency, ν. One case of interest when γ0 6= γ, where γ is given by

Eq. (3.21), is the startup problem. Suppose that for τ < 0, the RF field is

zero and the plasma is confined in a 1d DC potential well (γ0 = 0.5p) with the

distribution function given by

f (x, v, τ < 0) = n0

√β0

2πexp

[−β0

(v2

2+px2

2

)]

At τ = 0, the RF field is switched on and the plasma distribution function

now evolves in time.

72

CHAPTER 4. SPATIALLY LINEAR DC AND LINEAR RF FIELD — II73

4.1 Plasma distribution function

For τ ≥ 0, the time dependent distribution function is given by Eq. (3.10),

f(x, v, τ) = n0

√β0

2πexp

(−β0

2η(τ) (v − ξ(τ)x)2 − β0γ0

x2

η(τ)

)(4.1)

and the density is given by Eq. (3.11),

n(x, τ) =n0√η(τ)

exp

(−β0γ0

x2

η(τ)

)(4.2)

where the functions, η(τ), ξ(τ) are given by Eq. (3.8),

η(τ) =φ2 (τ)ψ′20 + 2γ0ψ

2 (τ)φ20

φ20ψ

′20

ξ(τ) =φ (τ)φ′ (τ)ψ′20 + 2γ0ψ (τ)ψ′ (τ)φ2

0

φ2 (τ)ψ′20 + 2γ0ψ2 (τ)φ20

In Eq. (3.12) it was shown that the function, η(τ) can be written as,

η(τ) =γ + γ0

ψ′20

(1 +

γ − γ0

γ + γ0


(4.3)

where γ = ψ′20/2φ2

0. Similarly, the expression for the function, ξ(τ), becomes

ξ(τ) =γφ (τ)φ′ (τ) + γ0ψ (τ)ψ′ (τ)

γφ2 (τ) + γ0ψ2 (τ)

=η′(τ)

2η(τ)

=−ν γ−γ0

γ+γ0sin(2ντ) + q sin(2τ) +O (q2)

1 + γ−γ0

γ+γ0cos(2ντ)− q cos(2τ) +O (q2)

(4.4)

As can be seen in the above expressions, the main factor that decides the

contribution of the term at the slow frequency, ν, compared to that at the RF


frequency, ω = 2, is (γ − γ0)/

(γ + γ0). From Eq. (3.20), we have,

c2 − c−2 ≈ 0.5qν (4.5)

Substituting Eq. (4.5) in Eq. (3.21), we get,

γ =1

2

(ν + 2

c2 − c−2

c0 + c2 + c−2

+ ...

)2

≈ 0.5p+ qp+ 0.5pq2 + 0.25q2 (4.6)

Thus,

γ − γ0

γ + γ0

=(0.5p+ qp+ 0.5pq2 + 0.25q2)− 0.5p

(0.5p+ qp+ 0.5pq2 + 0.25q2) + 0.5p

= qp+ 0.5pq + 0.25q

p+ qp+ 0.5pq2 + 0.25q2

≈ q

(1 +

q

4p

)using q2 p < 1 (4.7)

Using Eq. (4.7) and ψ′0 ≈√p, Eq. (4.3) becomes,

η(τ) = 1 + q

(1 +

q

4p

)cos(2ντ)− q cos(2τ) +O

(q2)

(4.8)

and Eq. (4.4) becomes,

ξ(τ) =−q√p

(1 + q

4p

)sin(2ντ) + q sin(2τ) +O (q2)

1 + q(1 + q

4p

)cos(2ντ)− q cos(2τ) +O (q2)

(4.9)

In Eq. (4.8), we can see that the magnitude of the component of η (τ) at the

slow frequency, 2ν, is greater than that at the RF frequency, ω = 2. However,

if q p, we can neglect the term q/4p and then these two components have the

same magnitude, q. However, in Eq. (4.9), the magnitude of the component

of ξ (τ) at the frequency 2ν is smaller than that at ω = 2, since p < 1. Thus,

though both the functions η (τ) and ξ (τ) have components at both the slow


frequency and RF frequency, the slow time evolution of the plasma temperature

(which depends on η (τ)) is much more important compared to that of the drift

velocity (which depends on ξ (τ)).

1

2

10

1e-3 1e-2 1e-1

dens

ity c

ompo

nent

at s

low

freq

uenc

y, 2

ν

q

w

p=0.001

p=0.004

p=0.008

p=0.5q2

p=0.25q

Figure 4.1: Plot of the magnitude of the coefficients of cos 2ντ and cos 2τ in theexpression for A as given in Eq. (4.8), normalized by q. It can be clearly seenthat magnitude of the response at w lies below the response at 2ν. The solidthick line labeled p = 0.5q2 serves to divide two prominent region of plasmaresponse. When p = 0.5q2, the RF response is as large as the DC responseand as we approach this region, plasma behavior is highly nonlinear. There isanother solid thick line labeled p = 0.25q. This also demarcates two regions inthe p− q space. As q crosses this line, there is a visible change in the slope ofthe curves. The plots clearly show that as q becomes large compared to p, thefirst nonlinearity to set in is of the order of q2

/4p. Also, the curves are well in

agreement with the expressions derived in Eq. (4.11).

Figure. 4.1 shows the magnitude of the coefficients of cos 2ντ and cos 2τ

in the expression for A as given in Eq. (4.8), normalized by q. For q p,

the plasma oscillates with the same magnitude at both the high and the low

frequency, even though the total Electric field only contains a high frequency


component. The solid thick line labeled p = 0.5q2 serves to divide two promi-

nent regions of plasma response. To the left of this curve, the frequency com-

ponents are more or less linear but after this region is crossed, the response

becomes highly nonlinear. The curves shown in the figure are also in agree-

ment with Eq. (4.8). There is also another solid thick line on the curve labeled

p = 0.25q. This also demarcates two regions on the p − q space. We can see

that when q crosses this solid line, there is a visible change in the slope of the

curves. Thus, as q becomes larger, the first nonlinearity to set in is of the order

of q2/4p.

4.2 Validation of Linear Response Theory

If q p , the RF field is a small perturbation on the DC field. And thus, one

would expect the modification of the distribution function, f (x, v, τ ≥ 0) −

f (x, v, τ < 0) also to be small. Hence, one would expect linear Vlasov theory

to correctly predict the distribution function to O (q). However, though linear

Vlasov theory would give a solution, we still need to prove that the obtained

solution is indeed a correct representation of the exact solution. Since, in our

work we have obtained exact solutions for the plasma distribution function, it

gives us an opportunity to verify the correctness of the solutions predicted by

linear Vlasov theory.

4.2.1 Linear estimate from exact theory

In this section, we consider the case when q p 1. From Eq. (3.10), we

know that,

f(x, v, τ) = n0

√β0

2πexp

(−β0

2η(τ) (v − ξ(τ)x)2 − β0γ0

x2

η(τ)

)(4.10)


Thus,

f1(x, v, τ) = f(x, v, τ)− f0(x, v, τ = 0)

= n0

√β0

2πexp

(−β0

2η(τ) (v − ξ(τ)x)2 − β0γ0

x2

η(τ)

)−n0

√β0

2πexp

[−β0

(v2

2+ γ0x

2

)]= n0

√β0

2πexp

(−β0

2

(ηv2 +

(ηξ2 +

2γ0

η

)x2 − 2ηξxv

))−n0

√β0

2πexp

[−β0

(v2

2+ γ0x

2

)]= n0

√β0

2π

[exp

(−β0

(v2

2+ γ0x

2

))× exp

(−β0

2

((η − 1) v2 +

(ηξ2 +

2γ0

η− 2

)x2 − 2ηξxv

))]−n0

√β0

2πexp

[−β0

(v2

2+ γ0x

2

)]≈ f0 (x, v, τ = 0)

(A(τ)v2 +B(τ)x2 + C(τ)xv

)(4.11)

where,

A (τ) = −β0

2(η − 1)

B (τ) = −β0

2

(ηξ2 +

2γ0

η− 2

)(4.12)

C (τ) = β0ηξ

Equations (4.8) and (4.9) under the assumption q p < 1, give

η(τ) ≈ 1 + q cos(2√pτ)− q cos(2τ) (4.13)

ξ(τ) ≈−q√p sin(2

√pτ) + q sin(2τ)

1 + q cos(2√pτ)− q cos(2τ)

(4.14)


where terms only up to O (q) have been retained. Substituting the above in

Eq. (4.12), we get,

A (τ) ≈ −β0

2(q cos(2

√pτ)− q cos(2τ))

B (τ) ≈ −0.5β0p (−q cos(2√pτ) + q cos(2τ))

C (τ) ≈ β0 (−√pq sin 2√pτ + q sin 2τ) (4.15)

4.2.2 Linear Vlasov Equation

The linearized Vlasov Equation is:

∂f1

∂t+ v

∂f1

∂x+eE1

m

∂f0

∂v+eE0

m

∂f1

∂v= 0 (4.16)

On substituting, f1 = g1f0, where g1 = Al (τ) v2 + Bl (τ)x

2 + Cl (τ)xv and

solving, we get,

Al =2qβ0

−4 + 4pcos 2

√pτ +

−2qβ0 cos 2τ

−4 + 4p(4.17)

Bl =−2pqβ0

−4 + 4pcos 2

√pτ +

2pqβ0 cos 2τ

−4 + 4p

Cl =4qβ0

−4 + 4psin 2

√pτ +

−4qβ0 sin 2τ

−4 + 4p

where the coefficient of cos 2ντ is chosen such that Al(τ = 0) = 0. For p 1,

we have,

Al ≈ −0.5qβ0 cos 2√pτ + 0.5qβ0 cos 2τ (4.18)

Bl ≈ 0.5pqβ0 cos 2√pτ − 0.5pqβ0 cos 2τ

Cl ≈ −√pqβ0 sin 2√pτ + qβ0 sin 2τ


which is the same expression for A,B,C that we have obtained in Eq. (4.15)

from our exact solutions for q p 1. It has been numerically verified that

the 1 − p factor in the denominator in Eq. (4.17) is required for theory to

agree with numerics. This factor is not there in Eq. (4.15) because in the

approximations that lead to this equation, we have neglected terms of that

order.

The Electric field that is applied to the plasma is unbounded, since it is

proportional to x. However, since the plasma’s scale length is proportional to

1/√

p, the normalized magnitude of the RF field at the nominal edge of the

plasma is qx = O(

q√p

) 1. This perturbation is therefore an acceptable

linear perturbation of a non-neutral plasma.

As shown above, the linear Vlasov theory is still valid. The response at

2ν represents the transient response. The RF field has been switched on at

t = 0. Thus, the plasma has experienced a small kick. The response at 2ν is

due to this finite kick. But since q p, the response at 2ν is still comparable

in magnitude to the response at ω = 2. If p q, then the plasma experiences

a much harder kick and the response at 2ν will then be much larger than the

response at ω = 2. In conventional wave theory, after linearizing the fluid

equations, we do a Fourier transform to obtain steady state solutions. But

the above results show that the Fourier transform must be done very carefully,

because the transient response at frequencies other than the RF frequency may

not die down to zero asymptotically with time.

4.3 Quiet Start

In Sec. (4.2), we considered a plasma that was initially confined by a pure

DC field and an RF field was switched on at τ = 0. Since the electric field for

τ < 0 was DC, the plasma distribution and density were also time independent


and given by the usual Boltzmann-Gibbs distribution. After the RF field was

switched on, we showed that the distribution function will have fluctuations

at the high frequency, ω, and also breathe at the slow frequency, ν.

One possible way to prevent the appearance of the slow breathing term in

the distribution function could be to increase the RF field magnitude slowly

instead of switching it on abruptly at τ = 0. In this case, for τ ≥ 0, the force

equation of the particle is given by

x = [−p+ 2q(τ) cos 2τ ]x (4.19)

where q(τ) is a smooth function of time such that

q(τ) =

0 τ ≤ 0

q1 τ ≥ τ1

and for 0 < τ < τ1, q(τ) slowly increases from q(0) = 0 to q(τ1) = q1.

Since Eq. 4.19 is linear, it has a time dependent exact invariant known as

the Ermakov-Lewis invariant [38] given by

I (x, v, τ) =1

2(ρ (τ) v − ρ′ (τ)x)

2+

x2

2ρ2 (τ)(4.20)

where ρ (τ) is the solution to the equation

ρ′′ − [−p+ 2q(τ) cos 2τ ] ρ− 1

ρ3= 0 (4.21)

The plasma distribution function corresponding to the force equation, Eq.

(4.19), is any arbitrary function of the invariant, Eq. (4.20). However, choos-

ing the distribution function to be an exponential of the invariant leads to a

maximum entropy state. Thus, the distribution function of the plasma is given


by

f (x, v, τ) = exp

[−β0

(1

2(ρ (τ) v − ρ′ (τ)x)

2+

x2

2ρ2 (τ)

)](4.22)

If we want the distribution function to be invariant over the RF time scale,

it is necessary for ρ(τ) to be a periodic function of τ with period, 2π/ω. As

can be seen, Eq. (4.22) is of the same form as Eq. (3.10). The periodicity

of the solution ρ(τ) depends on the initial conditions, ρ(0), ρ′(0). We choose

ρ′(0) = 0 since we want the plasma to have zero drift velocity at τ = 0. Thus,

the only free parameter we have is ρ(0). Comparing Eq. (4.22) with Eq.

(4.1), we can see that ρ (τ) = 1/√

2γ0ρ2 (0)/η (τ). Since η (0) = 1, we have

ρ (0) = 1/

4√

2γ0. In Sec. (3.1.3), we have shown that when q (τ) is a constant,

the plasma distribution function is periodic when γ0 = γ, where γ is given by

Eq. (3.21)

γ =ν2

2

(1 + 2q +O

(q2))

Thus, when q(τ) is a constant, ρ(0) = 1/

4√

2γ, yields a periodic solution to Eq.

(4.21).

For an arbitrary specification of q(τ), Eq. (4.21) will not have periodic

solutions for any choice of ρ(0). However, if q(0) ≈ 0 and the time scale

over which q(τ) changes is much larger compared to 2π/ω, then a choice of

ρ(0) = 1/

4√p gives periodic solutions for ρ(τ). This is shown in Fig. (4.2).

In Fig. 4.2, we have shown the numerical solution for Eq. 4.21 for the case

when q(τ) = q1

(1 + tanh

[(τ − τ0)

/w])

for q1 = 0.01 and p = 0.1. The two

curves correspond to two different values of w = 0.5π, 5π. As can be seen,

for w = 0.5π, ρ(τ) is not asymptotically periodic. But when w is increased to

w = 5π, ρ(τ) tends to a periodic function after a small region of non-periodicity.

Figure 4.3 shows the same thing for q(τ) = q1 exp[−w

(1/τ − 1

/τ0)2]

. Thus,

the individual choice of q(τ) is not important and can be arbitrary. What

matters is the temporal gradient of q(τ), which should be small.


1.74 1.75 1.76 1.77 1.78 1.79 1.8

1.81

0 20 40 60 80 100 120 140

ρ

τ

(a) w=0.5 π

1.75

1.76

1.77

1.78

1.79

1.8

0 20 40 60 80 100 120 140

ρ

τ

(a)

(b)w=5 π

Figure 4.2: This figure shows the solution of Eq. (7.2) for initial condition

ρ(0) = 1/

4√p and q(τ) = q1

(1 + tanh

[(τ − τ0)

/w])

with q1 = 0.01 and

p = 0.1. The two curves correspond to τ0 = 20π and two different values ofw = 0.5π, 5π. As can be seen, for w = 0.5π, ρ(τ) is not asymptotically periodic.But when w is increased to w = 5π, ρ(τ) tends to a periodic function after asmall region of non-periodicity.


1.76

1.77

1.78

1.79

1.8

0 20 40 60 80 100 120 140

ρ

τ

(a) w= π

1.76

1.77

1.78

1.79

0 20 40 60 80 100 120 140

ρ

τ

(a)

(b)w=100 π

Figure 4.3: This figure shows the solution of Eq. (7.2) for initial condition

ρ(0) = 1/

4√p and q(τ) = q1 exp

[−w

(1/τ − 1

/τ0)2]

with q1 = 0.01 and p =

0.1. The two curves correspond to τ0 = 50π and two different values of w =π, 100π. As can be seen, for w = π, ρ(τ) is not asymptotically periodic. Butwhen w is increased to w = 100π, ρ(τ) tends to a periodic function after asmall region of non-periodicity.


1.76

1.77

1.78

1.79

1.8

0 20 40 60 80 100 120 140 160

ρ

τ

(a) w=5 π

1.76

1.77

1.78

1.79

0 50 100 150 200 250 300 350

ρ

τ

(a)

(b)w=50 π

Figure 4.4: This figure shows the solution of Eq. (7.2) for initial conditionρ(0) = 1

/4√p and q(τ) = q1τ

/w with q1 = 0.01 and p = 0.1. The two curves

correspond to two different values of w = 5π, 50π. As can be seen, for w = 5π,ρ(τ) is not asymptotically periodic. But when w is increased to w = 50π, ρ(τ)tends to a periodic function after a small region of non-periodicity.

Figures 4.2 and 4.3 show the solutions of Eq. (4.21) when q(τ) is a smooth

function of time, τ . These functions are however not easy to implement in

any real experiment. It is of interest to see if the same idea of ρ(τ) being

asymptotically period holds even for continuous but non-smooth choices for

the function, q(τ).

Figure (4.4) shows the solution of Eq. (4.21) for initial condition ρ(0) =

1/

4√p and q(τ) = q1τ

/w with q1 = 0.01 and p = 0.1. The two curves corre-

spond to two different values of w = 5π, 50π. As can be seen, for w = 5π,

ρ(τ) is not asymptotically periodic. But when w is increased to w = 50π, ρ(τ)

tends to a periodic function after a small region of non-periodicity. Thus, it is

possible to obtain an asymptotically periodic solution for Eq. (4.21) even for


non-smooth choices of q(τ). This can have important implications for experi-

ments since it is much easier to implement a linearly increasing q(τ) compared

to a tan-hyperbolic function.

4.4 Conclusions

In this chapter, we have considered the response of a DC confined plasma to

an RF field that is switched on at τ = 0. We have shown that the plasma

response at the slow frequency, 2ν, to be as important as that at the RF

frequency, ω = 2.

For q p < 1, we have shown that the linear estimate obtained from the

exact solution agrees with the predictions of linear Vlasov theory.

Though abruptly switching on the RF field at τ = 0 leads to plasma evolu-

tion at the slow frequency, we have shown that if the RF field magnitude is in-

creased sufficiently slowly, then the plasma can be taken from one ω-invariant

distribution function to another ω-invariant distribution function. We have

also numerically shown that the function, q(τ), need not be smooth and even

the first time derivative can be discontinuous. However, in between two points

of discontinuity of the first time-derivative of q (τ), the function should change

slowly enough.

Chapter 5

Spatially nonlinear DC and

linear RF field

In the previous chapter, we have found the distribution function of the plasma

corresponding to the applied field in a Paul trap, and studied its properties.

While doing this, we neglected the field induced by the plasma itself and only

the externally applied linear field was taken into consideration. To obtain

self-consistent solutions for plasma dynamics in Paul traps, we need to simul-

taneously solve the equations

∂f

∂τ+ v

∂f

∂x+

[−px+ 2qx cos 2τ +

eEi(x, τ)

m

]∂f

∂v= 0

∂Ei(x, τ)

∂x= 4πen(x, τ) = 4πe

∞

−∞f (x, v, τ) dv

As of now, there are no known methods of simultaneously solving the above

two equations. One approach to solving the above equations is a recursive

method. In Step 1, we assume Ei = 0 and solve for f . We use this f to find

Ei and plug it back into the Vlasov equation and solve for the new f . And

then again use this new f to find the new Ei, and the process is repeated to

get a sequence of solutions f1, f2, f3, ... However, even carrying out this process

86

CHAPTER 5. SPATIALLY NONLINEAR DC AND LINEAR RF FIELD87

beyond the first step is extremely complicated. That is mainly because though

the applied field is monochromatic, the distribution function f1(x, v, τ) has all

harmonics of ω = 2 and thus, the Ei computed will contain all harmonics too

and most all of the terms would also be spatially nonlinear . Thus, obtaining

exact expressions for f2 is not possible. What we do in this chapter is to

retain only the lowest order nonlinear terms in Ei. We, then, use this modified

electric field expression to obtain expressions for the distribution function.

5.1 Stroboscopic Map

When characterizing particle motion under the effect of nonlinear electric fields,

one important consideration is whether the orbits are regular or chaotic. This

will determine whether we can solve for the particle trajectories as analytic

functions of time. A numerical technique that is used to differentiate between

regular and chaotic orbits is the Poincare Map [39].

For a trajectory in n-dimensional phase space, the set of points formed by

the intersection of this trajectory with a hypersurface of dimension, n − 1, is

known as the Poincare map of the trajectory. This hypersurface need not be

planar, but must be chosen so that the flow of the trajectory is transverse

to the surface at all points. For certain kinds of problems, instead of taking

intersections with a hypersurface, we could also consider the set of points

obtained by sampling the particle orbit at fixed intervals of time. Such a

map is known as the Fixed Time Poincare map or Stroboscopic map. The

stroboscopic map method is particularly useful while studying systems with

periodic forcing, such as a plasma under the effect of RF fields.

Figure 5.1 shows stroboscopic plots of the particle orbits for the fields given

by Eq. (1.12),

x = h(x) + g(x) cosωt


with g(x) being proportional to x and h(x) having a cubic nonlinearity. As

can be seen, some initial conditions lead to simple closed curves and some

others result in a chain of islands. The particle paths corresponding to these

initial conditions are regular [29]. There are also initial conditions, for which

the stroboscopic plot fills an area in phase space. These plots correspond to

particle orbits that are chaotic. Regions in phase space where the stroboscopic

plot changes from being a simple closed curve to a chain of islands is a site

for chaos. For regions in phase space where the stroboscopic plots are simple

closed curves, one could employ the multiple scale analysis used in solving the

standard Mathieu’s equation to solve for the case of nonlinear DC fields as

well [33].

In a Paul trap, since the applied field is time varying, the plasma dis-

tribution function cannot be time independent. However, since the field is

time-periodic, it is reasonable to expect the distribution function to be pe-

riodic, at any given point in phase space, with the same time-period as the

applied field. We will call such a distribution function that is periodic in time

with period, 2π/ω, as an ω-invariant distribution function. In this chapter,

we show that analytic expressions for such an ω-invariant distribution function

can be obtained for regions in phase space where the stroboscopic plot of the

particle trajectory is a simple closed curve.

The results of this thesis fall into two categories. There are results that

describe the self-consistent response of the plasma to an RF field. There are

also results that describe the (non-selfconsistent) plasma response for a given

total electric field. The two sets of results are valid over different ranges of

phase space, but both are important as the latter results are important when

comparing with literature where computations are only for the plasma response

to a prescribed electric field.


-10

-8

-6

-4

-2

0

2

4

6

8

10

-25 -20 -15 -10 -5 0 5 10 15 20 25

v

x

Figure 5.1: This figure shows the stroboscopic plot of the phase space tra-jectory of particles with different initial conditions under the force equationgiven by Eq. (5.4) keeping terms up to O (q2). It can be seen that some orbitsare simple closed curves and some orbits form islands in phase space. Thenumerical values of the various parameters were pe = −0.01, qe = 0.16 ≈ q,ωp = 0.5ν0 and x = 1.


5.2 The force equation

In an RF trap, the applied field is linear. The equation of motion of charged

particles under such a field is x = [−pe + 2qe cos(2τ)]x, where pe and qe are

the normalized DC and RF field strengths respectively and τ = ωt/

2 is the

normalized time. As shown in Eq. (3.10), the distribution function of the

plasma under such a field can be exactly solved for. Using these exact solutions,

it has been shown in our earlier work that the field induced by the charged

particles (charge e and mass m) is

Ei(x, τ) = 2πen0

√π

β0γ0

erf[xx

(1 +

qe2

cos 2τ +O(q2e

))](5.1)

where 1/β0 is the initial temperature of the plasma, γ0 ≈ 0.5ν2

0 (1 + 2q) and

x ≈ 1/√

β0γ0 is the spatial extent of the plasma. Equation (5.1) can be Taylor

expanded in x to get,

eEi

m= ω2

p

[1 +

qe2

cos 2τ]x−

ω2p

3x2

[1 +

3qe2

cos 2τ

]x3 + ... (5.2)

The first term on the right is the way the space charge modifies the externally

applied linear electric field. As can be seen, there is both a change in the DC

field and a change in RF confining field. Thus, the first term on the right

merely modifies strength of the applied field without modifying its shape. The

second term is the first nonlinear factor and is proportional to x3.


0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

erf(

x)

x

(1)

(2)

(3)

Figure 5.2: This is a plot of the error function. Curve (1) shows the exacterror function. Curve (2) is the linear approximation and curve (3) is the plotwith the cubic nonlinearity taken in. This shows that 2

(x− x3

/3) /√

π is agood approximation to the exact error function, erf(x), up to x = 1.

In Fig. (5.2), curve (3) is the cubic approximation to the error function,

erf(x/x). Since it is a good approximation right out to x = x, (5.2) is a suitable

description of the plasma response to the applied field in the bulk region of

the plasma. Thus, the first nonlinearity to appear in (5.1) due to space charge

effects is of order x3. Taking into account this cubic nonlinearity, the modified

equation for the motion of an ion in the RF trap takes the form

d2x

dτ 2+ px = 2qx cos 2τ −

ω2p

3

(xx

)2

x−ω2

p

2

(xx

)2

qx cos 2τ (5.3)

where p = pe − ω2p and q = qe

(1 + 0.25ω2

p

)≈ qe since ωp ω = 2. As can be

seen from Eq. (5.3), the qx3 cos 2τ term is of a higher order than the x3 term

since q 1. It is worth noting that q is of order unity in typical experiments;


however, theoretical treatments can make progress only when q is assumed to

be a small parameter. We will assess the importance of neglected terms in real

experimental conditions later, in section 5.7. Neglecting the qx3 cos 2τ term in

Eq. (5.3), we obtain

d2x

dτ 2+ px = 2qx cos 2τ − q2αx3 (5.4)

where q2α = ω2p

/3x2. Equation (5.4) can be solved by using the MLP method

described in Sec. (2.1.3). Though we have neglected the qx3 cos 2τ term in

this chapter, but this term is taken up in Chapter 6 and discussed in detail.

5.3 Solving the force equation

We need to solve Eq. (5.4),

d2x

dτ 2+ px = 2εx cos 2τ − ε2αx3

Using the conventional way of solving Mathieu’s equation, we write

x = x0 + εx1 + ε2x2 + ...

and

p = ν2 + εw21 + ε2w2

2 + ...

Substituting this in the equation of motion, we get to O (ε0),

x0 + ν2x0 = 0

⇒ x0 = A cos (ντ + φ)


To O (ε1), we get,

x1 + ν2x1 + w21x0 = 2x0 cos 2τ

To eliminate secular terms, we demand w21 = 0 and this gives,

x1 + ν2x1 = A [cos(ντ + 2τ + φ) + cos(ντ − 2τ + φ)]

⇒ x1 = A

[cos(ντ + 2τ + φ)

−(ν + 2)2 + ν2+

cos(ντ − 2τ + φ)

−(ν − 2)2 + ν2

]To O (ε2), we get,

x2 + ν2x2 + w21x1 + w2

2x0 = 2x1 cos 2τ − αx30

= 2A


−(ν + 2)2 + ν2+


−(ν − 2)2 + ν2

]cos 2τ

−αA3 cos3(ντ + φ)

= Acos(ντ + 4τ + φ) + sin(ντ + φ)

−(ν + 2)2 + ν2

+Acos(ντ + φ) + sin(ντ − 4τ + φ)

−(ν − 2)2 + ν2

−αA3

4[3 cos(ντ + φ) + cos(3ντ + 3φ)]

To eliminate secular terms, we demand,

w22 =

[1

−(ν + 2)2 + ν2+

1

−(ν − 2)2 + ν2

]− 3αA2

4

=1

2(ν2 − 1)− 3αA2

4

This gives,

x2 + ν2x2 = A


−(ν + 2)2 + ν2+


−(ν − 2)2 + ν2

]− αA3

4cos(3ντ + 3φ)


⇒ x2 = A


[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2]+


[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2]

]−αA

3

4

cos(3ντ + 3φ)

−8ν2

Thus, we have

p = ν2 +q2

2(ν2 − 1)− 3αq2A2

4

Since ν 1, the above equation for ν can be approximated to give

ν2 = p+q2

2+

3αq2A2

4(5.5)

Thus, the solution to Eq. (5.4) correct up to second order is

x(τ) = A cos(ντ + φ)− qA

4


1 + ν+


1− ν

]+q2A

32


(1 + ν) (2 + ν)+


(1− ν) (2− ν)

]−q

2αA3

4

cos(3ντ + 3φ)

−8ν2(5.6)

v(τ) = −νA sin(ντ + φ)

−qA4

[(ν + 2) sin(ντ + 2τ + φ)

1 + ν+

(ν − 2) sin(ντ − 2τ + φ)

1− ν

]−q

2A

32

[(ν + 4) sin(ντ + 4τ + φ)

(1 + ν) (2 + ν)+

(ν − 4) sin(ντ − 4τ + φ)

(1− ν) (2− ν)

]+3ν

q2αA3

4

sin(3ντ + 3φ)

−8ν2(5.7)

with ν given by Eq. (5.5). Equation (5.5) shows that under the effect of the

nonlinear space charge field, the slow frequency of the particles depends on

oscillation amplitude and is no longer a constant as was the case of purely


linear fields. Also, the perturbation scheme breaks down unless

q2αA2

32ν2 1

since otherwise the O (q2) correction would be as large as the O (q0) term.

5.4 Time averaged motion

As mentioned before, equations such as Eq. (5.4) that belong to the category

of equations where a high frequency forcing function is present are usually

analyzed by the using the theory of averaging. Time averaging the expressions

for x(t) and v(t) given in Eq. (5.6) and Eq. (5.7) yields


xa(τ) =1

2π

τ+π

τ−π

x(t)dt

⇒ 2πxa(τ) = A

τ+π

τ−π

cos(νt+ φ)dt

+qA

[ τ+π

τ−πcos(νt+ 2t+ φ)dt

−(ν + 2)2 + ν2+

τ+π

τ−πcos(νt− 2t+ φ)dt

−(ν − 2)2 + ν2

]

+q2A

τ+π

τ−πcos(νt+ 4t+ φ)dt

[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2]

+q2A

τ+π

τ−πcos(νt− 4t+ φ)dt

[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2]

−q2αA3

4

τ+π

τ−πcos(3νt+ 3φ)dt

−8ν2

= Asin(ντ + νπ + φ)− sin(ντ − νπ + φ)

ν

+qAsin(ντ + νπ + 2τ + 2π + φ)− sin(ντ + 2τ − νπ − 2π + φ)

[−(ν + 2)2 + ν2] [ν + 2]

+qAsin(ντ + νπ − 2τ − 2π + φ)− sin(ντ − νπ − 2τ + 2π + φ)

[−(ν − 2)2 + ν2] [ν − 2]

+q2Asin(ντ + νπ + 4τ + 4π + φ)− sin(ντ − νπ + 4τ − 4π + φ)

[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2] [ν + 4]

+q2Asin(ντ + νπ − 4τ − 4π + φ)− sin(ντ − νπ − 4τ + 4π + φ)

[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2] [ν − 4]

+q2αA3

32ν2

sin(3ντ + 3νπ + 3φ)− sin(3ντ − 3νπ + 3φ)

3ν

= A2 sin(νπ) cos(ντ + φ)

ν

+qA2 sin(νπ + 2π) cos(ντ + 2τ + φ)

[−(ν + 2)2 + ν2] [ν + 2]

+qA2 sin(νπ − 2π) cos(ντ − 2τ + φ)

[−(ν − 2)2 + ν2] [ν − 2]

+q2A2 sin(νπ + 4π) cos(ντ + 4τ + φ)

[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2] [ν + 4]

+q2A2 sin(νπ − 4π) cos(ντ − 4τ + φ)

[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2] [ν − 4]

+q2αA3

32ν2

2 sin(3νπ) cos(3ντ + 3φ)

3ν


⇒ xa(t) = A cos(ντ + φ)sin(νπ)

νπ

+qA


[−(ν + 2)2 + ν2] [ν + 2]+


[−(ν − 2)2 + ν2] [2− ν]

]sin(νπ)

π

+q2Acos(ντ + 4τ + φ)

[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2] [ν + 4]

sin(νπ)

π

+q2Acos(ντ − 4τ + φ)

[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2] [4− ν]

sin(νπ)

π

+q2αA3

32ν2cos(3ντ + 3φ)

sin(3νπ)

3νπ

≈ A cos(ντ + φ) +q2αA3

32ν2cos(3ντ + 3φ) (5.8)

where we have used the fact that νπ 2π ⇒ sin(νπ) ≈ 0 and sin(νπ)/νπ ≈

1. Similarly, one can solve for va(t), to get,

xa(τ) =1

π

τ+π/2

τ−π/2

x(t)dt

≈ A cos(ντ + φ) +q2αA3

32ν2cos(3ντ + 3φ)

va(τ) =1

π

τ+π/2

τ−π/2

v(t)dt

≈ −νA sin(ντ + φ)2 sin (νπ/2)

νπ− 3q2αA3

32νsin(3ντ + 3φ) (5.9)

where the residual high-frequency ripple has been neglected. Additionally, a

factor 2 sin (νπ/2)/νπ has been dropped from the first term on the right side

of the equations. This is because the second terms on the right side of Eqs.

(5.6) and (5.7) suffer from the problem of small denominators and those terms

are actually of order q2αA2/ν2 compared to the first term. Using Eq. (5.4)

with A = O(x), this is of order ω2p/ν

2 and hence of order unity by Eq. (1.20).

Thus, these terms have to be taken into account. The factor of 2 sin (νπ/2) /νπ

that multiplies the leading terms, thus, leads to higher order corrections.

The time-averaged response to Eq. (5.4) is conventionally obtained by


averaging the exact Hamiltonian. The exact Hamiltonian corresponding to

Eq. (5.4) is

H =v2

2+px2

2− qx2 cos 2τ +

q2αx4

4(5.10)

The time averaged ponderomotive Hamiltonian corresponding to the exact

Hamiltonian, Eq. (5.10), is given by

Hp =v2

2+px2

2+q2x2

4+q2αx4

4(5.11)

The particle orbits corresponding to Eq. (5.11), obtained by perturbation

analysis [32], are found to be

xp (τ) ≈ Ap cos(ντ + φp) +q2αA3

p

32ν2cos(3ντ + 3φp)

va (τ) ≈ −νAp sin(ντ + φp)−3q2αA3

p

32νsin(3ντ + 3φp) (5.12)

which is the same as the expressions for time averaged orbits, Eq. (5.9). It

must be noted that the derivation of the ponderomotive Hamiltonian includes

the notion of averaging over a “fast period”, but no sin νπ/νπ term appears

due to the implicit assumption that ν → 0. The is discussed further in Sec.

7.2.


For collisionless plasmas, the distribution function is simply a function of any

invariant associated with single particles. As is well known, time independent

invariants cannot exist when the applied fields depend explicitly on time. How-

ever, for periodically driven systems, it is reasonable to expect such a function

to be invariant on the same time scale, 2π/ω, as the driving force. For a given

time varying nonlinear electric field, such an ω-invariant distribution will exist


only if the stroboscopic plot of the orbits in phase space are simple closed

curves.

A stroboscopic plot of the particle orbit leads to a countable subset of

discrete points, separated by fixed time intervals of 2π/ω = π, on the phase

space orbit. The numerical analysis of Eq. (5.4) shows that the stroboscopic

plot of the particle orbit is dense on a simple closed curve only where the

gradients in the field are small enough. This can be seen in Fig. (5.1). For

other regions in phase space, we see island structures and, possibly, chaos.

Analytic expressions for the distribution function can only be expected to

hold in the regular region, and thus, we cannot expect a global distribution

function for the entire phase space that is ω-invariant. However, in regions of

phase space where the stroboscopic plot yields a simple closed curve, such a

function can be obtained. Such curves correspond to an irrational value of the

slow frequency ν. Of course, every rational value of ν even in these regions

results in island formation. However, their widths are so small that their effect

on the dynamics is negligible [29].

Figure 5.3 shows the particle orbit (curve 1) together with the stroboscopic

curve (curve 2). The time-averaged orbit (curve 3) is also shown for reference.

It is clear that the actual orbit differs considerably from the stroboscopic curve.

An odd aliasing effect is also seen in the stroboscopic curve in that it touches

the actual orbit at different phases of the fast oscillation at different slow time.

This results in a curve that differs from the time-averaged curve shown in the

figure, even though both have the same shape up to O (q2).

5.6 Stroboscopic Analysis

To obtain the expression for points on the stroboscopic map of the orbit, we

start from a time, τ , and sample the particle orbit at equal time steps of


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-4 -3 -2 -1 0 1 2 3 4

v

x

(1)

(2)

(3)

(4)

v

Figure 5.3: This plot shows the stroboscopic map of the particle orbits inphase space. Curve (1) is the full orbit as obtained by integration of Eq. (5.4).Curve (2) is the stroboscopic plot obtained by a sampling of the particle orbitat a fixed time step of 2π

/ω starting from τ = 0. The big crosses correspond

to a few points on this level curve. These points form a countable set that isdense on the level curve given by Eq. (5.28). Curve (3) is the time averagedorbit and corresponds to the time averaged ponderomotive Hamiltonian, Eq.(5.11). Curve (4) is also a stroboscopic plot like curve (2), but, in this case,the sampling begins at τ 6= 0. The arrow labeled v shows the direction ofsampled velocity, vs, at one particular instant of time. It can be clearly seenthat vs is not along the stroboscopic level curve, which explains the reasonwhy vs 6= dxs

/dt in Eq. (5.15) and Eq. (5.16).


2π/ω = π. This is equivalent to replacing τ in Eq. (5.6) and Eq. (5.7) by

τ + nπ where n ∈ Z,

x(τ + nπ) = A cos(ντ + νnπ + φ)

+qAcos(ντ + νnπ + 2τ + 2nπ + φ)

−(ν + 2)2 + ν2

+qAcos(ντ + νnπ − 2τ − 2nπ + φ)

−(ν − 2)2 + ν2

+q2Acos(ντ + νnπ + 4τ + 4nπ + φ)

[−(ν + 2)2 + ν2] [−(ν + 4)2 + ν2]

+q2Acos(ντ + νnπ − 4τ − 4nπ + φ)

[−(ν − 2)2 + ν2] [−(ν − 4)2 + ν2]

−q2αA3

4

cos(3ντ + 3νnπ + 3φ)

−8ν2(5.13)

and similarly we can obtain the expression for v(τ +nπ). We can re-write Eq.

(5.13) and the expression for v(τ + nπ) as,

x(τ + nπ) =

[A− qA

2

cos(2τ)

1− ν2+q2A

16

(2 + ν2) cos(4τ)

[1− ν2] [2− ν2]

]cos(ντ + νnπ + φ)[

−qA2

ν sin(2τ)

1− ν2+q2A

16

3ν sin(4τ)

[1− ν2] [2− ν2]

]sin(ντ + νnπ + φ)

−q2αA3

4

cos(3ντ + 3νnπ + 3φ)

−8ν2

v(τ + nπ) =

[−νA+

qA

2

−ν cos(2τ)

1− ν2

]sin(ντ + νnπ + φ)

−q2A

16

(ν3 − 10ν) cos(4τ)

[1− ν2] [2− ν2]sin(ντ + νnπ + φ)

+qA

2

(2− ν2) sin(2τ)

1− ν2cos(ντ + νnπ + φ)

−q2A

16

(8 + ν2) sin(4τ)

[1− ν2] [2− ν2]cos(ντ + νnπ + φ)

+3νq2αA3

4

sin(3ντ + 3νnπ + 3φ)

−8ν2(5.14)


Since the stroboscopic points, x(τ + nπ) and v(τ + nπ), are dense on a

simple closed curve in phase space, we can also define a continuous time, τs,

on these level curves of the stroboscopic map. This allows us to determine the

invariant underlying the curve. The continuous time expressions for strobo-

scopic coordinates (xs, vs) in phase space are given by replacing nπ by τs in

Eq. (5.14),

xs(τ + τs) =

[A− qA

2

cos(2τ)

1− ν2+q2A

16

(2 + ν2) cos(4τ)

[1− ν2] [2− ν2]

]cos(ντ + ντs + φ)[

−qA2

ν sin(2τ)

1− ν2+q2A

16

3ν sin(4τ)

[1− ν2] [2− ν2]

]sin(ντ + ντs + φ)

−q2αA3

4

cos(3ντ + 3ντs + 3φ)

−8ν2(5.15)

vs(τ + τs) =

[−νA+

qA

2

−ν cos(2τ)

1− ν2

]sin(ντ + ντs + φ)

−q2A

16

(ν3 − 10ν) cos(4τ)

[1− ν2] [2− ν2]sin(ντ + ντs + φ)

+qA

2

(2− ν2) sin(2τ)

1− ν2cos(ντ + ντs + φ)

−q2A

16

(8 + ν2) sin(4τ)

[1− ν2] [2− ν2]cos(ντ + ντs + φ)

+3νq2αA3

4

sin(3ντ + 3ντs + 3φ)

−8ν2(5.16)

Here, for τs = 0, xs(τ) and vs(τ) are points on the continuous orbit. For fixed

τ , xs(τ + τs) and vs(τ + τs) trace out the stroboscopic curve passing through

that phase space point, xs(τ), vs(τ). vs(τ + τs), the instantaneous velocity at

a stroboscopic point (shown in the Fig. 5.3 by an arrow), is clearly not along

the stroboscopic plot, i.e., vs 6= dxs

/dτs. Hence, for fixed τ , the stroboscopic

coordinates, (xs, vs), as functions of τs, are not the spatial coordinate and

velocity in the usual sense.

To obtain the ω-invariant energy expression (periodic in τ) for the actual


particle orbit, we need to eliminate τs from Eq. (5.15) and Eq. (5.16). For

τ = 0, xs(τs) and vs(τs) can be simply scaled to obtain the same expressions as

Eq. (5.9). As explained in Sec. (5.4), the orbits in Eq. (5.9) can be obtained

as the solutions of the time independent Hamiltonian given in Eq. (5.11).

Thus, that Hamiltonian is the invariant for the time averaged dynamics. Since

xs (τs) , vs (τs) are scaled versions of Eq. (5.9), a scaled version of Eq. (5.11) is

the ω-invariant for this problem. Hence, a scaled version of Eq. 5.11 represents

the invariant for these stroboscopic level curves corresponding to τ = 0.

Figure (5.3) shows the presence of ω-invariant curves (curve 3). However,

there are also other ω-invariant curves (curve 4) present. We need a unified

treatment to handle all these invariants. We begin by rewriting Eqs. (5.15)

and (5.16):

xs(τ + τs) = Ac1(τ) cos (ντ + ντs + φ)

+νqAc2(τ) sin (ντ + ντs + φ)

+q2αA3

32ν2cos (3ντ + 3ντs + 3φ) (5.17)

vs(τ + τs) = −νAc3(τ) sin (ντ + ντs + φ)

+qAc4(τ) cos (ντ + ντs + φ)

−3q2αA3

32νsin (3ντ + 3ντs + 3φ) (5.18)

where c1, c2, c3, c4 are periodic functions of τ with the same period as the RF


field, 2π/ω,

c1 (τ) = 1− q

2

cos(2τ)

1− ν2+q2

16

(2 + ν2) cos(4τ)

[1− ν2] [2− ν2]

≈ 1− q

2cos(2τ) +

q2

16cos(4τ)

c2 (τ) = −1

2

sin(2τ)

1− ν2+

q

16

3 sin(4τ)

[1− ν2] [2− ν2]

≈ −1

2sin(2τ) +

3q

32sin(4τ)

c3 (τ) = 1 +q

2

cos(2τ)

1− ν2+q2

16

(ν2 − 10) cos(4τ)

[1− ν2] [2− ν2]

≈ 1 +q

2cos(2τ)− 5q2

16cos(4τ)

c4 (τ) =1

2

(2− ν2) sin(2τ)

1− ν2− q

16

(8 + ν2) sin(4τ)

[1− ν2] [2− ν2]

≈ sin(2τ)− q

4sin(4τ) (5.19)

Eqs. (5.17) and (5.18) can be written in matrix notation as

xs

vs

= M1

cos (ντ + ντs + φ)

sin (ντ + ντs + φ)

+ M2

cos (3ντ + 3ντs + 3φ)

sin (3ντ + 3ντs + 3φ)

(5.20)

where

M1 =

Ac1 (τ) νqAc2 (τ)

qAc4 (τ) −νAc3 (τ)

(5.21)

and

M2 =

q2αA3/32ν2 0

0 −3q2αA3/32ν

(5.22)

The determinant of M1 is given by

det (M1) = −νA2(c1c3 + q2c3c4

)(5.23)

Now, since q 1 and c1, c3 > 0 for all τ , we have det (M1) 6= 0. Thus, M1 is


invertible and its inverse is given by

M1−1 =

1

−νA2 (c1c3 + q2c3c4)

−νAc3 −νqAc2

−qAc4 Ac1

(5.24)

Multiplying Eq. (5.20) by M1−1 and a subsequent scaling transforms xs(τ +

τs), vs(τ + τs) to the expressions in Eq. (5.9)

X = A cos(ντ + ντs + φ)

+q2αA3

32ν2cos(3ντ + 3ντs + 3φ)

V = −νA sin(ντ + ντs + φ)

−3q2αA3

32νsin(3ντ + 3ντs + 3φ) (5.25)

where

X =xs + q c2(τ)

c3(τ)vs

c1 (τ) + q2 c2(τ)c4(τ)c3(τ)

and

V =vs − q c4(τ)

c1(τ)xs

c3 (τ) + q2 c2(τ)c4(τ)c1(τ)

Comparing Eq. (5.25) with Eq. (5.9), it is clear that X(τ + τs) = xa(τ + τs)

and V (τ+τs) = va(τ+τs) to O (q2). Thus, the dynamical structure connecting

X and V is the same as that governing xa and va. Thus, the complete time

dependent invariant corresponding to the particle orbit has the same form as

the ponderomotive Hamiltonian, Eq. (5.11). This leads to an expression for


the ω−invariant energy of the particle,

Eω(x, v, τ) =V 2

2+pX2

2+q2X2

4+q2αX4

4

=1

2

vs − q c4(τ)c1(τ)

xs[c3(τ) + q2 c2(τ)c4(τ)

c1(τ)

]2

(5.26)

+1

2

(p+

q2

2

) xs + q c2(τ)c3(τ)

vs[c1(τ) + q2 c2(τ)c4(τ)

c3(τ)

]2

+q2α

4

xs + q c2(τ)c3(τ)

vs[c1(τ) + q2 c2(τ)c4(τ)

c3(τ)

]4

Keeping only terms up to O (q2), the expression for ω-invariant energy corre-

sponding to Eq. (5.4) is given by

Eω(x, v, τ) =1

2

v − q c4(τ)c1(τ)

x[c3(τ) + q2 c2(τ)c4(τ)

c1(τ)

]2

(5.27)

+1

2

(p+

q2

2

) x[c1(τ) + q2 c2(τ)c4(τ)

c3(τ)

]2

+q2α

4

x[c1(τ) + q2 c2(τ)c4(τ)

c3(τ)

]4

It should be noted that Eω(x, v, τ) is invariant over the detailed orbit (curve

1) in Fig. (5.3). For time instants τ = nπ, the above expression for the

ω-invariant energy becomes

Eω (x, v, τ = nπ) =1

2

(v

1 + 0.5q − 0.3125q2

)2

(5.28)

+1

2

(p+

q2

2

)(x

1− 0.5q + 0.0625q2

)2

+q2α

4

(x

1− 0.5q + 0.0625q2

)4


As can be clearly seen, the above equation is a scaled version of the time

averaged ponderomotive Hamiltonian, Eq. (5.11).

In principle, any function of Eω(x, v, τ) is an ω-invariant distribution func-

tion of the plasma under the effect of the force equation, Eq. (5.4). Since Eω

is quadratic in v, a distribution of the form

f(x, v, τ) = n0

√1

2πT0

exp

(−Eω(x, v, τ)

T0

)(5.29)

has a velocity distribution that is Maxwellian at every point. Here, Eω is given

by Eq. (5.27) and n0 and T0 are the density and temperature respectively at

the origin. Since this distribution function is Maxwellian (up to O(q2)) at

all times and at all spatial locations, it is immune to point collisions, and

is also a solution to the problem when Coulomb collisions are present. The

temperature,

T (τ) = T0

[c3(τ) + q2 c2(τ)c4(τ)

c1(τ)

]2

(5.30)

is spatially uniform but oscillates at the RF frequency, ω = 2. A similar

result was obtained in our earlier work for the linear field case also [29]. The

time averaged distribution function can be obtained by doing an average of

Eq. (5.29) over the RF period. The instantaneous density can be obtained by

integrating Eq. (5.29) with respect to velocity,

n(x, τ) =

∞

−∞f(x, v, τ)dv

= n0

[c3 + q2 c2c4

c1

](5.31)

× exp

[− β0

1

2

(p+

q2

2

) x[c1 + q2 c2c4

c3

]2

+q2α

4

x[c1 + q2 c2c4

c3

]4]


As can be seen in the above equation, when space charge effects are taken into

account, the density is no longer a Gaussian.

The field induced by the plasma can be obtained by integrating the expres-

sion for plasma density, Eq. (3.11). There is no closed form expression for the

result of this integration. However, since we are interested in a region close

to the origin, we can expand the exponential in Eq. (3.11) and integrate the

resulting power series keeping only those terms that are important,

E(x, τ) = −4πe

x

0

n(x′, τ)dx′

≈ −4πen0

[c3 + q2 c2c4

c1

](5.32)

×

x− β0

1

6

(p+

q2

2

)x3[

c1 + q2 c2c4c3

]2 +q2α

20

x5[c1 + q2 c2c4

c3

]4

In terms of the modified plasma extent, x = 1/√

β0γ, where γ = 0.5p+ 0.25q2

the above equation can be written as,

E(x, 0) = −4πen0x

[x

x− x3

6x3− q2α

20β0γ2

x5

x5

]

If the x5 term becomes important in the above equation, then our original

force equation, Eq. (5.4), becomes invalid. Thus, the results of this thesis are

valid when the term containing x5 in the above equation can be neglected, i.e.,

when 6q2α/

(20β0γ2) < 1. This implies

ωP < 0.8ν0 (5.33)


5.7 Discussion

5.7.1 Space charge effects in RF traps

In experiments on RF traps, one very important issue is the maximum density

of plasma that can be trapped for a given value of pe and qe. Since the plasma

in such traps consists of ions of a single species, it is clear that the plasma is

trying to push itself out. For confinement, we must have p + 0.5q2 > 0. This

condition translates to ωp < ν0 =√

(pe + 0.5q2e). This is the same order of

magnitude as the bound prescribed in Eq. (5.33). Thus, for most RF trap

experiments, the x5 term in Eq. (5.32) is not very important and hence, our

analysis is valid for most practical purposes.

It has been experimentally shown that the choice of pe = −0.03 and qe =

0.55 leads to the most stable configuration in a RF trap and can trap the

maximum number of ions [27]. For these values of applied fields, a choice of

ωp = ν0 gives a plasma density of the order of 105cm−3 for protons in a trap

operating at 3MHz, which is a widely used operating frequency in RF traps.

And this density is of the same order of magnitude as the maximum possible

density of plasma observed in such traps [27]. It must be noted that a value of

qe = 0.55 is very high compared to the condition qe 1 for which conventional

ponderomotive theory and also our analysis up to O (q2e) holds. Thus, care

must be taken while applying theoretical results to actual experiments.

Equation (3.11) predicts a density that is different from a Gaussian. How-

ever, if Eq. (5.33) is satisfied, then the deviation from a Gaussian is small.

And, as we have shown, when the plasma density exceeds the bound prescribed

in Eq. (5.33), the plasma is very close to the stability boundary beyond which

it can no longer be confined. Hence, this explains the reason for which exper-

iments on RF traps observe a Gaussian density [18].

It is important to note that this maximum density that can be confined also


depends on mass. The bound is on plasma frequency, which scales as√n0

/m.

Thus, for a given maximum ωp, a larger particle mass results in higher density.

5.7.2 Distribution function

Conventionally, the time averaged distribution function for a plasma subject

to high frequency electric fields, has been assumed to be a function (expo-

nential) of the time averaged ponderomotive Hamiltonian [16]. The idea is

that Vlasov’s Equation preserves f along its orbit, and hence the proper time-

averaged distribution for the problem is to require f to be constant on the

time-averaged trajectories.

Strictly speaking, f is preserved on the exact orbit and the time-averaged

distribution is the time average of that exact f . However, the exact orbit

for a non-autonomous problem has no time-independent invariants, and that

f is therefore time-dependent. As we argued in Sec. (5.5), the only time-

independent invariant that can exist in a system with a periodic disturbance

is one that is periodic, i.e., an invariant that is time-independent if sampled

every 2π/ω. Thus, the only correct prescription for a system with a periodic

disturbance is an f that is a function of such a periodic invariant. Precisely

such an invariant was obtained to O(q2) in the previous section, and Eq. (5.29)

is the distribution based on it.

The only proper time-averaged distribution function f is therefore the one

that is obtained by time-averaging the exact f obtained in the previous section.

It is not correct to time average the orbits and declare the distribution function

to be constant on that time-averaged orbit. There are two consequences to the

method followed in this paper:

1. The exact distribution on the actual orbit has been obtained, and it is

found that to O(q2), the exact distribution is Maxwellian at all times and

at all spatial locations. Since this is the case, this solution is also the


correct solution (to O(q2)) for the Vlasov Boltzmann equation for this

problem, where collisions are due to point coulomb collisions only. This

result was obtained earlier for the exactly linear problem, and is shown

here to be true even when self-consistent effects are taken into account.

2. The instantaneous temperature is observed to be spatially uniform but

oscillates at the RF frequency. This behavior is identical to that observed

for the strictly linear problem [40]. There have been conjectures in the

literature [21], [6] that the plasma temperature should fluctuate at the

RF frequency, but this analysis is, to our knowledge, the first proof of

these conjectures.

The transformation (xs, vs) to (X,V ) given below Eq. (5.25) represents a map-

ping from the instantaneous orbit to the slow time orbit. Given a stroboscopic

orbit that is a closed curve, there is always a mapping. This mapping can be

expanded to the desired order in xs and vs.

There are many such transformations from (xs, vs) to slow time variables

(X,V ). This is because for different initial points on the instantaneous orbit,

different stroboscopic curves are obtained. If one of them lies on a closed curve,

then all lie on closed curves. But only the orbit starting with (xs(0), vs(0)) is

related to the ponderomotive orbit via a scaling. For the O(q2) analysis done

here, this map is a linear map of the form

xs

vs

=

R11 qR12

qR21 R22

xs(0)

vs(0)

and hence the time-dependent invariant is quadratic in velocity at all points

on the orbit. This powerful result is what makes this distribution immune to

collisions as mentioned above.

When we proceed to higher order, we still have a time-dependent invariant


and a time-independent stroboscopic curve. It is also known that the pon-

deromotive energy is quadratic in velocity asymptotically [39]. The issue is

whether the detailed distribution function is Maxwellian or not at all instants

of time. As seen from the transformation above, xs and vs get mixed up at the

next higher order. Hence, a nonlinearity in space in the form of φP (x) appears

in the form of a non-quadratic velocity dependence of the invariant. Thus,

to higher order, the distribution function cannot be Maxwellian at all points

in time. However, it is possible that it is Maxwellian at stroboscopic times,

since higher order averaging theory has demonstrated this for the ponderomo-

tive Hamiltonian. It is also clear that the higher-order accurate distribution

function is not immune to collisions in the way that the O (q2) distribution

function is immune. Thus, this opens a channel of relaxation of plasmas un-

der the influence of an RF field. The other channel, already discussed in Sec.

(3.3.6), is if more than one species is present in the plasma. Multiple species

systems relax in the presence of collisions even in linear theory.

We have shown that to the lowest order in the nonlinearity, the distribution

function is a Maxwellian. This was possible because of two reasons. Firstly, the

ponderomotive Hamiltonian is quadratic in velocity. Secondly, the stroboscopic

curve could be scaled to get the time averaged ponderomotive orbit and this

scaling depends only on the field parameters and not the initial conditions of

the particle. However, as shown in the next chapter, for the case where the RF

field itself is spatially nonlinear, the distribution function is non-Maxwellian

even for the lowest order of nonlinearity.

It should be noted that the analysis in this paper is self-consistent only for

x ≤ x, since the error function has been approximated by a cubic to derive

that response. However, the orbits, distribution function and density for a

cubic electric field that have been calculated are valid for a much larger range.

Assuming that the validity of these expressions is at least up to where the


cubic term competes with the linear term, the calculations are valid as long as

q2αA2

32ν2< 1

From Sec. (5.2), we have q2α = ω2p

/3x2 and from Eq. (5.5) we have ν2 = ν2

0 +

0.75q2αA2 where ν20 = pe + 0.5q2

e . Substituting these in the above inequality,

we getA2ω2

p

96(ν2

0x2 + 0.25ω2

pA2) < 1

which is satisfied for all values of A. However, it is also necessary that in Eq.

(5.5), we must have 0.75q2αA2 < ν20 . This translates to the condition

A

x< 2

ν0

ωp

Thus, for ωp = 0.5ν0, we get A < 4x. Figure (5.4) compares the stroboscopic

map of the particle orbits obtained by numerical integration of Eq. (5.4)

and the analytical expression given by Eq. (5.6). As can be seen, as the

initial condition of the particle moves away from the origin in phase space, the

analytical expression become less and less accurate.

5.8 Conclusions

In this chapter, plasma response to an electrostatic field which is a superpo-

sition of nonlinear DC and linear RF field has been studied. The expressions

for particle orbits, Eq. (5.6), corresponding to the force equation, Eq. (5.4),

are perturbatively solved for. These solutions are then time averaged and the

resulting time averaged orbits, Eq. (5.9), are compared to the predictions of

conventional ponderomotive theory. It is found that the expression for the

time averaged particle orbit, Eq. (5.9), obtained from the exact Hamiltonian

is the same as the orbit predicted by time averaged ponderomotive Hamilto-


-10

-8

-6

-4

-2

0

2

4

6

8

10

-25 -20 -15 -10 -5 0 5 10 15 20 25

v

x

numericsanalytics

Figure 5.4: Comparison of stroboscopic map of the particle orbit obtainedby numerical integration of Eq. (5.4) and the analytical expressions given byEq. (5.6). As can be seen, as the initial condition of the particle moves awayfrom the origin in phase space, the analytical expression become less and lessaccurate.


nian, Eq. (5.11), in the limit ν → 0. But, as is well known in the context of

oscillation center theories, for ponderomotive theory to be able to predict the

time averaged orbit, we need to know the ponderomotive initial conditions,

xp(0), vp(0) corresponding to the actual initial conditions, x0, v0. For this, we

need to perturbatively solve the exact equation, Eq. (5.4).

It has been shown that the idea of time averaged distribution function is

different from that of obtaining time averaged particle orbits or time aver-

aged Hamiltonian. The time averaged distribution function is not constant on

the time averaged particle orbits in phase space. The ω-invariant distribution

function, Eq. (5.29), is a function of the resultant ω-invariant energy expres-

sion, Eq. (5.27). And, thus, the time average of this ω-invariant distribution

function is different from the exponential of the time averaged ponderomotive

Hamiltonian, Eq. (5.11). The temperature is also found to periodically os-

cillate with time but is constant with space. However, the plasma has been

found to be a Maxwellian only up to O (q2). This could be a possible reason

for the observed heating in RF traps.

Apart from theoretical relevance, the results obtained in this work are

very important from the experimental point of view. We have shown that the

condition, Eq. (5.33), where the plasma density deviates from being a Gaussian

is very close to the maximum plasma density that has been experimentally

observed to be trapped in a RF trap. Also, the maximum plasma density that

can be confined in such traps has been shown to linearly scale with the mass

of the confined ions.

Chapter 6

Spatially nonlinear RF Fields

In the previous two chapters, we have considered dynamics of a plasma subject

to a combination of DC fields and RF fields. Though we have considered the

case of nonlinear DC fields, the RF field has so far been considered to be only

linear. In this chapter, we introduce a nonlinear RF field, while keeping the

DC field linear, and study the dynamics of a plasma in such fields. The general

equation of motion of particles in the type of fields we are going to consider

now is

x+ px = qg(x) cos 2τ (6.1)

where g(x) is a smooth function of x. We will consider the special cases,

g(x) = βx3. This choice of g(x) is partly motivated by the nonlinear RF term

in Eq. (5.3). More importantly, a cubic spatial profile for g(x) mimics a wall

as opposed to a distributed retarding field like βx.

We begin by solving the force equation and then show that the predictions

of ponderomotive theory for the orbits matches with the time average orbits

obtained from exact solutions. Then, we construct the stroboscopic map of

the particle trajectory and use it to construct the distribution function of the

particle.

116

CHAPTER 6. SPATIALLY NONLINEAR RF FIELDS 117

6.1 The particle trajectory

The force equation under consideration in this chapter is

x+ px = qβx3 cos 2τ (6.2)

where β = 1/x2, with x being a measure of the plasma extent. As shown

in Appendix (A), using the modified Lindstedt-Poincare method described in

Sec. (2.1.3), we can solve Eq. (6.2) to obtain expressions for the particle orbit

in phase space,

x(τ) ≈ A cos(ντ + φ)

−qβA3

32[cos (3ντ + 2τ + 3φ) + cos (3ντ − 2τ + 3φ)]

−3qβA3

32[cos (ντ + 2τ + φ) + cos (ντ − 2τ + φ)]

+30q2β2A5

4096[cos (ντ + 4τ + φ) + cos (ντ − 4τ + φ)]

+60q2β2A5

4096ν2cos (3ντ + 3φ)

+15q2β2A5

4096[cos (3ντ + 4τ + 3φ) + cos (3ντ − 4τ + 3φ)]

+4q2β2A5

4096ν2cos (5ντ + 5φ) (6.3)

+3q2β2A5

64[cos (5ντ + 4τ + 5φ) + cos (5ντ − 4τ + 5φ)]


and

v(τ) ≈ −νA sin(ντ + φ)

+qβA3

32[(3ν + 2) sin (3ντ + 2τ + 3φ) + (3ν − 2) sin (3ντ − 2τ + 3φ)]

+qβA3

32[3 (ν + 2) sin (ντ + 2τ + φ) + 3 (ν − 2) sin (ντ − 2τ + φ)]

−30q2β2A5

4096[(ν + 4) sin (ντ + 4τ + φ) + (ν − 4) sin (ντ − 4τ + φ)]

−180νq2β2A5

4096ν2sin (3ντ + 3φ)

−15q2β2A5


−20νq2β2A5

4096ν2sin (5ντ + 5φ) (6.4)

−3q2β2A5


where

ν2 = p+15q2β2A4

64(6.5)

6.2 Time averaged motion

Following the same procedure as in Sec. (5.4), we obtain the time averaged

orbit corresponding to Eq. (6.3) to be

xa(t) = A cos(ντ + φ) +q2β2A5

1024ν2[15 cos (3ντ + 3φ) + cos (5ντ + 5φ)](6.6)

with ν given by Eq. (6.5). The exact Hamiltonian of the particle subject to

the force equation, Eq. (6.2), is given by

H =v2

2+px2

2− qβx4 cos 2τ

4(6.7)


Using Eq. (1.15), we know that the time averaged ponderomotive Hamiltonian

corresponding to the time periodic exact Hamiltonian, Eq. (6.7), is given by

Hp =v2

2+px2

2+q2β2x6

16(6.8)

This time averaged Hamiltonian leads to the force equation

x+ px = −3q2β2x5

8

We can solve this by using the Lindstedt-Poincare method. To begin with, we

transform to a time, ξ = ντ . Thus, we have

ξ2x′′ + px = −3q2β2x5

8

where

ξ2 = p+ qw1 + q2w2

and

x = x0 + qx1 + q2x2 + ...

Up to O (q0):

x′′0 + x0 = 0

⇒ x0 = A cos(ξ + φ)

There are no terms of O (q1). Up to O (q2), we have

px′′2 + w2x′′0 + px2 = −3β2x5

0

8

= −3β2A5 cos5(ξ + φ)

8

= −3β2A5

128[10 cos(ξ + φ) + 5 cos(3ξ + 3φ) + cos(5ξ + 5φ)]


⇒ w2 =15β2A4

64

⇒ x2 =

[15β2A5

1024pcos(3ξ + 3φ) +

β2A5

1024pcos(5ξ + 5φ)

]

xp = A cos(ντ + φ) +q2β2A5

1024p[15 cos(3ντ + 3φ) + cos(5ντ + 5φ)] (6.9)

with

ν2 = p+15q2β2A4

64(6.10)

It can be clearly seen that the predictions of the time averaging theory,

Eq. (6.9) and Eq. (6.10), are the same as the expressions obtained by time

averaging of the complete solutions, Eq. (6.6) and Eq. (A.2). However, as

we have shown in Chapter (3), the predictions of conventional ponderomotive

theory regarding the time averaged distribution function and density of the

plasma are incorrect. In what follows, we construct the distribution function

of the plasma when the particles are subject to the force equation, Eq. (A.1)

and show that the plasma is non-Maxwellian even to the lowest order in the

nonlinearity.

6.3 Stroboscopic Map

In chapter (5), we had obtained expressions for the stroboscopic curve corre-

sponding to the aperiodic particle orbit. Those expressions were for the general

case when the stroboscopic sampling is begun from any arbitrary τ > 0. These

expressions were then transformed (scaling+rotation) to those of the time av-

eraged motion, Eq. (5.12). Using this, we then constructed the time dependent

invariant, Eq. (5.27), for the force equation, Eq. (5.4).

Using Eq. (6.3) and following the procedure outlined in Sec. (5.6), the


expression for the stroboscopic variable, xs, up to O (q) is given by,

xs(τ + τs) = A cos(ντ + ντs + φ)

−qβA3

32[cos (3ντ + 3ντs + 2τ + 3φ) + cos (3ντ + 3ντs − 2τ + 3φ)]

−qβA3

32[3 cos (ντ + ντs + 2τ + φ) + 3 cos (ντ + ντs − 2τ + φ)]

= A cos(ντ + ντs + φ)

−qβA3

32[2 cos (3ντ + 3ντs + 3φ) cos 2τ + 6 cos (ντ + ντs + φ) cos 2τ ]

= A

(1− 3qβA2

16cos 2τ

)cos (ντ + ντs + φ) (6.11)

−2qβA3

32cos 2τ cos (3ντ + 3ντs + 3φ)

Similarly, for vs we have,

vs (τ + τs) = −νA sin (ντ + ντs + φ)

+qβA3

32(3ν + 2) sin (3ντ + 3ντs + 2τ + 3φ)

+qβA3

32(3ν − 2) sin (3ντ + 3ντs − 2τ + 3φ)

+3qβA3

32(ν + 2) sin (ντ + ντs + 2τ + φ)

+3qβA3

32(ν − 2) sin (ντ + ντs − 2τ + φ)

= −νA(

1− 3qβA2

16cos 2τ

)sin (ντ + ντs + φ)

+qβA3

8sin 2τ cos (ντ + ντs + φ)

+6νqβA3

32cos 2τ sin (3ντ + 3ντs + 3φ) (6.12)

+4qβA3

32sin 2τ cos (3ντ + 3ντs + 3φ)

with ν =√p. Having obtained expressions for the stroboscopic map, we now

need to construct the invariant corresponding to the particle orbit in phase

space.



In the previous section, we have obtained expression for the stroboscopic plot

of the particle trajectory. In order to be able to follow our earlier method (Sec.

(5.6)) of obtaining the invariant corresponding to these orbits, we would like

to find functions, X (xs, vs, τ) , V (xs, vs, τ), which on substituting Eqs. (6.11)

and (6.12), would lead to an expression that does not contain the cos 2τ and

sin 2τ terms.

Equations (6.11) and (6.12) can be written as

xs

vs

= A1 (τ)

cos (ντ + ντs + φ)

sin (ντ + ντs + φ)

+A2 (τ)

cos (3ντ + 3ντs + 3φ)

sin (3ντ + 3ντs + 3φ)

(6.13)

where

A1 (τ) =

A(1− 3qβA2

16cos 2τ

)0

qβA3

8sin 2τ −νA

(1− 3qβA2

16cos 2τ

) (6.14)

and

A2 (τ) =

−2qβA3

32cos 2τ 0

qβA3

8sin 2τ 6ν qβA3

32cos 2τ

(6.15)

Since det (A1) 6= 0, ∀τ ≥ 0, A1−1 exists and is given by

A1−1 =

1

det (A1)

−νA(1− 3qβA2

16cos 2τ

)0

− qβA3

8sin 2τ A

(1− 3qβA2

16cos 2τ

)

We can now multiply Eq. (6.13), on the left, with A1−1 so as to eliminate terms

containing cos 2τ and sin 2τ from the coefficients of cos (ντ + ντs + φ) and


sin (ντ + ντs + φ). However, this does not help in eliminating the cos 2τ and

sin 2τ terms from the coefficients of cos (3ντ + 3ντs + 3φ) and sin (3ντ + 3ντs + 3φ)

since

A1−1A2 =

1

det (A1)

2qνβA4

32cos 2τ 0

qβA4

8sin 2τ 6qνβA4

32cos 2τ

(6.16)

where we have retained terms only up to O (q). This is different from the case

of linear RF fields, since as can be seen from Eqs. (5.21) and Eq. (5.22),

M1−1M2 =

q2αA2/32ν2 0

0 3q2αA2/32ν2

(6.17)

where terms only up to O (q2) have been retained. As can be seen from Eqs.

(6.16) and (6.17), M1−1M2 is time independent, but A1

−1A2 is not. Unlike

the case of a linear RF field, for nonlinear RF fields, there can be no linear

transformation from (xs, vs) → (X,V ). Thus, the structure of the invariant,

and hence the distribution function, for the nonlinear RF field problem is much

more complicated than for the linear RF field case.

For the case of nonlinear RF field, though we cannot use our earlier method

of constructing the distribution function (Sec. (5.6)), we can construct the

plasma distribution function perturbatively. The invariant, I(x, v, τ), corre-

sponding to the force equation, Eq. (6.2), must satisfy the Vlasov equation

∂I

∂τ+ v

∂I

∂x+(−px+ qβx3 cos 2τ

) ∂I∂v

= 0 (6.18)

In order to solve Eq. (6.18) perturbatively, we need to know the form of the

invariant, I (x, v, τ) [41] to the order till which we want to obtain the solution.


6.4.1 O (q) Invariant

To solve Eq. (6.18) up to O (q), we begin with this form for I (x, v, τ),

I (x, v, τ) = g1 (τ) v2 + g2 (τ)x2

+q(g3 (τ) v2 + g4 (τ)xv + g5 (τ)x2

)(6.19)

+q(g6 (τ) v4 + g7 (τ)xv3 + g8 (τ)x2v2 + g9 (τ)x3v + g10 (τ)x4

)+O

(q2)

where g1 (τ) , ..., g10 (τ) are functions of τ to be determined. Using Eq. (6.19),

we get,

v∂I

∂x= 2g2 (τ)xv

+q(g4 (τ) v2 + 2g5 (τ)xv

)+q(g7 (τ) v4 + 2g8 (τ)xv3 + 3g9 (τ)x2v2 + 4g10 (τ)x3v

)+O

(q2)

x∂I

∂v= 2g1 (τ)xv

+q(2g3 (τ)xv + g4 (τ)x2

)+q(4g6 (τ)xv3 + 3g7 (τ)x2v2 + 2g8 (τ)x3v + g9 (τ)x4

)+O

(q2)


Substituting Eq. (6.19) in Eq. (6.18), and equating the coefficients of like

terms, we get

∂g1

∂τ= 0

∂g2

∂τ= 0

2g2 − 2pg1 = 0 (6.20)

∂g3

∂τ+ g4 = 0

∂g4

∂τ+ 2g5 − 2pg3 = 0

∂g5

∂τ− pg4 = 0 (6.21)

∂g6

∂τ+ g7 = 0

∂g7

∂τ+ 2g8 − 4pg6 = 0

∂g8

∂τ+ 3g9 − 3pg7 = 0

∂g9

∂τ+ 4g10 − 2pg8 + 2βg1 cos 2τ = 0

∂g10

∂τ− pg9 = 0 (6.22)

Solutions to Eq. (6.20) are

g1 =1

2

g2 =p

2(6.23)

Equation (6.21) can be written as,

d2g4

dτ 2= −4pg4 (6.24)

Since 4p 2, g4 (τ) does not oscillate at the RF frequency, ω = 2. And since

we are looking for periodic solutions to Eq. (6.24) with period 2π/ω, we choose

g4 = 0. Thus, the solution to Eq. (6.21) is g3 = 0 = g4 = g5.


Equation (6.22) can be written as

dG

dτ+ A1G =

0

0

0

−β cos 2τ

0

(6.25)

where

A1 =

0 1 0 0 0

−4p 0 2 0 0

0 −3p 0 3 0

0 0 −2p 0 4

0 0 0 −p 0

(6.26)

and

G =

g6

g7

g8

g9

g10

(6.27)

The eigen values of the matrix A1 satisfy the equation

(λ2 + 4p

) (λ3 + 10pλ+ 6p

)= 0 (6.28)

Equation (6.28) has real solutions only for p < 0. Thus, for p > 0, there is no


resonance. Periodic solutions to Eq. (6.22) are given by

g6 =3β

8 (1− 4p) (1− p)cos 2τ

g7 =3β

4 (1− 4p) (1− p)sin 2τ

g8 =−3β

4 (1− 4p)cos 2τ

g9 =−β

2 (1− 4p) (1− p)sin 2τ

g10 =p (2− 5p) β

8 (1− 4p) (1− p)cos 2τ (6.29)

Thus, the invariant corresponding to the force equation, Eq. (6.2), up to

O (q), is given by

I (x, v, τ) =v2

2+px2

2

+q

(3β

8 (1− 4p) (1− p)v4 cos 2τ +

3β

4 (1− 4p) (1− p)xv3 sin 2τ

)− 3qβ

4 (1− 4p)x2v2 cos 2τ (6.30)

+q

(−β

2 (1− 4p) (1− p)x3v sin 2τ +

p (2− 5p) β

8 (1− 4p) (1− p)x4 cos 2τ

)+O

(q2)

Figure (6.1) compares the invariant given by Eq. (6.30) and the results

of numerical integration of the force equation, Eq. (6.2). Curve (1) is the

stroboscopic plot of the orbits obtained by numerical integration of Eq. (A.1).

Curve (2) is the stroboscopic plot of the analytic solutions, up to O (q), to Eq.

(6.2). Curve (3) is the level curve corresponding to the invariant, Eq. (6.30),

corresponding τ = 0. Figure (6.2) shows the same set of curves for a particular

τ 6= 0. As can be seen in the figure, Eq. (6.30) gives good estimate of the

invariant for the rest of the curve, but near the turning points terms of O (q2)

become important and cannot be ignored.


Using Eq. (6.30), the distribution function of the plasma, up to O (q), can

be written as

f (x, v, τ) =n0√2πT0

exp

[− 1

T0

(v2

2+px2

2

)]× exp

[− q

T0

3β

8 (1− 4p) (1− p)v4 cos 2τ

]× exp

[− q

T0

3β

4 (1− 4p) (1− p)xv3 sin 2τ

]× exp

[3qβ

4T0 (1− 4p)x2v2 cos 2τ

](6.31)

× exp

[− q

T0

−β2 (1− 4p) (1− p)

x3v sin 2τ

]× exp

[− q

T0

p (2− 5p) β

8 (1− 4p) (1− p)x4 cos 2τ

]

where T0 is a constant and β = 1/x2. As can be clearly seen, due to the

presence of the terms that go like v4 and xv3, Eq. (6.31) gives a non-Maxwellian

distribution function even to O (q).

The distribution function given by Eq. (6.31) is periodic in time. This

expression can be time averaged to obtain

f (x, v) =n0√2πT0

exp

[− 1

T0

(v2

2+px2

2

)]×I0

[(− 3qβ

8T0 (1− 4p) (1− p)v4 +

3qβ

4T0 (1− 4p)x2v2

− pq (2− 5p) β

8T0 (1− 4p) (1− p)x4

)2

(6.32)

+

(− 3qβ

4T0 (1− 4p) (1− p)xv3 +

qβ

2T0 (1− 4p) (1− p)x3v

)21/

2]

where I0 is the modified Bessel function of the first kind. For x = 0, a time

average of the distribution function is shown in Fig. 6.3. As can be seen in the

figure, the time averaged distribution function agrees well with the Maxwellian

near v = 0, but deviates as v increases. In the presence of collisions, this fat-


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

v

x

(1)

(2)

(3)

Figure 6.1: This plot compares the invariant given by Eq. (6.30) and theresults of numerical integration of the force equation, Eq. (6.2). Curve (1) isthe stroboscopic plot of the orbits obtained by numerical integration of Eq.(A.1). Curve (2) is the stroboscopic plot of the analytic solutions, up to O (q),to Eq. (6.2). Curve (3) is the level curve corresponding to the invariant, Eq.(6.30), corresponding τ = 0. The parameters used were p = 0.011, q = 0.1,α = 0.1 and A = 2.5.


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-3 -2 -1 0 1 2 3

v

x

(1)

(2)

(3)

Figure 6.2: This plot shows the same set of curves as in Fig. (6.1) but for aparticular τ 6= 0. As can be seen in the figure, Eq. (6.30) gives good estimateof the invariant for the rest of the curve, but near the turning points terms ofO (q2) become important and cannot be ignored.


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Dis

trib

utio

n fu

ncti

on

v

Maxwellian<f>

Figure 6.3: This figure compares the time average of the distribution functiongiven by Eq. (6.31) to a Maxwellian. As can be clearly seen, the time averageddistribution function is very close to a Maxwellian in the plasma bulk butdeviates as v increases.

tail of the distribution function could lead to RF heating. For larger values

of v, the time averaged distribution function blows up to infinity, since the

modified Bessel function, I0, grows at rate that is faster than the decay of the

Gaussian term in Eq. (6.32). This is mainly because our perturbation scheme

is valid only for small values of v √

1/qβ.


6.4.2 O(q2)

Invariant

To solve Eq. (6.18), up to O (q2), we begin with a form

I (x, v, τ) =v2

2+px2

2(6.33)

+q

(3β

8 (1− 4p) (1− p)v4 cos 2τ +

3β

4 (1− 4p) (1− p)xv3 sin 2τ

)− 3qβ

4 (1− 4p)x2v2 cos 2τ

+q

(−β

2 (1− 4p) (1− p)x3v sin 2τ +

p (2− 5p) β

8 (1− 4p) (1− p)x4 cos 2τ

)+q2

(h1 (τ) v6 + h2 (τ) v5x+ h3 (τ) v4x2 + h4 (τ) v3x3

)+q2

(h5 (τ) v2x4 + h6 (τ) vx5 + h7 (τ)x6

)+O

(q3)

where h1 (τ) , ..., h7 (τ) are functions of τ to be solved for. Following the same

method as the previous section, we substitute Eq. (6.33) in Eq. (6.18) and

equate like terms to get the following equations

dh1

dτ+ h2 = 0

dh2

dτ+ 2h3 − 6ph1 = 0

dh3

dτ+ 3h4 − 5ph2 = 0

dh4

dτ+ 4h5 − 4ph3 + 4βg6 cos 2τ = 0

dh5

dτ+ 5h6 − 3ph4 + 3βg7 cos 2τ = 0

dh6

dτ+ 6h7 − 2ph5 + 2βg8 cos 2τ = 0

dh7

dτ− ph6 + βg9 cos 2τ = 0 (6.34)


The equations in Eq. (6.34) were solved using Mathematica [42] to obtain the

following periodic solutions,

h1 (τ) =3 (−138 + 100p+ 3p2) β2

512 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)cos 4τ

h2 (τ) =3 (−138 + 100p+ 3p2) β2

128 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)sin 4τ

h3 (τ) =3 (−138 + 100p+ 3p2) (3p− 8) β2

512 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)cos 4τ

h4 (τ) =(−138 + 100p+ 3p2) (p− 1) β2

16 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)sin 4τ

h5 (τ) = − 3β2

16 (1− 4p) (1− p)

+(−2880 + 5552p− 2042p2 − 132p3 + 27p4) β2

512 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)cos 4τ

h6 (τ) =(−576 + 1824p− 1730p2 + 596p3 − 9p4) β2

128 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)sin 4τ

h7 (τ) =(2− 3p) β2

16− 80p+ 64p2(6.35)

+(512− 1216p− 256p2 + 1442p3 − 596p4 + 9p5) β2

128 (−1 + p)2 (−1 + 4p) (16− 40p+ 9p2)cos 4τ

Thus, the distribution function of the plasma up to O (q2) is given by

f (x, v, τ) =n0√2πT0

exp

[− 1

T0

(v2

2+px2

2

)]× exp

[− q

T0

3β

8 (1− 4p) (1− p)cos 2τv4

]× exp

[− q

T0

3β

4 (1− 4p) (1− p)sin 2τxv3

]× exp

[3qβ

4T0 (1− 4p)cos 2τx2v2

]× exp

[− q

T0

−β2 (1− 4p) (1− p)

sin 2τx3v

]× exp

[− q

T0

p (2− 5p) β

8 (1− 4p) (1− p)cos 2τx4

](6.36)

× exp

[− q

2

T0

(h1 (τ) v6 + h2 (τ) v5x+ h3 (τ) v4x2 + h4 (τ) v3x3

)]× exp

[− q

2

T0

(h5 (τ) v2x4 + h6 (τ) vx5 + h7 (τ)x6

)]


where the functions h1 (τ) , ..., h7 (τ) are given by Eq. (6.35) and β = 1/x2.

The two unlabeled curves in Fig. (6.4) compare the stroboscopic plot of

the particle orbit in phase space obtained by numerical integration of Eq. (6.2)

and the level curve of the invariant predicted by Eq. (6.33) and Eq. (6.35).

This corresponds to the case when the stroboscopic sampling is begun at a

particular τ > 0. As can be seen, the two curves match very well. The curve

labeled “q” is the level curve corresponding to the O (q) invariant in Eq. (6.30).

Thus, we can clearly see an improvement as we go from O (q) to O (q2) for

the invariant expression. The parameters used in this figure are the same as

in Fig. (6.2).

6.5 Conclusions

We have considered a nonlinear RF field in this chapter and solved for the

equations of motion up to O (q2). We have found the time averaged orbit

corresponding to the particle trajectory and shown that these solutions match

with the predictions of the conventional time averaging theory. We then found

the expressions for the stroboscopic map, Eq. (6.11), corresponding to the case

when the sampling is begun at τ = 0. Unlike the case of linear RF field, we have

shown in Sec. (6.4) that there is no way of transforming (scaling+rotation)

the stroboscopic expressions, Eq. (6.11), into the expressions for the time

averaged orbit, Eq. (6.6). We then constructed the invariant and the plasma

distribution function corresponding to Eq. (6.2), up to O (q2), perturbatively.

It can be clearly seen that Eq. (6.36) gives a non-Maxwellian distribution

function.

Since Eq. (6.36) is non-Maxwellian, the results obtained are valid only

for collisionless systems. In the presence of collisions, there will be relaxation

processes set up. The effect of relaxation processes is to take the statistical


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-3 -2 -1 0 1 2 3

v

x

q

Figure 6.4: The two unlabeled curves in the plot compare the stroboscopicplot of the particle orbit in phase space obtained by numerical integration andthe level curve of the invariant predicted by Eq. (6.33) and Eq. (6.35). Thiscorresponds to the case when the stroboscopic sampling is begun at a particularτ > 0. As can be seen, the two curves match very well. The curve labeled “q”is the level curve corresponding to the O (q) invariant in Eq. (6.30). Thus, wecan clearly see an improvement as we go from O (q) to O (q2) for the invariantexpression. The parameters used in this figure are the same as in Fig. (6.2).


system towards a maximum entropy state, which we know to be a Maxwellian.

However, for the nonlinear RF field in a Paul trap, there are no invariants that

are quadratic in x and v. Hence, we can never use these invariants to build a

Maxwellian distribution function. Thus, though the relaxation processes will

tend to take the system towards a Maxwellian distribution, such a maximum

entropy state will never be reached. This is because the closer the distribution

comes to a Maxwellian, the weaker is the effect of collisions. However, the

left hand side of the Boltzmann equation is not zero for a Maxwellian since

the invariants of the motion are not quadratic in velocity for nonlinear RF

fields. So, there is a collisionless drive to push the distribution away from a

Maxwellian. In such circumstances, the distribution will come close, but not

exactly, to a Maxwellian, so that the residual collisional relaxation balances

the collisionless drive away from thermal equilibrium.

A system that has a non-thermodynamic steady state in the presence of

collisions must necessarily continuously increase its entropy. This might well

explain the observed RF heating seen in laboratory plasma experiments.

For the case of highly localized RF fields, a mechanism for particle heating

was proposed in [22]. In that paper, a clear distinction was made between the

slow and fast particles, and it was shown that the kinetic energy of the fast

particles increases almost linearly with time. It was also shown that for the

case of highly localized RF fields, the main bulk of the electron distribution

function is not altered significantly. This was mainly because it is the plasma

bulk that is responsible for supporting these highly localized fields. However,

the localized RF fields lead to a significantly fat-tailed distribution. In our

problem of a nonlinear RF field in a Paul trap, a similar behavior can be seen in

Fig. 6.3 where the nonlinear RF field induces a fat-tailed distribution without

causing much alteration in the bulk of the plasma. However, one difference is

that our solutions are steady-state, whereas in the solutions obtained in [22],


energy is continuously going into the tail.

Chapter 7

Discussions and Conclusions

7.1 Invariants of motion

Each of the four previous chapters (3-6) have been mainly about finding the

invariant corresponding to a given non-autonomous Hamiltonian and then us-

ing it to analyze the plasma dynamics. In chapters 3 and 4, the electric field

is linear and hence, one could directly use the Ermakov-Lewis invariant [38]

I (x, v, τ) =1

2(ρ (τ) v − ρ′ (τ)x)

2+

x2

2ρ2 (τ)(7.1)

where ρ (τ) is the solution to the equation

ρ′′ − [−p+ 2q(τ) cos 2τ ] ρ− 1

ρ3= 0 (7.2)

However, use of the invariant in Eq. 7.1 has limitations since solving Eq.

7.2 directly in closed form can be a formidable task. In chapter 4, we have

numerically solved Eq. 7.2 and carried forward the analysis. However, in

chapter 3, we have solved for the invariant explicitly using a slightly different

approach.

In chapter 5, the invariant for the Hamiltonian, Eq. (5.10), was found out

138

CHAPTER 7. DISCUSSIONS AND CONCLUSIONS 139

by transforming the expression for the stroboscopic map into the expression

for the time averaged orbits. This method did not work for the Hamiltonian,

Eq. (6.7), in chapter 6. An invariant corresponding to the Hamiltonian, Eq.

(6.7), was found perturbatively. As shown below, this perturbative method

can be used to find the invariant for Eq. (5.10) in chapter 5 too.

The invariant corresponding to the Hamiltonian, Eq. (5.10), should satisfy

the equation

∂I

∂τ+ v

∂I

∂x+[−px+ 2qx cos 2τ − q2αx3

] ∂I∂v

= 0 (7.3)

The invariant, I (x, v, τ), is of the form

I (x, v, τ) = g1 (τ) v2 + g2 (τ)x2

+q(g3 (τ) v2 + g4 (τ)xv + g5 (τ)x2

)+q2

(g6 (τ) v2 + g7 (τ)xv + g8 (τ)x2

)(7.4)

+q2(g9 (τ) v4 + g10 (τ) v3x+ g11 (τ) v2x2 + g12 (τ) vx3 + g13 (τ)x4

)Using Eq. (7.4), we get

v∂I

∂x= 2g2xv

+q(g4v

2 + 2g5xv)

+q2(g7v

2 + 2g8xv)

+q2(g10v

4 + 2g11v3x+ 3g12v

2x2 + 4g13vx3)

(7.5)


x∂I

∂v= 2g1xv

+q(2g3xv + g4x

2)

+q2(2g6xv + g7x

2)

+q2(4g9v

3 + 3g10v2x2 + 2g11vx

3 + g12x4)

(7.6)

Substituting Eqs. (7.4), (7.5) and (7.6) in Eq. (7.3) and equating the like

terms up to O (q0), we get

∂g1

∂τ= 0

∂g2

∂τ= 0

2g2 − 2pg1 = 0 (7.7)

which can be solved to give

g1 =1

2

g2 =p

2(7.8)

Equating like terms up to O (q1), we get,

∂g3

∂τ+ g4 = 0

∂g4

∂τ+ 2g5 − 2pg3 + 2 cos 2τ = 0

∂g5

∂τ− pg4 = 0 (7.9)


Equation (7.9) can be solved to give

g3 (τ) =2 cos 2τ

−4 + 4p

g4 (τ) =4 sin 2τ

−4 + 4p

g5 (τ) = −2p cos 2τ

−4 + 4p(7.10)

Equating like terms up to O (q2) we get,

∂g6

∂τ+ g7 = 0

∂g7

∂τ+ 2g8 − 2pg6 + 4g3 cos 2τ = 0

∂g8

∂τ− pg7 + 2g4 cos 2τ = 0

∂g9

∂τ+ g10 = 0

∂g10

∂τ+ 2g11 − 4pg9 = 0

∂g11

∂τ+ 3g12 − 3pg10 = 0

∂g12

∂τ+ 4g13 − 2pg11 − α = 0

∂g13

∂τ− pg12 = 0 (7.11)

which can be solved to give

g6 (τ) =6 cos 4τ

(−4 + 4p) (−16 + 4p)

g7 (τ) =24 sin 4τ

(−4 + 4p) (−16 + 4p)

g8 (τ) = − (2p+ 16) cos 4τ

(−4 + 4p) (−16 + 4p)+

2

−4 + 4p

g9 (τ) = 0 = g10 = g11 = g12

g13 (τ) =α

4(7.12)

Thus, the invariant corresponding to the Hamiltonian, Eq. (5.10), is given by


Eq. (7.4) where the functions g1 (τ) , ..., g13 (τ) are given by Eqs. (7.8), (7.10)

and (7.12). This expression for the invariant is exactly the same as in Eq.

(5.27) up to O (q2).

ω-invariance

For Paul trap experiments, what is important is not the detailed time evolution

of the distribution function but the averaged behavior over the time , 2π/ω.

For obtaining this time averaged distribution function, two approaches have

been taken in the past:

1. To time-average the Vlasov equation itself [16] and to solve the resulting

equation to get,

f (x, v) =n0√2πT0

exp

[−Hp

T0

](7.13)

where Hp is the time averaged ponderomotive Hamiltonian given by Eq.

(1.15). This approach that leads to Eq. (7.13) is incorrect because

the plasma distribution is a function of the complete time-dependent

invariant and not the time averaged ponderomotive Hamiltonian. As

shown in Fig. (3.4), a time average of this time-dependent distribution

function is double humped beyond a certain threshold in space, unlike

Eq. (7.13) which is Maxwellian for all points in space. This double

humped nature of the time averaged distribution function was reported

by Krapchev [17] too. Also, as shown in Sec. (3.3.1), the time-averaged

density obtained by averaging the complete time-dependent distribution

function is different from the density derived from Eq. (7.13).

2. To write the distribution function, f (x, v, τ), as a harmonic series in the

RF frequency, ω, and then to equate the like terms to obtain an infinite

set of recurrence relations. These set of recurrence relations are then

solved under certain assumptions. One of the basic assumption made


in this regard has been that the time averaged distribution function can

be written as a power series in even powers of the applied field. This

assumption made by Krapchev simplified the remaining algebra in his

paper because while obtaining the recurrence relations he assumed that

the spatial gradients of all harmonics of the distribution function are very

small. The basic idea of writing f (x, v, τ) as a harmonic series as done in

this approach is correct. However, the assumptions made to solve these

set of recurrence relations are incorrect. Firstly, the spatial gradients of

the harmonics of the distribution function are not very small. In our

approach, the gradients of the fields and the distribution function are

on the same footing as the actual quantities. As shown in Eq. (3.26), a

series expansion of the time averaged density has powers of the spatial

gradient of the applied field as well.

The correct distribution function of the plasma is a function of the invariants

associated with single particle motion. The invariants corresponding to par-

ticle motion in time-periodic fields can, in general, be aperiodic. However, if

we are interested in obtaining time-asymptotic expressions for the distribu-

tion function, it seems mandatory to make efforts towards obtaining an ω-

invariant expression for the invariant as has been done in Eq. (5.27). As men-

tioned earlier, the function, I (x, v, τ) is said to be ω-invariant if I (x, v, τ) =

I(x, v, τ + 2π

/ω). Such a distribution function is time stationary when sam-

pled at intervals of ∆τ = 2π/ω. Finding such an ω-invariant distribution

function is important mainly for two reasons:

1. It is reasonable to expect that the slow breathing of the distribution

function would die out asymptotically with time.

2. In conventional ponderomotive theory (which is the theory used to study

such systems), it has been assumed that the plasma response is periodic


in time with the same period as the applied field.

As mentioned above, when an ω-invariant distribution function was constructed

for a given field profile and its time average compared with the predictions of

conventional ponderomotive theory, discrepancies were found.

7.2 Canonical Adiabatic Theory

The conventional ponderomotive Hamiltonian, Eq. (1.15), that was derived

earlier belongs to a broader class known as adiabatic invariants [29]. Canonical

adiabatic theory deals with Hamiltonians in which variation in all but one of

the degrees of freedom, and in the time, is slow. Such a Hamiltonian can be

written as

H = H0 (J, ε~y, εt) + εH1 (J, θ, ε~y, εt) + ... (7.14)

where J, θ are the action-angle variables corresponding to the case ε = 0 and

~y = (~p, ~q) are the slow canonical variables, not necessarily in the action-angle

formulation. For ε = 0, the system is integrable since we have H = H0(J). For

ε > 0, the Hamiltonian, Eq. (7.14), is non-integrable, in general. However, if ε

is small enough, we can call the system as nearly integrable. For nearly inte-

grable systems, we can construct adiabatic invariants that are approximately

constant as the system evolves in time.

In canonical adiabatic theory, the objective is to transform (J, θ, ε~y, εt) to a

new set of canonical coordinates,(J, θ, ~εy

)such that the resulting Hamiltonian,

H = H0 + εH1 + ...

is independent of the fast phase variable, θ. To achieve this transformation, a


mixed-variable generating function, S, is introduced

S = Jθ +~p · ~q + εS1

(J, θ, ~εp, ε~q, εt

)+ ...

such that

H(J, θ, ~εy, εt

)= H (J, θ, ε~y, εt) + ε

∂S(J, θ, ~εp, ε~q, εt

)∂ (εt)

+ ...

To first order, we have

J = J + ε∂S1

∂θ

θ = θ − ε∂S1

∂J

pi = pi + ε∂S1

∂qi

qi = qi − ε∂S1

∂pi

and

H1

(J, θ, ~εy, εt

)= H1

(J, θ, ~εy, εt

)+ ω

∂S1

(J, θ, ~εy, εt

)∂θ

Since we require H1 to be independent of θ, we choose S1 such as to eliminate

the θ dependence on the right hand side of the above equation. Thus,

H1 = 〈H1〉θ (7.15)

and

ω∂S1

∂θ= −H1θ (7.16)

where

〈H1〉θ =1

2π

2π

0

H1dθ

is the averaged part of H1 and is evaluated by holding the slow variables as


constant and

H1θ = H1 − 〈H1〉θ

is the oscillating part of H1. And the adiabatic invariant, to first order, is

given as

J = J − ε∂S1

∂θ

= J +ε

ωH1θ (7.17)

Any arbitrary function of J is also an adiabatic invariant. The above method

was proposed by Poincare [43] and Von Zeipel [44]. Though the method is a

very elegant way of constructing the adiabatic invariant, the algebra at higher

orders becomes very complicated due to the use of mixed-variable generating

functions. This difficulty was overcome by the use of Lie Transform techniques

proposed by Hori [45], Garrido [46], Deprit [47] and Dewar [48]. In this new Lie

formalism, though the detailed steps are different from that of the Poincare-

Von Zeipel method, the basic idea of choosing S1 such as to eliminate θ from

H1 as in Eq. (7.15) and Eq. (7.16) is still the same.

The transformed Hamiltonian, up to O (ε), can be written as

H(J, θ, ~εy, εt

)= H0 (J, ~εy, εt) + ε 〈H1〉θ (7.18)

The right hand side of Eq. (7.18) is independent of θ and, thus, it seems that

J is an exact invariant. However, this is not true since H is the Hamiltonian

corresponding to the variables,(J , θ)

and not (J, θ). When we transform the

J to J using Eq. (7.17), we can see that H is no longer independent of θ

since H1θ depends on θ. This is the reason for which J in Eq. (7.17) is an

adiabatic invariant and not exact invariant. This is so because in the basic

averaging step, Eq. (7.15), we have assumed the slow variables to be constant.


This is very similar to the averaging process carried out to obtain Eq. (1.14).

7.3 Gyro-Kinetic Theory

The Vlasov equation for a plasma under the influence of a monochromatic RF

field is given by

∂f

∂t+ v

∂f

∂x+ ω0v0(x) cos (ω0t)

∂f

∂v= 0 (7.19)

In conventional ponderomotive theory, it has been an assumption that the time

averaged distribution function of the plasma can be written as [16]

f0 = f0 (HP ) (7.20)

where, Hp is the time averaged ponderomotive Hamiltonian given by Eq.

(1.15), for a particular case. Equation (7.20) follows from the conjecture that

the distribution function is constant over curves of the time averaged motion in

phase space. However, as shown earlier, this is not true since the distribution

function is constant over the detailed orbit, and not the time averaged orbit.

It is not clear whether the Eq. (7.19) can be solved exactly to obtain

f(x, v, t) for any arbitrary. However, under certain assumptions, the above

equation was solved by Krapchev [17], and an expression for the time averaged

distribution function was obtained

f0(v, x) =n0

2πvT

exp

(−1

2

v20

v2T

) ∞

−∞exp

(ipv

vT

)exp

(−p

2

4

)J0

(pv0

vT

)dp

(7.21)

The integral in the above equation was numerically evaluated by Krapchev and

it was found that f0(v, x) ceases to be a Maxwellian as we move away from the

origin in configuration space. And beyond a certain threshold in space, the


time averaged distribution function, f0, also becomes double humped. This

qualitative behavior has been confirmed by an exact analysis by the authors

[40]. Hence, Krapchev has found, and we have also shown, that Eq. (7.20) is

incorrect.

The motion of particles under an RF field is similar to that of particles

gyrating in a magnetic field. The fast gyrophase rotation of particles in gy-

rokinetic theory is the equivalent of fast time oscillations in the RF problem.

And the guiding center motion is the equivalent of the time averaged motion

in ponderomotive theory.

The Hamiltonian for a particle in the presence of a static magnetic field,

~B = ~∇× ~A is given by

H =1

2

(~p− ~A

)2

(7.22)

where ~p = ~v+ ~A(x, y, z) is the canonical momentum and we have taken e = 1 =

m = c for convenience. To solve for the distribution function corresponding to

the Hamiltonian, Eq. (7.22), it is more convenient to make a transformation

to the guiding center variables [49]. In this new coordinate system, f becomes

a harmonic series in the gyro-phase, θ, in the same way as the distribution

function for the RF problem is a harmonic series in the time, ωt. The zeroth

order distribution function, f0, is a function of (µ,E) where µ is the zeroth-

order adiabatic invariant associated with the gyration and E is the particle

energy. Thus, f0 is a Maxwellian. However, as we go to higher orders, the

distribution function becomes non-Maxwellian. This is due to two reasons.

Firstly, f1 depends on the gyro-phase, θ, which depends on the velocity in a

non-trivial way. Secondly, the first order correction to the adiabatic invariant

also leads to non-Maxwellianity. The expansion parameter in the gyrokinetic

theory is ρ/L which is like q

/p for the RF problem (which is a measure of

non-Maxwellianity as shown in Eq. (3.32)).

Thus, though there are similarities between the RF problem and the guiding


center problem, there are two fundamental differences:

1. There is a zeroth order contribution of the magnetic field to the distri-

bution in the form of the zeroth order adiabatic invariant, µ.

2. The exact distribution function in the case of the magnetic field is non-

Maxwellian whereas in the case of the RF problem, it is only the time

averaged distribution that is non-Maxwellian.

7.4 Three Dimensional Distribution Function

The equations of motion in the radial direction are

x = [−p+ 2q cos 2τ ]x

y = [−p+ 2q cos 2τ ] y (7.23)

We know that in the above system, angular momentum is conserved. Hence,

l = xvy − yvx is a constant. This implies vy = (l + yvx)/x. Also,

r2 = x2 + y2

⇒ rvr = xvx + yvy

= xvx +y

x(l + yvx)

=x2 + y2

xvx +

y

xl

=r2

xvx +

y

xl

⇒ vr =rvx

x+yl

rx


The expressions for the invariant in x and y directions are

η [vx − ξx]2 + γ0x2

η= Ex

η [vy − ξy]2 + γ0y2

η= Ey

where η and ξ are periodic functions of τ with period π. Thus, the full expres-

sion for the invariant in the radial direction is,

Eω = η [vx − ξx]2 + γ0x2

η+ η [vy − ξy]2 + γ0

y2

η

= η [vx − ξx]2 + γ0x2

η+ η

[l + yvx

x− ξy

]2

+ γ0y2

η

= η[v2

x + ξ2x2 − 2ξxvx

]+ γ0

x2

η

+η

[y2v2

x

x2+

(l

x− ξy

)2

+ 2yvx

x

(l

x− ξy

)]+ γ0

y2

η

= η[v2

x − 2ξxvx

]+ η

[y2v2

x

x2+ 2

yvx

x

(l

x− ξy

)]+γ0

x2

η+ γ0

y2

η+ ηξ2x2 + η

(l

x− ξy

)2

= η

(r2v2

x

x2− 2

[− lyrx

+ ξr

]rvx

x+

[− lyrx

+ ξr

]2

−[− lyrx

+ ξr

]2)

+γ0r2

η+ ηξ2x2 + η

(l2

x2+ ξ2y2 − 2

l

xξy

)= η

(rvx

x−[− lyrx

+ ξr

])2

− η

[l2y2

r2x2+ ξ2r2 − 2

ly

rxξr

]+γ0

r2

η+ ηξ2r2 + η

(l2

x2− 2

l

xξy

)= η (vr − ξr)2 + η

l2

r2+ γ0

r2

η(7.24)

The invariant in Eq. (7.24) has been reported earlier in the literature [41],

[50] and can arrived at using the well known Ermakov-Lewis invariant [38].

However, in these earlier papers, no explicit expressions for η and ξ were given.

In this work, we have solved for η and ξ and the solutions are given in Eq.


(3.8).Thus, the ω-invariant radial distribution function is

fr(r, vr, τ) =nr√2πTr

exp

[− 1

Tr

(η (vr − ξr)2 + η

l2

r2+ γ0

r2

η

)](7.25)

Thus, the complete 3D distribution function of the plasma in an RF trap under

the effect of the externally applied RF field is

f(r, vr, z, vz, τ) =n0

2π√TrTz

exp

[− 1

Tr

(ηr (vr − ξrr)

2 + ηrl2

r2+ γr

r2

ηr

)](7.26)

× exp

[− 1

Tz

(ηz (vz − ξzz)

2 + γzz2

ηz

)]

The plasma density is

n(r, z, τ) =n0√ηrηz

exp

[− 1

Tr

(ηrl2

r2+ γr

r2

ηr

)− 1

Tz

γzz2

ηz

](7.27)

The plasma density has a maxima in the radial direction at a distance R from

the axis, where

2γr

ηr

R− 2ηrl2

R3= 0

⇒ R =

(η2

r l2

γr

)1/4

(7.28)

Thus, the bulk of the plasma forms a kind of a torus. However, R keeps

oscillating with time about its initial value, R0 =(l2/γr

)1/4

. Thus, though

there is no rigid rotor rotation of the plasma, but the radial position of peak

density keeps oscillating.

7.5 Time averaging and Poincare Map

A comparative study of the time averaging method and the Poincare map was

done by Hadjidemetriou [51]. As stated in the paper by Hadjidemetriou, the

main shortcoming of Poincare map is that though it accurately describes a dy-


namical system, it is purely a numerical method. When the orbits are chaotic,

averaging methods can play no role in further investigation and Poincare maps

are the only way to study dynamics. But it was argued that where the Poincare

map leads to closed curves, there averaging methods could lead to a good an-

alytical understanding of the behavior of the solutions. The time averaging

methods are very useful because solving the time averaged Hamiltonian is

much easier than solving the exact time varying equation. However, as we

have shown in this paper, averaging methods applied directly to a time vary-

ing Hamiltonian have some important short-comings. One short-coming is that

the initial condition corresponding to the time averaged Hamiltonian is not the

actual initial condition for a particle orbit. And to predict this corresponding

initial condition for the time averaged Hamiltonian, one has to solve the exact

time varying equation. Though this limitation does not dispose off the utility

of time averaging methods but it is nevertheless a handicap.

As discussed in Sec. (7.1), if we are interested in obtaining expressions for

the distribution function, it seems mandatory to make efforts towards obtain-

ing an ω-invariant expression for the invariant as has been done in Eq. (5.27).

Thus, there is also an analytical thinking associated with the numerical meth-

ods of Poincare maps. Obtaining these ω-invariant expression for the invariant

is, however, not a simple thing in general. To obtain the invariant for a given

force equation, it is important to obtain analytic expressions for the invariants

corresponding to the stroboscopic level curves obtained by fixed-time sampling

of the particle orbits. In an earlier work [52], efforts were made to come up

with a scheme (numerical fit) to obtain analytic expressions for the strobo-

scopic map for orbits corresponding to a given equation of motion. We would

like to stress that though it is not very difficult to obtain analytic expressions

that would numerically fit to stroboscopic plots, absence of numerically signif-

icant errors does not guarantee the correctness of the expression. For example,


-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-3 -2 -1 0 1 2 3

v

x

(1)

(2)

Figure 7.1: The curve labeled (1) is the stroboscopic plot of the numericalsolutions of Eq. (5.4) and the curve labeled (2) is the level curve correspondingto the invariant 0.5v2 + 0.5ν2x2 with ν given by Eq. (5.5). As can be seen, itis very hard to say from a numerical plot that the invariant 0.5v2 + 0.5ν2x2 isincorrect.

one could say that 0.5v2 + 0.5ν2x2 is an invariant for Eq. (5.6) and Eq. (5.7),

where ν is given by Eq. (5.5). In Fig. (7.1), the curve labeled (1) is the

stroboscopic plot of the numerical solutions of Eq. (5.4) and the curve labeled

(2) is the level curve corresponding to the invariant 0.5v2 + 0.5ν2x2 with ν

given by Eq. (5.5). As can be seen in the figure, it is very hard to say from a

numerical plot that the invariant 0.5v2 + 0.5ν2x2 is incorrect.

As we have shown in Eq. (5.15), it is possible to define a “time” on the

stroboscopic plots. Putting τ = 0 in Eq. (5.15) leads to a scaled version of

the same orbit expressions, Eq. (5.12), as that predicted by time averaged

ponderomotive Hamiltonian. Thus, there seems to be some dynamical struc-

ture behind the stroboscopic plots (with a continuous time defined on these


curves) and sufficient care must be taken while obtaining the corresponding

ω-invariant energy expressions.

7.6 Conclusions

In this thesis, we have theoretically analyzed the dynamics of plasma in Paul

traps. To begin with, in chapter 3, we obtained exact analytic solutions for

the ω-invariant plasma distribution function, Eq. (3.10), corresponding to the

single particle force equation, Eq. (3.1). This distribution function was found

to be Maxwellian for all time at all spatial points. In Sec. In Sec. (3.3.1), the

time-periodic density obtained by integrating the distribution function over

the velocity-space was time-averaged and was shown to be different from the

predictions of conventional ponderomotive theory.

In chapter 4, we considered the problem of a plasma where the RF field is

abruptly switched on at τ = 0. We showed that the distribution function is no

longer ω-invariant and the slow breathing does not die out even in the presence

of collisions, since the distribution function is a Maxwellian. However, as the

results of numerical analysis in Fig. (4.4) show, if the RF field is slowly ramped

up, the plasma can be taken from one ω-invariant state to another.

In chapter 5, the electric field induced by the plasma in a Paul trap was

solved for using the results of chapter 3. The total induced field of the plasma

is quite complex, and to make any further progress the series has to be trun-

cated. Keeping this in mind, terms only up to the lowest order nonlinearity

were retained in the expression for the induced field. We solved this modified

single particle force equation, Eq. (5.4), using the modified Lindstedt-Poincare

method (Sec. (2.1.3)) to obtain solutions for the particle trajectories. Using

these solutions for the particle trajectories, we obtained expressions for the

stroboscopic map of the particle orbit in phase space. These expressions, Eq.


(5.15), are valid for the general case when the stroboscopic sampling is begun

at any time τ > 0 and are not restricted to the special case when the sampling

starts at τ = 0. We were, then, able to transform (rotation+scaling) these

stroboscopic expressions into those for the time averaged orbit, Eq. (5.9). As

shown in Sec. (5.6), this transformation enabled us to use the ponderomotive

Hamiltonian to construct the complete time dependent invariant and hence,

the ω-invariant distribution function up to O (q2). This distribution function

was found to be Maxwellian, like for the case of linear DC and linear RF field.

In chapter 6, we considered the case of plasma dynamics in a nonlinear RF

field. Like in the previous chapters, we used the modified Lindstedt-Poincare

method to solve for the particle trajectory. As shown in Sec. (6.4), we then

used perturbation methods to construct the invariant corresponding to the

single particle orbit and, hence, the ω-invariant distribution function. Unlike

the case of a linear RF field, the distribution function for the case of nonlinear

RF field was found to be non-Maxwellian even to the lowest order. This non-

Maxwellian nature of the distribution function could be a possible reason for

the observed heating in Paul traps.

This thesis has clarified to a large extent the theoretical aspects of plasma

dynamics in Paul traps. We have shown beyond doubt that conventional pon-

deromotive theory has some deficiencies and is in need of revision. One way to

take our work forward would be to see if the approach taken by Krapchev to

solve the Vlasov equation can be modified to get the results predicted in this

thesis.

Bibliography

[1] W. Paul, ”Electromagnetic traps for charged and neutral particles”, Rev.

Mod. Phys., Vol. 62, p. 531, 1990

[2] J. D. Prestage, G. J. Dick and L. Maleki, ”New ion trap for frequency

standard applications”, J. Appl. Phys., Vol. 86, p. 1013, 1989

[3] J. C. Schwartz, M. W. Senko and J. E. P. Syka, ”A two-dimensional

quadrupole ion trap mass spectrometer”, J. Am. Soc. Mass Spectrom.,

Vol. 13, p. 659, 2002

[4] D. H. E. Dubin and T. M. O’Neil, ”Trapped nonneutral plasmas, liquids,

and crystals (the thermal equilibrium states)”, Rev. Mod. Phys., Vol. 71,

p. 87, 1999

[5] D. J. Griffiths, ”Introduction to Electrodynamics”, Prentice Hall, 1999

[6] F. Vedel, ”On the dynamics and energy of ion clouds stored in an R.F.

quadrupole trap”, Int. J. Mass Spect. and Ion Proc., Vol. 106, p. 33, 1991

[7] N. W. McLachlan, ”Theory and Applications of Mathieu Functions”, Ox-

ford University Press, Oxford, 1947

[8] M. Abramowitz and I. A. Stegun, ”Handbook of mathematical functions”,

National Bureau of Standards, Washington D. C., 1972

[9] D. R. Nicholson, ”Introduction to Plasma Theory”, John Wiley & Sons,

New York, 1983

156

BIBLIOGRAPHY 157

[10] G. Z. Li, R. Poggiani, G. Testera, G. Torelli and G. Werth, ”Antihydrogen

production in a combined trap”, Hyperfine Interactions, Vol. 76, p. 343,

1993

[11] J. M. Finn, R. Nebel, A. Glasser and H. R. Lewis, ”Orbital resonances

and chaos in a combined rf trap”, Phys. Plasmas, Vol. 4, p. 1238, 1997

[12] J. Walz, S. B. Ross, C. Zimmermann, L. Ricci, M. Prevedelli and T. W.

Hansch, ”Confinement of electrons and ions in a combined trap with the

potential for antihydrogen production”, Hyperfine Interactions, Vol. 100,

p. 133, 1996

[13] J. F. J. Todd, ”Ion trap mass spectrometer-past, present, and future”,

Mass Spectrometry Reviews, Vol. 10, p. 3, 1991

[14] A. Steane, ”The ion trap quantum information processor”,App. Phys. B,

Vol. 64,p. 623, 1997

[15] F. Reif, ”Fundamentals of Statistical and Thermal Physics”, McGraw-Hill

Book Company, New York, 1965

[16] K. Avinash, A. K. Agarwal, M. R. Jana, A. Sen and P. K. Kaw, ”Space

charge effects in the Paul trap”, Phys. Plasmas, Vol. 2, p. 3569, 1995

[17] V. B. Krapchev, ”Kinetic theory of the ponderomotive effects in a

plasma”, Phys. Rev. Lett., Vol. 42, p. 497, 1979

[18] H. Schaaf, U. Schmeling and G. Werth, ”Trapped ion density distribution

in the presence of He-Buffer gas”, Appl. Phys., Vol. 25, p. 249, 1981

[19] D. A. Church and H. G. Dehmelt, ”Radiative cooling of an electrodynam-

ically contained proton gas”, J. Appl. Phys., Vol. 40, p. 3421, 1969

[20] R. G. Greaves and C. M. Surko, ”Practical limits on positron accumu-

lation and the creation of electron-positron plasmas”, In: F. Anderegg

BIBLIOGRAPHY 158

et al., Editors, Non-Neutral Plasma Physics IV, American Institute of

Physics, New York, 2002, p. 10 0-7354-0050-4/02.

[21] I. Siemers, R. Blatt, Th. Sauter and W. Neuhauser, ”Dynamics of ion

clouds in Paul traps”, Phys. Rev. A, Vol. 38, p. 5121, 1988

[22] G. J. Morales and Y. C. Lee, ”Effect of localized electric fields on the

evolution of the velocity distribution function”, Phys. Rev. Lett., Vol. 33,

p. 1534, 1974

[23] R. Breun, S. N. Golovato, L. Yujiri, B. McVey, A. Molvik, D. Smatlak,

R. S. Post, D. K. Smith and N. Hershkowitz, ”Experiments in a tandem

mirror sustained and heated solely by rf”, Phys. Rev. Lett., Vol. 47, p.

1833, 1981

[24] H. K. Forsen, ”Review of the magnetic mirror program”, J. Fusion Energy,

Vol. 7, p. 269, 1988

[25] R. Majeski, T. Tanaka, T. Intrator, N. Hershkowitz, K. Ko, W. Grossman

and A. Drobot, ”ICRF-edge plasma investigations in Phaedrus-B”, Fus.

Engg. Des., Vol. 12, p. 31, 1990

[26] R. Ifflander and G. Werth, ”Optical detection of ions confined in a rf

quadrupole trap”, Metrologia, Vol. 13, p. 167, 1977

[27] R. Alheit, C. Hennig, R. Morgenstern, F. Vedel and G. Werth, ”Obser-

vation of instabilities in a Paul trap with higher-order anharmonicities”,

App. Phys. B, Vol. 61, p. 277, 1995

[28] R. D. Knight and M. H. Prior, ”Laser scanning measurements of the

density distribution of confined 6Li+ ions”, J. Appl. Phys., Vol. 50, p.

3044, 1979

BIBLIOGRAPHY 159

[29] A. J. Lichtenberg and M. A. Lieberman, ”Regular and Chaotic Dynam-

ics”, Springer-Verlag, New York, 1991

[30] I. B. Bernstein, J. M. Greene and M. D. Kruskal, ”Exact nonlinear plasma

oscillations”, Phys. Rev., Vol. 108, p. 546, 1957

[31] C. M. Bender and S. A. Orszag, ”Advanced mathematical methods for

scientists and engineers”, McGraw-Hill Book Company, New York, 1978

[32] P. Amore and A. Aranda, ”Presenting a new method for the solution of

nonlinear problems”,Phys. Lett. A, Vol. 316, p. 218, 2003

[33] N. W. McLachlan, ”Non-linear differential equation having a periodic co-

efficient”, The Mathematical Gazette, Vol. 35, p. 32, 1951

[34] C. Joshi, ”The development of laser- and beam-driven plasma accelerators

as an experimental field”, Phys. Plasmas, Vol. 14, p. 055501, 2007

[35] W. Lu, C. Huang, M. Zhou, M. Tzoufras, F. S. Tsung, W. B. Mori and T.

Katsouleas, ”A nonlinear theory for multidimensional relativistic plasma

wave wakefields”, Phys. Plasmas, Vol. 13, p. 056709, 2006

[36] P. Mora and T. M. Antonsen Jr., ”Kinetic modeling of intense, short laser

pulses propagating in tenuous plasmas”, Phys. Plasmas, Vol. 4, p. 217,

1997

[37] V. E. Zakharov, ”Collapse of langmuir waves”, Sov. Phys.-JETP, Vol. 35,

p. 908, 1972

[38] H. R. Lewis, Jr., ”Classical and quantum systems with time-dependent

harmonic-oscillator-type Hamiltonians”, Phys. Rev. Lett., Vol. 18, p. 510,

1967

[39] J. Guckenheimer and P. Holmes, ”Nonlinear Oscillations, Dynamical Sys-

tems, and Bifurcations of Vector Fields”, Springer-Verlag, New York, 1983

BIBLIOGRAPHY 160

[40] K. Shah and H. Ramachandran, ”Analytic, nonlinearly exact solutions of

an rf confined plasma”, Phys. Plasmas, Vol. 15, p. 062303, 2008

[41] J. N. Bandyopadhyay, A. Lakshminarayan, and V. B. Sheorey, ”Algebraic

approach in the study of time-dependent nonlinear integrable systems:

Case of the singular oscillator”, Phys. Rev. A, Vol. 63, p. 042109, 2001

[42] Wolfram Research, Inc., Mathematica, Version 5.1, Champaign, IL, 2004

[43] H. Poincare, ”Les Methodes Nouvelles de la Mechanique Celeste”,

Gauthier-Villars, Paris, 1892

[44] Von Zeipel, ”Reserches sur le mouvement des petites planetes”, Ark. As-

tron. Mat. Fys. 11, No. 1, 1916

[45] G. Hori, ”Theory of general perturbation with unspecified canonical vari-

able”, Publications of the Astronomical Society of Japan, vol. 18, p.287,

1966

[46] L. M. Garrido, ”General interaction picture from action principle for me-

chanics”, J. Math. Phys., Vol. 10, p.1045, 1069

[47] A. Deprit, ”Canonical transformations depending on a small parameter”,

Celestial Mechanics and Dynamical Astronomy, Vol. 1, p. 12, 1969

[48] R. L. Dewar, ”Renormalised canonical perturbation theory for stochastic

propagators”, J. Phys. A, Vol. 9, p. 2043, 1976

[49] R. G. Littlejohn, ”Hamiltonian formulation of guiding center motion”,

Phys. Fluids, Vol. 24, p. 1730, 1981

[50] H. Qin and R. C. Davidson, ”An exact magnetic-moment invariant of

charged-particle gyromotion”, Phys. Rev. Lett., Vol. 96, p. 085003, 2006

BIBLIOGRAPHY 161

[51] J. D. Hadjidemetriou, ”The Poincare map and the method of averaging:

A comparative study”, Nonlinear Phenomenon in Complex Systems, Vol.

11, p. 149, 2008

[52] W. Eberl, M. Kuchler, A. Hubler, E. Luscher, M. Maurer and P. Meinke,

”Analytic representations of stroboscopic maps of ordinary nonlinear dif-

ferential equations”, Z. Phys. B, Vol. 68, p. 253, 1987

Appendix A

Trajectory of particle in

nonlinear RF field

Consider the force equation given by Eq. (A.1),

x+ px = qβx3 cos 2τ (A.1)

Using the modified Lindstedt-Poincare method described in Sec. (2.1.3), we

write p and x as a series expansion in q,

p = ν2 + qp1 + q2p2 + ...

and

x = x0 + qx1 + q2x2 + ...

Up to q0:

x0 + ν2x0 = 0

⇒ x0 = A cos (ντ + φ)

162

APPENDIX A. TRAJECTORY OF PARTICLE IN NONLINEAR RF FIELD163

Up to q1:

x1 + ν2x1 + p1x0 = βx30 cos 2τ

= βA3 cos3 (ντ + φ) cos 2τ

=βA3

4[cos (3ντ + 3φ) + 3 cos (ντ + φ)] cos 2τ

=βA3

8[cos (3ντ + 2τ + 3φ) + cos (3ντ − 2τ + 3φ)]

+3βA3

8[cos (ντ + 2τ + φ) + cos (ντ − 2τ + φ)]

This gives

p1 = 0

and

x1 =βA3

8

[cos (3ντ + 2τ + 3φ)

− (3ν + 2)2 + ν2+

cos (3ντ − 2τ + 3φ)

− (3ν − 2)2 + ν2

]+βA3

8

[3cos (ντ + 2τ + φ)

− (ν + 2)2 + ν2+ 3

cos (ντ − 2τ + φ)

− (ν − 2)2 + ν2

]


Up to q2:

x2 + ν2x2 + p1x1 + p2x0 = 3βx20x1 cos 2τ

= cos2 (ντ + φ)3β2A5

8

[cos (3ντ + 2τ + 3φ)− (3ν + 2)2 + ν2

+cos (3ντ − 2τ + 3φ)− (3ν − 2)2 + ν2

]cos 2τ

+cos2 (ντ + φ)9β2A5

8

[cos (ντ + 2τ + φ)− (ν + 2)2 + ν2

+cos (ντ − 2τ + φ)− (ν − 2)2 + ν2

]cos 2τ

=3β2A5

16

[cos (3ντ + 2τ + 3φ)− (3ν + 2)2 + ν2

+cos (3ντ − 2τ + 3φ)− (3ν − 2)2 + ν2

]cos 2τ

+9β2A5

16

[cos (ντ + 2τ + φ)− (ν + 2)2 + ν2

+cos (ντ − 2τ + φ)− (ν − 2)2 + ν2

]cos 2τ

+cos (2ντ + 2φ)3β2A5

16cos (3ντ + 2τ + 3φ)− (3ν + 2)2 + ν2

cos 2τ


16cos (3ντ − 2τ + 3φ)− (3ν − 2)2 + ν2

cos 2τ


16cos (ντ + 2τ + φ)− (ν + 2)2 + ν2

cos 2τ


16cos (ντ − 2τ + φ)− (ν − 2)2 + ν2

cos 2τ


⇒ x2 + ν2x2 + p1x1 + p2x0 =3β2A5

32

[cos (3ντ + 4τ + 3φ)− (3ν + 2)2 + ν2

+cos (3ντ + 3φ)− (3ν + 2)2 + ν2

]+

3β2A5

32

[cos (3ντ + 3φ)− (3ν − 2)2 + ν2

+cos (3ντ − 4τ + 3φ)− (3ν − 2)2 + ν2

]+

9β2A5

32

[cos (ντ + 4τ + φ)− (ν + 2)2 + ν2

+cos (ντ + φ)− (ν + 2)2 + ν2

]+

9β2A5

32

[cos (ντ + φ)− (ν − 2)2 + ν2

+cos (ντ − 4τ + φ)− (ν − 2)2 + ν2

]+

3β2A5

32

[cos (5ντ + 2τ + 5φ)− (3ν + 2)2 + ν2

+cos (ντ + 2τ + φ)− (3ν + 2)2 + ν2

]cos 2τ

+3β2A5

32

[+

cos (5ντ − 2τ + 5φ)− (3ν − 2)2 + ν2

+cos (ντ − 2τ + φ)− (3ν − 2)2 + ν2

]cos 2τ

+9β2A5

32

[cos (3ντ + 2τ + 3φ)− (ν + 2)2 + ν2

+cos (ντ − 2τ + φ)− (ν + 2)2 + ν2

]cos 2τ

+9β2A5

32

[cos (3ντ − 2τ + 3φ)− (ν − 2)2 + ν2

+cos (ντ + 2τ + φ)− (ν − 2)2 + ν2

]cos 2τ


⇒ x2 + ν2x2 + p1x1 + p2x0 =3β2A5

32

[cos (3ντ + 4τ + 3φ)− (3ν + 2)2 + ν2

+cos (3ντ + 3φ)− (3ν + 2)2 + ν2

]+

3β2A5

32

[cos (3ντ + 3φ)− (3ν − 2)2 + ν2

+cos (3ντ − 4τ + 3φ)− (3ν − 2)2 + ν2

]+

9β2A5

32

[cos (ντ + 4τ + φ)− (ν + 2)2 + ν2

+cos (ντ + φ)− (ν + 2)2 + ν2

]+

9β2A5

32

[cos (ντ + φ)− (ν − 2)2 + ν2

+cos (ντ − 4τ + φ)− (ν − 2)2 + ν2

]+

3β2A5

64

[cos (5ντ + 4τ + 5φ)− (3ν + 2)2 + ν2

+cos (5ντ + 5φ)− (3ν + 2)2 + ν2

]+

3β2A5

64

[cos (ντ + 4τ + φ)− (3ν + 2)2 + ν2

+cos (ντ + φ)

− (3ν + 2)2 + ν2

]+

3β2A5

64

[cos (5ντ + 5φ)− (3ν − 2)2 + ν2

+cos (5ντ − 4τ + 5φ)− (3ν − 2)2 + ν2

]+

3β2A5

64

[cos (ντ + φ)

− (3ν − 2)2 + ν2+

cos (ντ − 4τ + φ)− (3ν − 2)2 + ν2

]+

9β2A5

64

[cos (3ντ + 4τ + 3φ)− (ν + 2)2 + ν2

+cos (3ντ + 3φ)− (ν + 2)2 + ν2

]+

9β2A5

64

[cos (ντ + φ)− (ν + 2)2 + ν2

+cos (ντ − 4τ + φ)− (ν + 2)2 + ν2

]+

9β2A5

64

[cos (3ντ + 3φ)− (ν − 2)2 + ν2

+cos (3ντ − 4τ + 3φ)− (ν − 2)2 + ν2

]+

9β2A5

64

[cos (ντ + 4τ + φ)− (ν − 2)2 + ν2

+cos (ντ + φ)− (ν − 2)2 + ν2

]

This implies

p2 =9β2A4

32

[1

− (ν + 2)2 + ν2+

1

− (ν − 2)2 + ν2

]+

3β2A4

64

[1

− (3ν + 2)2 + ν2+

1

− (3ν − 2)2 + ν2

]+

9β2A4

64

[1

− (ν + 2)2 + ν2+

1

− (ν − 2)2 + ν2

]≈ −15β2A4

64

Thus,

ν2 = p+15q2β2A4

64(A.2)


and

x2 = 3β2A5

32

[cos(3ντ+4τ+3φ)

[−(3ν+2)2+ν2][−(3ν+4)2+ν2]+ cos(3ντ+3φ)

[−(3ν+2)2+ν2][−(3ν)2+ν2]

]+3β2A5

32

[cos(3ντ+3φ)

[−(3ν−2)2+ν2][−(3ν)2+ν2]+ cos(3ντ−4τ+3φ)

[−(3ν−2)2+ν2][−(3ν−4)2+ν2]

]+9β2A5

32

[cos(ντ+4τ+φ)

[−(ν+2)2+ν2][−(ν+4)2+ν2]+ cos(ντ−4τ+φ)

[−(ν−2)2+ν2][−(ν−4)2+ν2]

]+3β2A5

64

[cos(5ντ+4τ+5φ)

[−(3ν+2)2+ν2][−(5ν+4)2+ν2]+ cos(5ντ+5φ)

[−(3ν+2)2+ν2][−(5ν)2+ν2]

]+3β2A5

64

[cos(ντ+4τ+φ)

[−(3ν+2)2+ν2][−(ν+4)2+ν2]+ cos(5ντ+5φ)

[−(3ν−2)2+ν2][−(5ν)2+ν2]

]+3β2A5

64

[cos(5ντ−4τ+5φ)

[−(3ν−2)2+ν2][−(5ν−4)2+ν2]+ cos(ντ−4τ+φ)

[−(3ν−2)2+ν2][−(ν−4)2+ν2]

]+9β2A5

64

[cos(3ντ+4τ+3φ)

[−(ν+2)2+ν2][−(3ν+4)2+ν2]+ cos(3ντ+3φ)

[−(ν+2)2+ν2][−(3ν)2+ν2]

]+9β2A5

64

[cos(ντ−4τ+φ)

[−(ν+2)2+ν2][−(ν−4)2+ν2]+ cos(3ντ+3φ)

[−(ν−2)2+ν2][−(3ν)2+ν2]

]+9β2A5

64

[cos(3ντ−4τ+3φ)

[−(ν−2)2+ν2][−(3ν−4)2+ν2]+ cos(ντ+4τ+φ)

[−(ν−2)2+ν2][−(ν+4)2+ν2]

]


Thus, the particle trajectory is given by,

x(τ) = A cos (ντ + φ)

+3qβA3

8

[cos (ντ + 2τ + φ)− (ν + 2)2 + ν2

+cos (ντ − 2τ + φ)− (ν − 2)2 + ν2

]+

qβA3

8

[cos (3ντ + 2τ + 3φ)− (3ν + 2)2 + ν2

+cos (3ντ − 2τ + 3φ)− (3ν − 2)2 + ν2

]

+3q2β2A5

64

6[− (ν + 2)2 + ν2

] +1[

− (3ν + 2)2 + ν2] +

3[− (ν − 2)2 + ν2

]

× cos (ντ + 4τ + φ)[− (ν + 4)2 + ν2

]+

3q2β2A5

64

6[− (ν − 2)2 + ν2

] +1[

− (3ν − 2)2 + ν2] +

3[− (ν + 2)2 + ν2

]

× cos (ντ − 4τ + φ)[− (ν − 4)2 + ν2

]+

3q2β2A5

64

2[− (3ν − 2)2 + ν2

] +3[

− (ν + 2)2 + ν2] cos (3ντ + 3φ)[− (3ν)2 + ν2

]+

3q2β2A5

64

2[− (3ν + 2)2 + ν2

] +3[

− (ν − 2)2 + ν2] cos (3ντ + 3φ)[− (3ν)2 + ν2

]+3q2β2A5

64

[3

[−(ν+2)2+ν2] + 2

[−(3ν+2)2+ν2]

]cos(3ντ+4τ+3φ)

[−(3ν+4)2+ν2]

+3q2β2A5

64

[3

[−(ν−2)2+ν2] + 2

[−(3ν−2)2+ν2]

]cos(3ντ−4τ+3φ)

[−(3ν−4)2+ν2]

+3q2β2A5

64

[cos(5ντ+5φ)

[−(3ν+2)2+ν2][−(5ν)2+ν2] + cos(5ντ+5φ)

[−(3ν−2)2+ν2][−(5ν)2+ν2]

]+3q2β2A5

64

[cos(5ντ+4τ+5φ)

[−(3ν+2)2+ν2][−(5ν+4)2+ν2] + cos(5ντ−4τ+5φ)

[−(3ν−2)2+ν2][−(5ν−4)2+ν2]

]


⇒ x(τ) ≈ A cos(ντ + φ)

−qβA3

32[cos (3ντ + 2τ + 3φ) + cos (3ντ − 2τ + 3φ)]

−3qβA3

32[cos (ντ + 2τ + φ) + cos (ντ − 2τ + φ)]

+30q2β2A5

4096[cos (ντ + 4τ + φ) + cos (ντ − 4τ + φ)]

+60q2β2A5

4096ν2cos (3ντ + 3φ)

+15q2β2A5

4096[cos (3ντ + 4τ + 3φ) + cos (3ντ − 4τ + 3φ)]

+4q2β2A5

4096ν2cos (5ντ + 5φ) (A.3)

+3q2β2A5

64[cos (5ντ + 4τ + 5φ) + cos (5ντ − 4τ + 5φ)]


And the particle velocity is

v(τ) = −νA sin (ντ + φ)

−3qβA3

8

[(ν + 2) sin (ντ + 2τ + φ)

− (ν + 2)2 + ν2+

(ν − 2) sin (ντ − 2τ + φ)− (ν − 2)2 + ν2

]−qβA3

8

[(3ν + 2) sin (3ντ + 2τ + 3φ)

− (3ν + 2)2 + ν2+

(3ν − 2) sin (3ντ − 2τ + 3φ)− (3ν − 2)2 + ν2

]

−3q2β2A5

64

6[− (ν + 2)2 + ν2

] +1[

− (3ν + 2)2 + ν2] +

3[− (ν − 2)2 + ν2

]

×(ν + 4) sin (ντ + 4τ + φ)[− (ν + 4)2 + ν2

]−3q2β2A5

64

6[− (ν − 2)2 + ν2

] +1[

− (3ν − 2)2 + ν2] +

3[− (ν + 2)2 + ν2

]

×(ν − 4) sin (ντ − 4τ + φ)[− (ν − 4)2 + ν2

]−3q2β2A5

64

2[− (3ν − 2)2 + ν2

] +3[

− (ν + 2)2 + ν2] 3ν sin (3ντ + 3φ)[

− (3ν)2 + ν2]

−3q2β2A5

64

2[− (3ν + 2)2 + ν2

] +3[

− (ν − 2)2 + ν2] 3ν sin (3ντ + 3φ)[

− (3ν)2 + ν2]

−3q2β2A5

64

[3

[−(ν+2)2+ν2] + 2

[−(3ν+2)2+ν2]

](3ν+4) sin(3ντ+4τ+3φ)

[−(3ν+4)2+ν2]

−3q2β2A5

64

[3

[−(ν−2)2+ν2] + 2

[−(3ν−2)2+ν2]

](3ν−4) sin(3ντ−4τ+3φ)

[−(3ν−4)2+ν2]

−3q2β2A5

64

[5ν sin(5ντ+5φ)

[−(3ν+2)2+ν2][−(5ν)2+ν2] + 5ν sin(5ντ+5φ)

[−(3ν−2)2+ν2][−(5ν)2+ν2]

]−3q2β2A5

64

[(5ν+4) sin(5ντ+4τ+5φ)

[−(3ν+2)2+ν2][−(5ν+4)2+ν2] + (5ν−4) sin(5ντ−4τ+5φ)

[−(3ν−2)2+ν2][−(5ν−4)2+ν2]

]


⇒ v(τ) ≈ −νA sin(ντ + φ)

+qβA3


+qβA3

32[3 (ν + 2) sin (ντ + 2τ + φ) + 3 (ν − 2) sin (ντ − 2τ + φ)]

−30q2β2A5

4096[(ν + 4) sin (ντ + 4τ + φ) + (ν − 4) sin (ντ − 4τ + φ)]

−180νq2β2A5

4096ν2 sin (3ντ + 3φ)

−15q2β2A5


−20νq2β2A5

4096ν2 sin (5ντ + 5φ) (A.4)

−3q2β2A5


List of publications

1. K. Shah and H. Ramachandran, “Analytic steady state solutions for an

rf confined plasma”, Plasma 2007 conference, Gandhinagar (Gujarat,

India), 2007

2. K. Shah and H. Ramachandran, “Analytic, nonlinearly exact solutions

for an rf confined plasma”, Phys. Plasmas, Vol. 15, p. 062303, 2008

3. K. Shah and H. Ramachandran, “Space charge effects in rf traps: Pon-

deromotive concept and stroboscopic analysis”, Phys. Plasmas, Vol. 16,

p. 062307, 2009

172

Documents

Plasma Dynamics in Paul Traps