Analytic, nonlinearly exact solutions for an rf confined plasma

Analytic, nonlinearly exact solutions for an rf confined plasmaKushal Shah and Harishankar Ramachandran Citation: Phys. Plasmas 15, 062303 (2008); doi: 10.1063/1.2926632 View online: http://dx.doi.org/10.1063/1.2926632 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v15/i6 Published by the American Institute of Physics. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Analytic, nonlinearly exact solutions for an rf confined plasmaKushal Shaha� and Harishankar Ramachandranb�

Department of Electrical Engineering, IIT Madras, Chennai-600036, India

�Received 31 December 2007; accepted 22 April 2008; published online 6 June 2008�

RF confined electron plasmas are of importance in Paul traps �W. Paul, Rev. Mod. Phys. 62, 531�1990��. The stability of such plasmas is unclear and statistical heating arguments have beenadvanced to explain the observed heating in such plasmas �I. Siemers et al., Phys. Rev. A 38, 5121�1988��. This study investigates the nature of a one-dimensional collisionless electron plasma that isconfined by an rf field of the form �−B+A cos��t��x, where x is the space coordinate and � is therf frequency. Nonlinearly exact solutions are obtained. The distribution function and the plasmadensity are obtained in closed form and have constant shapes with time varying oscillations. Theseoscillations are at the rf frequency and its harmonics, modulated by a low frequency related to theelectron bounce time. The linear limit of weak fields is recovered. Analytic expressions are obtainedfor the required external field to make it consistent with prescribed distribution functions. Thesesolutions remain valid even in the presence of collisions. Solutions involving multiple species arealso obtained, though only for collisionless traps. It is found that the ponderomotive force responseneeds to be corrected to account for the temperature fluctuations. No stochastic heating is observedin this field configuration. © 2008 American Institute of Physics. �DOI: 10.1063/1.2926632�

I. INTRODUCTION

Interactions of plasmas with rf waves have been exten-sively studied in connection with Paul traps.1 RF confine-ment of plasmas has also been a topic of immense interest.2–8

To explain the interaction of rf waves with plasma, sta-tistical arguments have been offered.9–13 There are two kindsof stochastic approaches. One is to consider the plasma be-havior as a consequence of Brownian motion9 and the otheris the idea of Fermi acceleration.11,12 In the Brownian motionapproach, the motion of the electrons is assumed random,and approximate expressions for the density and distributionfunctions are derived using probabilistic arguments. In thecase where Fermi acceleration14 is considered, the rf field isassumed to die down very quickly with distance inside theplasma boundary, thus forming a reflecting structure resem-bling a wall and electrons are reflected off this “wall” result-ing in the heating of their distribution. These wall-like struc-tures near the plasma boundary are called rf sheaths and theircharacteristics have been well studied.15–18 As these reflectedelectrons return to the bulk of the plasma, they transfer theirenergy to the bulk plasma by undergoing collisions, whichresults in the heating up of the plasma as a whole.

Although these models are quite promising, it is notclear if they resemble plasma behavior completely in all itsaspects. The Brownian motion model requires the plasma tobe highly collisional, which rules out analysis of collisionlessplasmas. The Fermi acceleration model is for capacitive dis-charges, where the plasma is assumed to be collisionless19

near the plasma boundary and accounts for the heating ofelectrons due to fields which are localized within a sheath ofmarginal thickness near the plasma boundary.

There is, also, a third category of solutions which at-

tempt to solve the collisionless Vlasov equation. In theseattempts �Krapchev and Abhay20,21�, it is assumed that thetime-varying distribution function can be decomposed as aFourier series which has the same period as the rf wave.Under this assumption, the Vlasov equation is then solved toobtain expressions for the time averaged distribution func-tion and density.

In addition to these studies on plasmas confined by rffields, there have also been many investigations into the na-ture of rf-induced wave phenomena in plasmas.22,23

This paper analyzes the problem of an electron plasmaconfined partly by rf and partly by static potentials. Pure rfconfinement and purely static confinement are valid limits ofthis analysis. A very specific form of the total electric field isconsidered that leads to tractable equations. The analysis isfor a one-dimensional collisionless pure electron plasma witha nonuniform electrostatic standing wave. The field is as-sumed to be given by

E�x,t� = −m

e�− B + A cos��t��x , �1�

where A ,B are constants, w is the rf frequency, and m and −eare the mass and charge of an electron, respectively. Exactsolutions are obtained for arbitrary initial conditions. Con-ventionally, plasma response to fields of the kind as in Eq.�1� for B=0 is covered under the notion of the “ponderomo-tive effect.” To interpret the results obtained in this paper, wecompare them with conventional ponderomotive forcetheory, which is briefly discussed below, and the linear Vla-sov theory.

The ponderomotive effect24–31 describes the response ofa plasma to fields of the form E�x , t�=g�x�cos�wt�. Particlesunder a force of this form are pushed away from high fieldregions and a drift motion is superimposed on the high fre-

a�Electronic mail: [email protected]�Electronic mail: [email protected].

PHYSICS OF PLASMAS 15, 062303 �2008�

1070-664X/2008/15�6�/062303/13/$23.00 © 2008 American Institute of Physics15, 062303-1


http://dx.doi.org/10.1063/1.2926632

http://dx.doi.org/10.1063/1.2926632

quency response. The force equation for an electron in sucha field is given by

d2x

dt2 = −e

mg�x�cos�wt� .

We divide the path of the particle, x�t�, into two compo-nents x�t�=x0�t�+x1�t�, where x0 represents the drift and x1

represents the high frequency component. Now, Taylor ex-panding g�x� about x0, we get

d2x0

dt2 +d2x1

dt2 = −e

m�g�x0� + �dg

dx�

x0

x1�cos�wt� ,

where terms of order O�g��x0�x12� are neglected and it is as-

sumed that x1 is small compared to the scale length of thespatial variation of g�x�. If we average the above equationover the fast time scale, we obtain

� d2x0

dt2 = − � e

m

dg

dx�

x0

x1 cos�wt�� ,

where ..� represents average over the fast time scale. Theremaining part of the equation is

d2x1

dt2 = −e

mg�x0�cos�wt� ,

where x0�− x0�� and �−eg��x0� /m��x1 cos�wt�− x1 cos�wt��have been neglected as small.

This equation can be easily solved to give

x1 =e

mw2g�x0�cos�wt� .

Substituting this in the equation for x0 and averaging over thefast time scale, we obtain

d2x0

dt2 = − � e2

4m2w2

d

dxg2�x��

x0

�2�

which is the well-known ponderomotive force equation. Theabove derivation is valid if x1 is small compared to the scalelength of spatial variation of g�x�. For this to be true, thespatial gradient of the field must be small.

It is also widely accepted that the time-averaged densityof the electrons in the plasma under an electric field of theform E�x , t�=g�x�cos�wt� is given by

n�x� = n0 exp�−�P

kT , �3�

where n0 is the electron density in the absence of the electricfield and �P is the fictitious ponderomotive potential givenby

�P�x� =e2

4m2w2g2�x� .

This result was formally derived from kinetic theory byKrapchev.20

In the following sections, we analyze the response of aplasma to the electric field of Eq. �1�. Exact solutions of thekinetic equations are obtained. This paper has been organizedas follows: In Sec. II, we solve the equation of motion of theelectron under an electric field as given in Eq. �1� and obtainclosed form solutions for the path of the electron at anyfuture time for arbitrary initial conditions. Then, in Sec. III,we derive closed form analytic expressions for the time evo-lution of the distribution function and density of the plasmaunder this force field. Then in Sec. IV, we use the expres-sions found in Secs. II and III to obtain a self-consistentmodel of the plasma. Section V is devoted to some discus-sion on the results found and Sec. VI contains conclusions.Appendices A–D, in the EPAPS,41 contain the derivations ofsome of the results presented in this paper.

II. THE EQUATIONS OF MOTION

Consider an electric field given in Eq. �1�. It must benoted that this total electric field does not satisfy the Pois-son’s equation because �E /�x�� /�0. Thus, the solution isonly valid if the electric field has gradients along y and z.Such a field would cause particle flow along all three direc-tions unless there were some means of constraining particlemotion along x such as the presence of a strong magneticfield. This limits the extent to which the solutions in thispaper are “self-consistent.”

The force equation for an electron under this field is

d2x

dt2 = �− B + A cos�wt��x . �4�

By transforming to z=exp�jwt�, Eq. �4� becomes a lineardifferential equation whose coefficients are polynomials in z.In previous treatments for the case when B=0, this has led tothe assumption that under the effect of this force field, thetime evolution of the distribution function of the plasma is apower series in z, i.e., expressible as a Fourier series in timewith fundamental frequency w �Refs. 20 and 21�. Unfortu-nately, this is not the case as the solution is actually a gen-eralized power series in exp�jwt� and a new frequency, �,appears. This new frequency contains the ponderomotive ef-fect.

We make a transformation of variables, �=wt /2, writing2A /w2 as q and 4B /w2 as p to get

d2x

d�2 = �− p + 2q cos�2��x . �5�

Equation �5� is the Mathieu’s equation32 whose solutions arewell known. This equation does not have stable solutions forall p ,q. For a given q, there are values of p denoted bya0 ,a1 ,a2 , . . ., for which Eq. �5� has even periodic solutions

062303-2 K. Shah and H. Ramachandran Phys. Plasmas 15, 062303 �2008�


and there are also other values of p denoted by b1 ,b2 , . . .. forwhich the equation has odd periodic solution. These two setsof values �a0 ,a1 ,a2 , . . . � and �b1 ,b2 , . . . � form two countablyinfinite sets and are called the characteristic values. If q isreal, the characteristic values ar and br are also real and aredistinct. For q�0, we have a0�b1�a1�b2� . . .. For p=0,the solutions are stable if 0qqc, where qc�0.9. Thiscorresponds to the case when the particles see only the rffield. If p�0, then the dc field is trying to throw away theelectrons to infinity. For this case, any arbitrarily small posi-tive value of q will not be sufficient to confine the plasma.Thus, when p�0, there is both a nonzero positive lowerbound and an upper bound on the value of q for the solutionsof Eq. �5� to be stable. This corresponds to the case when thetotal dc field is trying to destabilize the plasma but the rffield comes in and makes the particle orbits bounded andhence stable. For p�0, both the dc and the rf field contributetowards confining the plasma.

Mathieu’s equations have been used before1 to analyzeplasma dynamics in Paul traps. However, the analysis wasrestricted to finding the stability boundary. In anotherpaper,33 Mathieu’s equations were used to study stochasticheating of orbits in the presence of large amplitude standingwaves. The analysis was, however, not carried forward toobtain time evolution expressions for the distribution func-tion and density of the plasma.

Equation �5� has two linearly independent solutions,34,35

�� and ��, which are given by

�� = �r=−�

�

c2r cos�� + 2r�� 6�

and

�� = �r=−�

�

c2r sin�� + 2r�� , �7�

where c2r for r=0, �1, �2, . . . are constant coefficients re-lated by the following recurrence relation:

�p − �� + 2r�2�c2r = q�c2r−2 + c2r+2� �8�

and � is given as the solution of an asymptotic series,

p = �2 +1

2��2 − 1�q2 +

5�2 + 7

32��2 − 1�3��2 − 4�q4

+9�4 + 58�2 + 29

64��2 − 1�5��2 − 4��2 − 9�q6 + O�q8� . �9�

For p ,q 1, we can approximately solve Eq. �9� to get

� ��p +q2

2. �10�

As an example, consider the case of q=0.16 andp=−0.01. This case corresponds to a field given by

E�x,�� = − �m/e��− 0.01 + 0.32 cos�2��x .

The high frequency response should be roughly �x�−0.08x0 cos�2��. Thus, this is a regime where ��x� �x0�, asrequired for conventional ponderomotive theory to be valid.For this case, the expressions for � and are

�� cos�� − 0.03789 cos�� + 2��

− 0.04211 cos�� − 2�� + 0.0003688 cos�� + 4��

+ 0.0004322 cos�� − 4�� ,

�11�� sin�� − 0.03789 sin�� + 2��

− 0.04211 sin�� − 2�� + 0.0003688 sin�� + 4��

+ 0.0004322 sin�� − 4�� ,

where �=0.0529.The solid line in Fig. 1 displays the plot of the numerical

solution of Eq. �5� for q=0.16, p=−0.01 and initial condi-tions x0=0.91918,v0=0. The dashed line is the analytic so-lution as given in Eq. �11� considering the first three terms.This reveals the simple structure lying behind these complexlooking orbits. There is a small mismatch between the twoplots which is due to the fact that the full analytic solutioncontains an infinite number of terms and accuracy can beimproved by including more terms in the expression. Thisregime is clearly one where the �x due to the high frequencywave is small compared to the distance between turningpoints. The phase space plot for this case is shown in Fig. 2.When q becomes large, ��x��x0� and the assumptions lead-ing to the derivation of the ponderomotive force expressionbecome invalid.

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200

x

t

FIG. 1. The solid line shows the plot of the path of the particle as pernumerical integration of Eq. �5�. The dashed line shows the path as predictedby the mathematical expression given in Eq. �11�, 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 little discrepancy can be reduced further byconsidering higher order terms in the mathematical expression.

062303-3 Analytic, nonlinearly exact solutions… Phys. Plasmas 15, 062303 �2008�


The general solution to Eq. �5� is

x�� = D�� + E�� , �12�

where D and E are constants that depend on the initial con-ditions, namely the particle position and velocity at �=0.

As can be seen in Eq. �11�, the coefficients in the expres-sion for � and that really matter are c0, c2, and c−2. Forr�0, �c2r /c0��O�q�r� /4r2�. Thus, c0 dominates over theother coefficients since q 1. The particle trajectory is a lowfrequency sinusoid �both cosine and sine components� withfrequency equal to �, which is irrationally related to w ingeneral �in our normalization w=2�, and two low-amplitudehigh frequency components, of frequency w+� and w−�,superimposed on that and other higher frequencies as can beseen from the expressions in Eqs. �6� and �7�. The low fre-quency path is the one predicted by the ponderomotive forceexpression. To see this equivalence, we consider the pon-deromotive force expression as given in Eq. �2�. For the casewhen g�x� corresponds to Eq. �1� with p=0, the ponderomo-tive equation for the low frequency path becomes

d2x

d�2 = −1

2q2x . �13�

This has already been derived in the Introduction. The gen-eral solution to the above equation will be a linear combina-tion of cos �� and sin ��, where �=q /�2, which agrees withEq. �10� for p=0. Thus, Eq. �13� is only approximate and theexact expression for � is the asymptotic series given in Eq.�9� with p=0.

The expressions for v is

v =dx

d�= D�� + E�, �14�

where � represents differentiation with respect to �. Putting�=0, in Eqs. �12� and �14�, we get

x0 = D�0 + E0,

�15�v0 = D�0� + E0�,

where the subscript 0 refers to the value of the correspondingfunctions at �=0. From Eqs. �6� and �7� it is clear that �contains only cosine terms and contains only sine terms.This means that �� will contain only sine terms and � willcontain only cosine terms. Thus, �0�=0 and 0=0. Substitut-ing this in Eq. �15�, we solve for D and E, to get

D = x0/�0,

�16�E = v0/0�.

Substituting this in Eqs. �12� and �14� and solving for x0 andv0, we obtain

x0 =1

0��x� − v� ,

�17�

v0 =1

�0�v� − x�� ,

where �� and �� are given by Eqs. �6� and �7�. It isimportant to note that Eq. �17� is linear in x and v since �and are functions of time alone. Since �00� is the nonzeroWronskian for Eq. �5�, the denominators in Eq. �17� arestrictly nonzero.

The distribution function of the plasma at any time � cannow be written as

f�x,v,�� = f0�x0�x,v,��,v0�x,v,�� ,

where f0�x0 ,v0� is the distribution function of the plasma at�=0. It must be noted that f0�x0 ,v0� can be any arbitraryfunction of x0 and v0.

III. TIME EVOLUTION OF THE DISTRIBUTIONFUNCTION AND DENSITY

The time evolution of the distribution function and den-sity of the plasma has a strong dependence on its expressionat �=0. We consider a form of the initial distribution func-tion given by

f0�x0,v0� = n0� �0

2�exp�− �0�1

2v0

2 + �0x02 � , �18�

where �0�0 is any non-negative real number which definesthe scale length of the plasma for ��0, n0 is the plasmadensity at x0=0, and �0 is a measure of the plasma tempera-ture. We could have as well chosen any arbitrary distributionfunction, f0�x0 ,v0�, instead of the Maxwellian, but wechoose this since it is related to the idea of thermal equilib-rium. For the above distribution function, the density of theplasma at �=0 is

n0 exp�− �0�0x02� . �19�

We have shown in Appendix A41 that the analytic ex-pression for the time evolution of the distribution function ofthe electrons is given by

-0.2

-0.1

0

0.1

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

v

x

FIG. 2. This is the phase space plot of the trajectory shown in Fig. 1. We cansee that near x=1 and x=−1, the particle is undergoing high frequencyoscillations. This is the region of phase space for which the ponderomotivetheory holds.



f�x,v,�� = n0� �0

2�exp�−

�0a

2�020�

2�v −cx

a 2�

�exp�−�0�0�0

20�2

ax2

⇒ f�x,v,�� = n0� �0

2�exp�−

�0

2��v − ��x�2�

�exp�− �0�0x2

�� , �20�

where

a�� = �20�2 + 2�02�0

2,

�21�c�� = ��0�

2 + 2�0��02,

�� =�20�

2 + 2�0�022

�020�

2 ,

�22�

�� =c��a��

=��0�

2 + 2�0��02

�20�2 + 2�02�0

2 ,

and �, , �0, and 0� were defined in the previous section.� , are purely functions of � and �0 ,0� are nonzero con-stants. It should be noted that the method used in AppendixA41 applies to arbitrary f0�x0 ,v0�. We can now integrate Eq.�20� with respect to v to get the time evolution of the densityof the plasma,

n�x,�� = n0 exp�−�0�0�0

20�2

�20�2 + 2�02�0

2x2 �� 0

20�2

�20�2 + 2�0�0

22

⇒ n�x,�� =n0

��exp�− �0�0

x2

�� , �23�

n�x ,�� remains a Gaussian in x whose width fluctuates intime. This fluctuation has both high-frequency and low fre-quency components. It has been shown in Eq. �D9�, Appen-dix D,41 that for q2 p 1,

�� =a

�020�

2 = 1 + q�1 +q

4p�cos 2�� − q cos 2� + O�q2� .

This corresponds to the oscillation seen when an rf field ofamplitude 2q is suddenly applied to a confined plasma at �=0. It has been shown in Eq. �D4� that for q 1,

�� =� + �0

0�2 �1 +

� − �0

� + �0cos�2�� − q cos�2�� + O�q2�� ,

where ��0�0�−1/2 is the scale-length of the initial plasma and�=0�

2 /2�02�0.5p+0.25q2. For �=�0, �� becomes inde-

pendent of �. For general p ,q, in Appendix B,41 we showthat for

� �p

2+

q2

4�24�

the functions �� and �� can be written as a Fourier serieswith fundamental frequency w=2. Also, �� is purely a sineseries and �� is purely a cosine series, which can be writ-ten as

�� = �r=0

�

er sin�2r�� ,

�� = �r=0

�

dr cos�2r�� .

Thus, for this value of �, the distribution function and den-sity become

f�x,v,�� = n0� �0

2�exp�−

�0

2 �r=0

�

dr cos�2r��

��v − x�r=0

�

er sin�2r��2��exp�−

�0�

�r=0� dr cos�2r��

x2� �25�

and

n�x,�� =n0

��r=0� dr cos�2r��

exp�−�0�

�r=0� dr cos�2r��

x2� .

�26�

Thus, for �0=��0.5p+0.25q2, the time-varying terms in thedistribution function and density have no dependence on �and thus, the fluctuations in f�x ,v ,�� and n�x ,�� are only aFourier series with fundamental frequency w. This corre-sponds to a solution that is stationary in “slow time,” onethat oscillates at the rf frequency and its harmonics. Thissolution corresponds to an initial distribution that is invarianton curves of constant E=1 /2v2+�x2, which for ��0.5p+0.25q2 confirms the “ponderomotive energy” concept forp=0. For the case of p�0, this expression of � can beviewed as a definition of the “generalized ponderomotiveenergy” concept where the particles see both the dc as wellas the rf field.1

Although ��0.5p+0.25q2 corresponds to the usual no-tion of ponderomotive energy, this expression for � is notquite exact. As is shown in Eq. �B11�, Appendix B,41 a moreaccurate expression for � is

� =1

2�� +

2�c2 − c−2�c0 + c2 + c−2

�2

=�2

2�1 + 2q + O�q2�� . �27�

For the case of p=0 and q�0, �=q /�2+O�q3�. Thus, forthis case,

� =q2

4�1 + 2q + O�q2�� . �28�



It has been numerically verified that the above expression for� is much more accurate than �=0.25q2 and the second termin the brackets in Eq. �28� is of the order of q and is 0.3815for q=0.16 and p=0. So, the improved expression for pon-deromotive energy for the linear case when p=0 is

E =1

2v2 +

q2

4�1 + 2q + O�q2��x2. �29�

Each of the three plots in Fig. 3 show two curves for thespatial variation of density corresponding to Eq. �23� at twodifferent times �=0 and �=2k� /�. Here, k�N and is cho-sen such that 2k� /� is close to � / �2��. These three plots arefor three different values of �0, when q=0.16 and p=−0.01.Figure 3�a� is for �0=0.5p+0.25q2, Fig. 3�b� is for �0=� asgiven by Eq. �27�, and Fig. 3�c� is for �0=20p+10q2. It canbe seen in Fig. 3�a� that for �0=0.5p+0.25q2, which corre-sponds to the conventional ponderomotive theory, the densityis not absolutely invariant on the slow time scale. But if �0 ischosen according to Eq. �27�, the density does not change onthe slow time scale corresponding to frequency �, which canbe seen in Fig. 3�b�. This confirms the accuracy of Eq. �27�.Figure 3�c� shows that if �0 is arbitrarily chosen, then thedensity changes significantly on the slow time scale. Thissubstantial change in density can be understood by consider-ing the evolution of the distribution function in phase spacewhich can be seen in Fig. 4. The density of particles at theturning points at �=0 is the density at x=0 at time �� /2�. For arbitrary loading, this density differs from thedensity at x=0 at time �=0. This results in fluctuation off�x ,v ,�� at each point x, resulting in density fluctuations.

IV. SELF-CONSISTENT SOLUTIONSAND PLASMA CONFINEMENT

In the previous section, we have derived analytic expres-sions for the time evolution of the distribution function of aplasma under the electric field given by Eq. �1�. The actualelectric field in the plasma is given by

E�x,�� = Ee�x,�� + Ei�x,�� ,

where E�x ,�� is the total field given by Eq. �1�, Ee�x ,�� is theexternally applied electric field, and Ei�x ,�� is the self-fieldinduced by the plasma electrons. Ei�x ,�� is given by

dEi

dx= − 4�en�x,�� , �30�

where n�x ,�� is the density of the electrons and the ions havebeen neglected since we are considering a pure electronplasma. The solution for the distribution function of theplasma obtained in the previous section is self-consistent if

Ee + Ei = −m

e�− B + A cos�2��x

⇒ Ee�x,�� = −m

e�− B + A cos�2��x − Ei�x,�� .

�31�

So, if an rf field is applied satisfying this equation, theplasma response becomes self-consistent. Equation �30�therefore becomes analogous to the nonlinear Poisson’sequation in BGK mode theory. We repeat that this solution

does not satisfy �E /�x=4��. Thus, gradients of E� must existalong y and z directions requiring some mechanism to keepparticle motion along x.

If we excite the plasma with the above external field,then the Ee�x ,�� and Ei�x ,�� will add together to give anoscillating electric field, linearly varying in space, which canconfine the plasma.

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)

FIG. 3. This shows the density plots with q=0.16, p=−0.01 at two differenttimes �=0 and �=k�, where k�N is such that k� is close to � /2�. Thethree plots are for different values of �0. �a� �0=0.5p+0.25q2; �b� �0=�,where � is given in Eq. �27�; �c� �0=20p+10q2��. In �a�, it can be seenthat the two curves are quite close to each other. This shows that, approxi-mately, �=0.5p+0.25q2 is the value of �0 at which the density function doesnot have a large � dependence. The curve in �b� clearly shows that Eq. �28�gives a much more accurate expression for the � which leads to a moreaccurate expression for the ponderomotive energy of a particle under thelinearly varying oscillatory electric field. The overlap of the curves at �=0and �=k� is so good that it is not visible in this graph. In �c� it can beclearly seen that for this value of �0, the density function has a strongdependence on �.

-40 -20 0 20 40-3

0

3

0.20.4

0.60.8

x

v

FIG. 4. This is the contour plot of the distribution function of the plasma forthe case �0=20p+10q2 with q=0.16, p=−0.01. The two superimposed con-tour plots correspond to the two times of the curves shown in Fig. 3�c�. Thisclearly shows that the drastic change in the distribution function is thereason for the huge change in the density function over the � time scale.



We have already shown in Eq. �23� that n�x ,�� is sym-metric in x and is given by

n�x,t� =n0

��exp�− �0�0

x2

�� ,

where �� is given by

�� =�20�

2 + 2�0�022

�020�

2 .

Clearly, Ei is antisymmetric and hence is zero at x=0. Thus,the expression for Ei is

Ei�x,�� =− 4�en0

��

0

x

exp�− �0�0x2

��dx

= − 2�en0� �

�0�0erf��0�0

x

�� , �32�

where erf is the error function. Thus, the right-hand side ofEq. �32� is fully specified for the given distribution function.

The Fourier components of the electric field, Ei�x ,�� areplotted in Fig. 5. As can be seen, the component at the rffrequency w dominates other higher harmonics. The differ-ence between the components at w and 2w is two orders inmagnitude.

Thus, only the fundamental and the bounce frequencycomponents of Ee�x ,�� have significant amplitude,

Ee�x,�� = E0�x� + E��x�cos�� + E1+�x�cos��2 + ��

+ E1−�x�cos��2 − �� + ¯ .

If �0 is such that Eq. �27� is satisfied, the � dependencevanishes and for all practical purposes, Ee�x ,�� is a nonuni-form monochromatic electric field with a dc component. For

this value of �0, �� is a cosine series and hence is a sym-metric function of time and Ee�x ,�� becomes

Ee�x,�� = E0�x� + E1�x�cos�2�� .

Thus, when � is given by Eq. �27�, the Fourier decomposi-tion of Ee�x ,�� is a pure cosine series and hence the compo-nent of Ei�x ,�� at frequency w is in phase with Et�x ,��. De-pending on the value of −mB /e desired in Eq. �1�, differentE0�x� profiles are required.

V. DISCUSSION

A. The ponderomotive potential

The standard definition of the “ponderomotive potential”is based on first order theory. As a result, the value of � is inerror for larger field gradients. For electric fields linearlyvarying in space, the theory of Mathieu functions provides abetter expression for the ponderomotive force, that includesthe stability boundary. The expressions obtained in the pre-vious sections describe exact solutions of the one-dimensional problem for the case of constant gradient fields.This is important since it permits us to validate theories inthe literature against this exactly solvable case.

For the case of a linearly varying field, conventionalponderomotive force expressions approximately predict thelow frequency path or the path along which the particle willdrift. Based on this expression, the expression for the spatialvariation of the time-averaged plasma density is derived. Intwo papers published in 1979, Krapchev and Abhay20,21

showed that under the action of an electric field−�m /e�wv0�x�cos�wt�, the time-averaged density of the pureelectron plasma will be n=n0 exp�v0

2�x� /2vT2�, where vT is the

thermal velocity of the plasma electrons. This is in agree-ment 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 gra-dients are weak.

When these expressions are applied to the field distribu-tion studied in this paper, they predict that the time-averageddensity should vary as

n = n0e−�0q2x2/4.

From the expressions derived in this paper in Eq. �23�, theactual instantaneous density is given by

n =n0

��exp�− �0�0

x2

�� .

For �0=�, the time-averaged expression for this density isgiven by �Eq. �C8�, Appendix C41�,

n�x� = n0 exp�−�0

2

q2x2

2��1 −

q

2−

�0

2

q3x2

2+ O�q2�� .

�33�

While the exponential dependence is in agreement with pon-deromotive theory, the additional multiplicative factors arenot. These factors arise from the fact that the electron re-sponse is adiabatic and hence temperature is not a constant.In conventional ponderomotive theory, an implicit assump-

1e-4

1e-2

1

0 5 10 15 20 25

mag

(E(x

,w))

x

DCw

2w

3w

FIG. 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. �32�. This is for the case when � is given by Eq. �27� and q=0.16, p=−0.01. As can be clearly seen, leaving the dc component, the component at� is dominating. And we also have small contributions from components atfrequencies 2�, 3� ��=2 in our normalization�. The remaining harmonicsare lower in magnitude than the ones shown. The component at 2� is twoorders of magnitude lower than that at �. So, the field given by Eq. �32� isessentially a nonuniform monochromatic electric field for all practicalpurposes.



tion is made that the rf field is connected to a heat bath heldat �0. Hence this factor is not seen. However, such an as-sumption is questionable at rf frequencies. When the exactproblem is solved, we find the additional factors. It is star-tling that there is a correction factor of O�q�, which is alower order than the ponderomotive effect itself. While thisresult has only been obtained for a very specific field profile,the source of the deviation from conventional theory sug-gests that a similar correction term would appear in anybounded system. Of course in the case of a linear gradient,this term is particularly significant, since the plasma responseis exactly adiabatic at all collision frequencies, as is seen inthe subsection discussion collisional response.

A question of interest is what changes are required inplasma fluid equations to obtain Eq. �33�. The ponderomo-tive force is seen by all particles and as seen in Eqs. �6� and�7�, the shape of the particle orbit is independent of the par-ticle’s initial conditions. Hence, on taking the moment, thisforce remains in its single particle form. However, as seen inEq. �20�, the temperature oscillates in “fast time” yielding achanged value for the average temperature as seen in Eq.�33�. The equation of state therefore needs to be changed to

P=n�T, where fluid behavior is assumed to be isothermal.The time averaged temperature is �Eq. �C6�, Appendix C41�

T = T0�1 + q + O�q2�� , �34�

where T0=1 /��0 is the initial temperature of the plasma.Substituting this into the momentum equation, the long termsteady state density becomes

n = n0 exp�−�P

�T� ,

where �P is the ponderomotive potential energy of the eachelectron. The above expression for density is the well known

Boltzmann relation. Since rf interaction preserves fluid, n0

can be obtained from the initial conditions to yield

n�x� = n0�T0

Texp�−

�P

�T .

Substituting �P=0.25q2x2�1+2q� from Eq. �29� and T fromEq. �34�, we obtain to first order

n�x� = n0� 1

1 + q + O�q2�

�exp�− �0

0.25q2x2�1 + 2q + O�q2��1 + q + O�q2� �

= n0 exp�−�0

2

q2x2

2 �1 −

q

2−

�0

2

q3x2

2+ O�q2�� ,

�35�

which is in agreement with Eq. �33�.One difficulty with this modification of the fluid equa-

tions is that the solutions obtained in this paper exhibit adia-batic behavior for all rf frequencies and all collision frequen-

cies. However, we conjecture that non idealities would relaxthe slow time behavior to make the isothermal equation of

state, P=n�T valid.In general, the exact plasma response also contains a

term at the slow frequency, 2�. This corresponds to the non-linear, transient response. Conventional ponderomotivetheory looks for steady state solutions, i.e., it assumes thatthe distribution function of the plasma can be expanded in aFourier series which has the same fundamental frequency, w,as the applied field. However, unless � is given by Eq. �27�,a � dependent term appears, and modifies the density pertur-bation. This is further discussed in the next subsections.

B. Relation to BGK theory

As can be seen in Fig. 5, the externally applied field hasa significant dc component. This is the electric field that isrequired to balance the self-field induced by the plasma elec-trons. If we choose a value of � for a particular p, q otherthan that given by Eq. �27�, then the applied field will havecomponents at frequency �, in addition to a dc componentand components at the rf frequency w and its harmonics.Existing treatments of this problem deal only with the fieldseen by particles, and thus neglect the need to balance thisself-field.

The derivation in Sec. III was for a electron distributionfunction whose velocity variation was a Maxwellian. How-ever, any arbitrary distribution could have been chosen thatdepended on x and v through Ep=0.5v2+�0x2. Let the distri-bution function of the plasma at �=0 be f0�x0 ,v0�=F�0.5v2

+�0x2�, where F is a smooth but otherwise arbitrary functionof its argument. This distribution will evolve with time, andat an arbitrary time �, it can be written as

f�x,v,�� = F��2

�v − ��x�2 + �0x2

�� ,

where �� and �� were defined in Eq. �22�. The density ofthe plasma at any arbitrary time is thus

n�x,�� = �−�

�

f�x,v,��dv

= �−�

�

F� ��2

�v − ��x�2 + �0x2

��dv

=� 1

��

−�

�

F�u2

2+ �0

x2

��du ,

where

u = ��v − ��x� =� 1

��n�� 1

��x,� = 0� .

Corresponding to this density expression we can now findthe induced field and, hence, the total external field required.

The same derivation shows that any such function of EP

yields a distribution that is time stationary with fluctuationsat the rf frequency and its harmonics when � is given by Eq.



�27�. Each such distribution function corresponds to a differ-ent applied electric field. There is, thus, an interesting corre-spondence to BGK theory.

In the case of a BGK mode,36 if we specify the electrondistribution function as a function of energy, we can find thepotential required to confine the plasma according to thatparticular distribution. If the distribution of the electrons isgiven by f�E�, this leads to the nonlinear Poisson’s equation,

d2��x�dx2 = 4�e�

−e�

� f�E�dE�2m�E + e��x��

.

In this paper, we started with an initial distribution f0�0.5v02

+�x02� at �=0. We assumed that the total field seen by the

plasma must be −�m /e��−B+A cos�wt��x. For this field, theparticle paths are given by the Mathieu’s functions. The dis-tribution function and the density were obtained by themethod given in Sec. III. From this we found the inducedfield,

Ei�x,t� = − 4�e�0

x

n�x�,t�dx�,

and hence, the total external field required to confine theplasma

Ee�x,�� = Et�x,�� − Ei�x,�� ,

Ee�x ,��, and f�EP ,�� constitute a self-consistent solution ofthe plasma evolution equations. There is a lack of consis-

tency in the fact that �E /�x�4��. It requires that E� varyalong y or z, yet the particles be constrained to move onlyalong x, perhaps by a magnetic field. For any given distribu-tion of this form, we can determine the electric field that is tobe imposed to make the problem self-consistent. The con-verse is, however, not possible. In general, Ee�x ,�� has aninfinite number of harmonics, each of which has an indepen-dent spatial profile. Essentially what this means is thatEe�x ,�� can be an arbitrary function of time at each point inspace. At most, f�EP ,�� can match to one spatially varyingharmonic of Ee�x ,��. Thus, for an arbitrary specification ofEe�x ,��, it is not possible to determine a distribution functionthat is invariant under the application of that field. The anal-ogy with BGK theory is therefore limited.

The presence of plasma means that a dc field becomespresent. This needs interpretation. BGK modes also corre-spond to time-stationary distributions that are confined by dcfields. If a dc field is also present here, what difference isthere between the two?

The two types of solutions are clarified in Fig. 6. Thefigure shows the confining fields when a Maxwellian plasmais present. In order to compare, the static and the rf solutionsare assumed to have the same time-averaged density profiles�Gaussian in shape�. The static solution corresponding to aMaxwellian has a linear confining electric field �shown as thecurve labeled “Effective static Field” in the figure�. Chargesexecute simple harmonic motion in this field. The externallyapplied field is the difference of the desired linear field andthe field induced by the charge density of the plasma. This is

the curve labeled “Applied static Field” in Fig. 6. The self-consistent field itself is shown as the line labeled “InducedField.”

When rf is used to confine the plasma, the applied fieldis determined by the two parameters p and q. When q is zero�no rf�, the resulting density corresponds exactly to the staticsolution above. When p is zero, the external applied field isassumed to have a dc component that exactly cancels theself-repulsion of the plasma. Then electrons see only an rffield, which is the case studied in Ponderomotive force stud-ies in the literature. When both p and q are present, we havesome very interesting possibilities. Figure 6 shows the caseof q=0.16 and p=−0.01. The p value is assumed to be suchthat it corresponds exactly the self-repulsion of the plasma atx=0. At larger x there is a mismatch, since the self-field ofthe plasma grows as the error function while the total field isrequired to be linear. Thus an additional repelling field isnecessary and is externally applied. This is shown as thenegative dotted-dashed line in Fig. 6. For p that is morenegative than this value, the dotted-dashed line will acquire alinear slope as well. For less negative p, a positive �confin-ing� field develops that turns into a repelling field at greaterdistances.

The possibility of a repelling dc applied field is quiteinteresting. As is well known, static fields cannot confinecharges. Thus, if perpendicular confinement is achieved elec-trostatically, parallel confinement must be achieved by othermeans such as rf. Yet, we see that a pure rf solution stillrequires a confining dc field which invalidates the solution.The solution to this problem is to use an extracting fieldalong x. This ensures that the static field is consistent with

-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)

FIG. 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�−�0�0x2�, then the“Effective static Field” corresponds to the electric field for which the poten-tial goes like �0�0x2. The “Induced Field” is the field induced by theexp�−�0�0x2� electron density in the absence of ions. The “Applied staticField” is the sum of these two fields, and is the external electric field whichhas to be applied to get this particular density profile. Now, if the same timeaveraged profile has to be achieved by using an rf field, then the total fieldseen by the plasma has a much steeper slope and is shown by the straightline labeled “Effective rf Field.” The curves labeled “Induced Field” and“Applied static Field” are not straight lines. These curves are purely quali-tative and are not to scale.



the averaging theorem of Laplace’s equation. To confine theplasma along x, we apply an rf field that not only keeps theparticles in via ponderomotive force, but also overcomes thedc repelling field that is present. Since there are small self-consistent contributions at the rf frequency and its harmon-ics, the applied field is slightly modified from 2qx cos�2��.The rf field amplitude is shown as the nearly straight linelabeled “Applied rf Field” in Fig. 6. The figure is only in-tended to convey the qualitative differences between staticequilibrium and the rf solutions obtained in this paper, and isnot to scale.

C. Collisional effects

The self-consistent solution we have obtained for thedistribution function of the plasma has a Maxwellian veloc-ity distribution at all spatial positions x and at all times �. Asis well known, the Maxwellian annihilates the point collisionoperator. Thus, the solution obtained in the previous sectionby solving the collisionless Vlasov equation is, in fact, aglobal thermal equilibrium. This shows that even in the pres-ence of collisions, these solutions are still valid, because thesystem is already in the maximum entropy state by virtue ofbeing described by a Maxwellian. This situation is quite dif-ferent from that of a local thermal equilibrium where thedistribution 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 thereforea solution where the plasma is confined by an rf field withoutundergoing heating.

If the plasma interaction is adiabatic, the temperature, T,and volume, V, of the plasma must satisfy the condition

TVCp/Cv−1 = constant,

where for a one-dimensional system Cp /Cv=3. The timeevolution of the distribution function of the plasma is evalu-ated in Eq. �20� and found to be

⇒ f�x,v,�� = n0� �0

2�exp�−

�0

2��v − ��x�2�

�exp�− �0�0x2

�� .

From the above expression, we can see that T�1 /�� andV��0.5��. Thus, T�V−2. This gives

TVCp/Cv−1 � V−2V3−1 = constant.

The distribution function obtained in Eq. �20� clearly satis-fies the adiabaticity condition.

Even more intriguing are the solutions for which � isvery different from the value given in Eq. �27�. These aretime-varying, exact solutions in which the plasma undergoeslarge-scale reorganization over the � /2� time scale. Thesesolutions also have Maxwellian velocity distributions at alltimes and all spatial locations! Thus, these time-varying so-lutions are also exact solutions of the one-dimensional,Vlasov–Boltzmann equation. Such solutions behave verysimilarly to the quasistatic compressions and rarefactions ofideal 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 rffrequencies �with harmonics at r��, r=0,1 , . . .�. Thebreathing of the plasma at the bounce frequency � is a self-consistent, exact response of the collisional plasma.

D. Multispecies plasmas

In this paper, we have considered a single species elec-tron plasma. If we had ions in addition to electrons the prob-lem changes somewhat. Each plasma species satisfies Eq. �1�with a different ms /qs ratio. Hence, Eq. �5� is satisfied byeach species for p and q values that are scaled by ms /qs. Thelow frequency � corresponding to the ions will therefore bedifferent from that corresponding to the electrons. Nonethe-less, a self-consistent, collisionless solution can be obtained,where

Ee�x,�� = −m

e�− B + A cos 2�� − Ei�x,�� ,

where Ei�x ,�� is now the electric field induced by all thespecies present in the plasma. This solution will not howeverbe invariant at the slow time. Each species has its ownbreathing frequency �s. By choosing q it is possible to elimi-nate the slow time variation of any one species, but the otherspecies will continue to oscillate. However, for the specialcase of a two species plasma consisting of two oppositelycharged species, it is possible to find p and q such that bothspecies are time invariant at the slow time scale. Considerelectrons and singly charged ions of mass mi. Then,

pe

pi=

qe

qi= −

mi

me= − k�say� ,

where k�1. Now, we can approximately write � from Eq.�23� as

� =pi

2+

qi2

4=

pe

2+

qe2

4

⇒ pi +1

2qi

2 = − kpi +1

2k2qi

2

⇒ pi�1 + k� =1

2�k2 − 1�qi

2

⇒pi

qi2 =

k − 1

2.

For this choice of p and q, both the electron and ion distri-butions will be stationary in slow time.

Whatever the solution found, the presence of multiplespecies makes the solution susceptible to collisions. Clearly,when different species breathe at different frequencies, thedistributions collisionally drag on each other. Even in thecase where low frequency oscillations are eliminated for bothspecies, the high frequency oscillations of ions and electronsare opposite in phase. This is obvious since the forces areopposite in direction. Hence there is a high frequency vi

−ve present, which means that collisional drag operates onthese oscillations. Since these oscillations are driven by the



applied rf field, the collisional stability of such solutions andthe possibility of collisionally driven heating of the distribu-tions are issues that need investigation.

The special case of infinitely massive ions is interesting.The ion profile can be chosen to cancel the self-consistentfield of the electrons. No dc confining field is now necessary�but is always permitted�. The breathing solutions found arenot collisionally valid, since the ions act as point scatterersthat conserve energy but isotropize momentum. The station-ary solutions are also collisionally invalid since the high fre-quency oscillations now see a momentum drag.

E. Linear response theory

Let us consider a situation when the total field seen bythe plasma electrons is static for ��0, i.e., E�x ,��=mpx /e.At �=0, an rf field is switched on, and thus for ��0,E�x ,��=−mx�−p+2q cos�2�� /e and the plasma behavior isgoverned by the analysis as given in this paper. If the rf fieldis very weak, the plasma response can be expected to obeythe expressions given by linear theory.37

For ��0, the plasma is static and though the particlesare moving around, the plasma density and distribution func-tions are independent of time,

f�x,v,� � 0� = n0� �0

2�exp�− �0�1

2v2 +

p

2x2 � . �36�

When the rf field is switched on, the plasma response will nolonger be static but will evolve in time. As shown in thepaper, the plasma response has two dominating frequencycomponents, one at the rf frequency, �=2 and the other atthe bounce frequency, 2�, where � is given by Eq. �10�. InAppendix D,41 it has been verified that the exact solutionsobtained in the paper satisfy the linear Vlasov equation �forq2 p 1�,

�f1

�t+ v

�f1

�x+

− eE0

m

�f1

�v+

− eE1

m

�f0

�v= 0,

�37�f1�x,v,�� = f�x,v,� � 0��Av2 + Bx2 + Cxv� ,

where

A � −�0

2�q cos�2�� − q cos�2�� ,

B � − 0.5�0p�− q cos�2�� + q cos�2�� , �38�

C � �0�− �pq sin 2�� + q sin 2�� .

The linear limit corresponds to the rf field having anamplitude much smaller than the confining static field, i.e.,q p �which automatically implies q2 p�. In this limit, Eq.�37� shows that not only is the response at � linear in q, butso too is the response at 2�.

Figure 7 shows the magnitude of the response at the twofrequencies versus q at x=0. For q p, the plasma oscillateswith the same magnitude at both the high and the low fre-quency, even though the total electric field only contains ahigh frequency component. The solid thick line labeled p

=0.5q2 serves to divide two prominent regions of plasmaresponse. To the left of this curve, the frequency componentsare more or less linear but after this region is crossed, theresponse becomes highly nonlinear. The curves shown in thefigure are also in agreement with Eqs. �37� and �38�. There isalso another solid thick line on the curve labeled p=0.25q.This also demarcates two regions on the p−q space. We cansee that when q crosses this solid line, there is a visiblechange in the slope of the curves. Thus, as q becomes larger,the first nonlinearity to set in, is of the order of q2 /4p.

The electric field that is applied to the plasma is un-bounded, since it is proportional to x. However, since theplasma’s scale length is proportional to 1 /�p, the normalizedmagnitude of the rf field at the nominal edge of the plasma isqx=O�q /�p� 1. This perturbation is therefore an accept-able linear perturbation of a non-neutral plasma.

As shown above, the linear Vlasov theory is still valid.The response at 2� represents the transient response. The rffield has been switched on at t=0. Thus, the plasma hasexperienced a small kick. The response at 2� is due to thisfinite kick. But since q p, the response at 2� is still com-parable in magnitude to the response at �=2. If p q, thenthe plasma experiences a much harder kick and the responseat 2� will then be much larger than the response at �=2. Inconventional wave theory, after linearizing the fluid equa-tions, we do a Fourier transform to obtain steady state solu-tions. But the above results show that the Fourier transformmust be done very carefully, because the transient responseat frequencies other than the rf frequency may not die downto zero asymptotically.

1

2

10

1e-3 1e-2 1e-1

dens

ityco

mpo

nent

atsl

owfr

eque

ncy,

2ν

q

w

p=0.001

p=0.004

p=0.008

p=0.5q2

p=0.25q

FIG. 7. This figure shows the plot of the magnitude of the coefficients ofcos 2�� and cos 2� in the expression for A as given in Eq. �38�, normalizedby 0.5q. It can be clearly seen that magnitude of the response at w lies belowthe response at 2�. The solid thick line labeled p=0.5q2 serves to divide twoprominent regions of plasma response. When p=0.5q2, the rf response is aslarge as the dc response and as we approach this region, plasma behavior ishighly nonlinear. There is another solid thick line labeled p=0.25q. Thisalso demarcates two regions in the p−q space. As q crosses this line, thereis a visible change in the slope of the curves. The plots clearly show that asq becomes large compared to p, the first nonlinearity to set in is of the orderof q2 /4p. Also, the curves are well in agreement with the expressions de-rived in Eq. �37�.



F. Experimental realizability

It is not clear if the analysis carried out in this paper isrealizable in practice. One can think of a plasma confined bya strong magnetic field along x that is subjected to an exter-nal rf field. However, the presence of plasma charges meansthat a curl-free electric field cannot achieve the desired con-finement. This is so, since the total potential is required to bequadratic in x which limits the kind of density profiles thatcan be compensated. Suppose

��x,r�� ,t� = C�r�

� ,t�x2

2,

where C�r�� , t� is some function of its arguments. This im-

plies

�2� =x2

2��

2 C�r�� ,t� + C�r�

� ,t�

= 4�en�x,r�� ,t�

⇒ n�x,r�� ,t� = C1�r�

� ,t� + x2C2�r�� ,t� ,

where the functions C1�r�� , t� and C2�r�

� , t� depend on

C�r�� , t�. As the analysis in this paper shows, interesting

plasma density profiles are Gaussian functions of x andhence do not fall into the above class of functions.

Possibly, a fully electromagnetic wave could compensatefor a Gaussian density profile, but the authors have not beenable to discover such an arrangement. The solutions in thispaper are, therefore, presented as purely theoretical ideas.

VI. CONCLUSIONS

In this paper, we have obtained nonlinearly exact solu-tions for the one-dimensional rf confinement problem, for thecase of spatially linear electric fields. The electric field con-sidered is a combination of a dc field and an rf field, both ofwhich vary linearly in space. Exact expressions for the dis-tribution function and the particle density have been obtainedin closed form. The assumption of a linear spatial depen-dence is very restrictive. However, it permits very strongresults to be obtained. We find that single species plasmasconfined in this field do not experience any stochastic heat-ing. The solutions are valid even in the presence of strongcollisions, where a Brownian motion or Fokker–Planck ap-proach might be expected to yield more accurate results.

The nature of the ponderomotive force has been clarifiedto a certain extent in this study. Unless the plasma is alreadyin a distribution of the form f�EP� when the rf field is turnedon, the plasma response is not well approximated by theponderomotive force equation. Instead, the plasma executescomplex breathing cycles at a new frequency. The reason forthis discrepancy in behavior lies in the implicit assumptionof harmonic plasma ponderomotive response to an rf field.The plasma response to an rf field is not in general har-monic. Instead a bounce frequency appears, which is, in gen-eral, irrationally related to the rf frequency.

An implicit assumption made in the conventional theoryof the ponderomotive effect15,16 is that the nonuniform rf

field does not effect the velocity space distribution of theplasma. But for the case of a linear field profile, we havefound that this is not true. The temperature is found to oscil-late on the rf time scales adiabatically and the time averageof the temperature is found to depend on the rf field strength�Eq. �33��. This is very significant and it is expected thatdeviations from conventional theory will be present for thecase of spatially nonlinear rf fields also.

Among the surprising findings is a class of nonstationarysolutions that do not relax in the presence of collisions.These are maximum entropy states that breathe at the elec-tron bounce frequency. The plasma response shows spectrallines at r��k� for r�Z and k=0,1 ,2 and is not quasistaticby any definition of that term. Yet the solutions behave adia-batically in that they move from thermodynamic state tothermodynamic state along the isoentropic curves, in finitetime.

The correction needed in fluid theory to account for thenew terms in Eq. �33� were derived in Eq. �35�. The deriva-tion assumed that the fluid is isothermal under dc conditions.However, for the specific field profile studied in this paper,the fluid remains adiabatic even in dc conditions. We conjec-ture that nonlinear profiles would introduce sufficient mixingto drive the fluid to isothermal dc behavior, and that the casestudied is not representative.

The main reason why the linear field problem yieldedclosed form solutions is also its weakness; all the particles inthe plasma respond with the same frequency regardless oftheir initial position. Their orbits are not the same. However,fast and slow particles both respond identically. This is ratherlike a complex variant of simple harmonic motion.

When the field is spatially nonlinear, particle trajectoriesbecome much more complex. A single frequency no longerdescribes the electron bounce dynamics. However, numericalsimulations still suggest that the orbits do not diverge and arestill well described by r��k� for r ,k�Z. The difference isthat � is now not only a function of EP, but also of theposition along isoenergy curves. The analysis of plasma re-sponse in such fields is, of course, much more challenging.

The stability of both the time-independent and the time-dependent solutions obtained in this paper have yet to bedetermined and would shed light on the relaxation paths ofthese systems.

In this paper, we have considered only a monochromaticelectric field. The case when many waves of different fre-quencies are present has also been considered in thepast.38–40 Turbulent growth of waves due to wave-wave in-teraction was shown to behave as diffusion in velocity-spacein the quasilinear approximation. But it is beyond the scopeof the present work to comment on this.

The Appendices of this paper are available at the EPAPSrepository.41

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Analytic, nonlinearly exact solutions for an rf confined plasma