The motion of a single particle under the influence of the ponderomotive force directed perpendicular to the external magnetostatic field is analysed. By solving the exact equation of motion for a specific applied electromagnetic field, the resultant ponderomotive drift is compared with the prediction of a single-particle theory using the oscillation-centre approximation. The regime of validity of this theory is discussed. It is shown that, for certain values of the amplitude and frequency of the electromagnetic field, the particle motion is unstable and therefore the concept of a single-particle ponderomotive force is meaningles

Festeau-Barrioz, M. C.

Sawley, M. L.

Václavík, J.

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J. Plasma Physics (1987), vol. 37, part 3, pp. 497-506 497Printed in Great BritainSingle-particle motion under the influence of theperpendicular ponderomotive forceBy M. C. FESTEAU-BARRIOZf, M. L. SAWLEYAND J. VACLAVlKCentre de Recherches en Physique des Plasmas, Association Euratom - ConfederationSuisse, Ecole Polytechnique Federale de Lausanne, 21, Av. des Bains,CH-1007 Lausanne, Switzerland(Received 7 July 1986 and in revised form 11 September 1986)The motion of a single particle under the influence of the ponderomotive forcedirected perpendicular to the external magnetostatic field is analysed. Bysolving the exact equation of motion for a specific applied electromagnetic field,the resultant ponderomotive drift is compared with the prediction of asingle-particle theory using the oscillation-centre approximation. The regime ofvalidity of this theory is discussed. It is shown that, for certain values of theamplitude and frequency of the electromagnetic field, the particle motion isunstable and therefore the concept of a single-particle ponderomotive force ismeaningless.1. IntroductionIt is well known that a high-frequency electromagnetic field may exert on acharged particle a time averaged (ponderomotive) force. The resultant particlemotion is generally analysed by means of the oscillation-centre approximation(Motz & Watson 1967), in which the position of the particle is split into slowlyand rapidly varying parts, x = xs + Xy. Here xg describes the time averagedmotion of the oscillation centre, and x^ the fast motion about the oscillationcentre. Assuming that |x^| <^  |xg|, the equation of motion can be separatedaccording to slow and fast time scales. Solving the equation of motion on theslow time scale yields the time averaged motion of the particle under theinfluence of the ponderomotive force. In the presence of an external magneto-static field Bo, the ponderomotive force parallel to Bo gives rise to accelerationin the direction along Bo, while the perpendicular ponderomotive force leads toa mutually perpendicular particle drift.The concept of the ponderomotive force has received much attention in thefield of plasma physics owing to its application to the confinement andacceleration of plasmas (Motz & Watson 1967). Recent studies have investi-gated, for example, RF plugging (Watari et al. 1978; Fader et al. 1981) andlow-frequency mode stabilization (Ferron et al. 1983; Yasaka & Itatani 1984)in mirror devices. Since the collective nature of a plasma under the influenceof an RF field may play an important role in determining the plasma dynamics,a single-particle treatment may not, however, always be appropriate. Indeed,| Present address: Centre de Calcul, Ecole Polytechnique Federate de Lausanne, Ecublens,1015 Lausanne, Switzerland.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms of498 M. C. Festeau-Barrioz, M. L. Sawley and J. Vdclavikit has been shown (Kh'ma 1968; Statham & ter Haar 1983) that the pondero-motive force obtained from a fluid description of the plasma does not agree withthat obtained using the oscillation-centre approximation described above. Adetailed discussion of the origin of this fundamental difference between the fluidand single-particle ponderomotive forces may be found elsewhere (Sawley &Vaclavik 1986; Vaclavik, Sawley & Anderegg 1986).It may be asked whether the fluid result for the slow drift caused by theperpendicular ponderomotive force could be obtained from single-particletheory if the oscillation-centre approximation is not used. That this is notpossible can be shown most convincingly by comparing the fluid result with thesolution of the exact equation of motion for a single particle. In the presentpaper, a specific example of single-particle motion under the influence of aperpendicular ponderomotive force is considered. By numerical integration ofthe equation of motion, we compare the exact solution with that obtained fromthe application of the oscillation-centre approximation. We are thus able toshow that single-particle theory is not capable of yielding the fluid result.The use of the oscillation-centre approximation to yield the slow motion ofthe particle tacitly assumes that the amplitude of the fast motion is small.However, as this paper will show, the fast motion can in fact become unstablefor certain values of frequency and amplitude of the RF field. Under theseconditions, the concept of a single-particle ponderomotive force is, of course,meaningless.2. Mathematical formulation2.1. Equation of motionIn this paper we shall consider the motion of a single particle of charge q andmass m under the combined influence of an electromagnetic field and a uniformmagnetostatic field (Bo = ezB0). For the sake of simplicity we shall consider thespecific example for which the total electric and magnetic fields may be writtenasE(x,t)=-eyBobx, B(t) = ezB0(l + b), (1)where b = b cos at and b is constant. We shall ignore the self-consistentcontribution of the total fields due to the motion of the charged particle.The equation of motion for the particle in the above fields is(2)y= -ftSx-ft(l + S)z, (3)z = 0, (4)where the cyclotron frequency ft = qBJm. The particle motion perpendicularto Bo is thus described mathematically in terms of two coupled differentialequations. It should be noted that these equations are nonlinear in the RF fieldamplitude but, for the present choice of fields, linear in the particle position.In the following, we shall normalize quantities by adopting ft"1 and pCl'1 astime and distance units. The most appropriate choice for the normalizingvelocity p will become apparent in § 2.3. We shall use the normalized quantitiesX = Clx/p, Y = ily/p, T = Slt and v = w/ft. (5)use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms ofSingle-particle motion under perpendicular ponderomotive force 4992.2. Oscillation-centre approximationSplitting the particle motion perpendicular to Bo into slowly and rapidly varyingparts,(2) and (3) may be separated according to slow and fast time scales, as is usualin the oscillation-centre approximation (Motz & Watson 1967). The fast timescale equation is solved to 0(b) to yield2bXs {l-v)T (l + u)T(7)In general, a linear combination of sin T and cos T must be added to solutions(7) in order to satisfy the initial conditions for the particle fast motion. However,this yields no contribution to the slow (time averaged) motion, which is givento O(b2) bydXs_ dYs_ j W _I T ' 0 ' ~dT ~~2(1^) *' ( )From (8) it can be seen that the ponderomotive force perpendicular to Bo givesrise to a time averaged drift in the y direction. It may be readily shown thatthis drift can be written, as is usual, in terms of the single-particle ponderomotivepotential:*!*£* where ,6 =dT~dXs2.3. Exact solutionEquations (2) and (3) may be numerically integrated to yield the exact solutionfor the particle motion perpendicular to Bo. However, using the Hamiltonianformulation, since y is an ignorable co-ordinate the y component of thecanonical momentum, Py = mp, is a constant of motion. It is straightforwardto show that (2) and (3) may therefore be written, in normalized form, as(10)(dY/dT) = l-(l+b)X. (11)Equation (10), which describes the x component of the particle motion, may berecognized as an inhomogeneous Hill equation.Equations (10) and (11) must be supplemented by an appropriate set of initialconditions. For definiteness, we considerdXX(0) = l, ^ ; ( 0 ) = 0 and 7(0) = 0. (12)This initial value problem may be solved by standard Runge-Kutta numericalintegration techniques, to yield the exact nonlinear motion of the particle underthe influence of the electromagnetic field.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms of500 M. C. Festeau-Barrioz, M. L. Sawley and J. Vdclavik- 6 - 4 - 2 0 2X 1-0 100 200 300 4000 100 200 300 400TFIGURE 1. Particle motion for v = 09 and 6 = 0-1.3. Results3.1. Particle motionThe initial value problem defined by (10)-(12) has been numerically solved fora number of different values of v and b. The solution for a particular example(v = 0-9, 6 = 0-1) is shown in figure 1. Under the influence of the fields, theparticle exhibits a fast orbital motion that is amplitude modulated at anintermediate frequency vm. The slow time scale (ponderomotive) drift of theparticle in the (negative) y direction is also apparent.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms ofSingle-particle motion underjperpendicular ponderomotive force 5011-2-X 1 0 - X 1 -20-X 0 --20- 6 0 20- 14X104--4 X 10"10 -4X104 -2X104 2X104 4X104 6X104FIGURE 2. Particle motion for 6 = 0-1 and for the six indicated values of v. The three exampleson the left represent stable orbits, while the three on the right are unstable. The total timeis the same (T = 290) for all examples.The effect of changing the frequency v of the electromagnetic field isdemonstrated in figure 2. For the six cases shown, the amplitude of the fieldis b — 0-1. It may be observed that the particle motion is strongly modified bya change of frequency. For certain of the values shown (v = 0-5, 10, 20), themotion is unstable; the amplitude of the fast component increasing withoutbound. Comparing the cases v = 0-9 and 11 , it may be seen that the directionof the ponderomotive drift changes sign as the frequency crosses the cyclotronfrequency (v = 1). For both of these cases, the amplitudes of the fast motionand ponderomotive drift are larger than for the case v = 0- 2 for which thefrequency is further from the cyclotron frequency.Figure 3 illustrates the effect on the particle motion of a change in the fieldamplitude while keeping the frequency constant (v = 0-9). As the field amplitudeis increased, the amplitude of both the fast motion and the ponderomotive driftincreases. For 6 = 0-6, the motion of the particle is no longer stable. Also evidentfrom figure 3 is that as b is increased from 0-1 to 0-5, the frequency vm at whichthe fast motion is modulated decreases.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms of502 M. C. Festeau-Barrioz, M. L. Sawley and J. Vdclavik-6 _4 „ -2 010-- 1 080 -60 -40 v -20 0 20-800 -600 -400 v -200 0 200-2X106-8X 106 -6X 106 -4X 106 -2X 106 0 2X106YFIGURE 3. Particle motion for v = 0-9 and for the four indicated values of 6. The total timeis the same (T = 290) for all examples.3.2. Comparison with the oscillation-centre approximationExcept for particular values of field frequency v, the numerically calculatedparticle motion agrees qualitatively with that predicted using the oscillation-centre approximation provided that the field amplitude b is sufficiently small.A more quantitative comparison can be obtained from figures 4r-Q.In figure 4 is plotted, for three values of v, the modulation frequency vm,use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms ofSingle-particle motion under perpendicular ponderomotive force 5030-8FIGURE 4. Normalized modulation frequency as a function of 6, for O, v = 0-85;x, v = 0-9; +, v = 0-95.normalized to the values \(i — v) determined using the oscillation-centreapproximation, as a function of b. For 6 ^ 1 , vm m\(\_ — v) as is expected.However, as b is increased, a sharp drop in um is observed, being mostpronounced for v close to 1. The decrease of vm to zero results from approachingthe instability boundary: further increase in the field amplitude increases thegrowth rate of the unstable motion.The effect of the field amplitude on the magnitude of the ponderomotive driftvelocity is shown in figure 5. In this figure are plotted values of theponderomotive drift velocity d Ys/dT, for three values of v, obtained from bothnumerical integration and the oscillation-centre approximation, (8). Again itmay be seen that for 6-^1 the two calculations yield similar results. However,as b increases, so does the discrepancy between the results. When the particlemotion is unstable, the concept of a ponderomotive drift becomes meaningless.Figure 6 shows a plot, for b = 02 , of the frequency dependence of the valueof the ponderomotive drift velocity obtained from numerical integration,normalized to that calculated using the oscillation-centre approximation. Thepoints plotted in this figure are for values of v and b for which the fast motionof the particle is stable. It is found that unstable orbits occur in the vicinity ofv = (2/m) for integer m. (13)The width of these regions of instability increases with the value of b, and islargest for small values of m.From figure 6 it may also be seen that the values of the ponderomotive driftvelocity obtained by the two methods diverge ifv = (1/w) for integer n. (14)Clearly, for these values of frequency, the application of the oscillation-centreapproximation is not valid. Note that the ponderomotive drift velocity arisinguse, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms of504 M. C. Festeau-Barrioz, M. L. Sawley and J. Vdclavik0-5FIGURE 5. Ponderomotive drift velocity as a function of b, for O, v = 0-7; x , v = 0-8; +,v = 0-9. , corresponding curves obtained using the oscillation-centre approximation, (8).FIGURE 6. Normalized ponderomotive drift velocity as a function of v for b = 0-2. , valuesof v satisfying (14); , values of v satisfying (13) for odd m.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms ofSingle-particle motion under perpendicular ponderomotive force 505from stable orbits near the frequencies given by (13), with m odd, are welldescribed using the oscillation-centre approximation.The origin of the regions of instability can be appreciated by noticing thatfor b sufficiently small, (10) becomes an inhomogeneous form of the Mathieuequation (McLachlan 1964):+ (a-2qcos2Z)X = 0, (15), vT 4 46where Z =—-, a =—z and q= -.2 v2 v2The well-known stability diagram for the Mathieu equation reveals unstableregions which, for small q, are in the vicinity of a = m2 for integer m. Thiscondition can be seen to be equivalent to (13).In the oscillation-centre approximation, the equation of motion is solved tosecond order in b to obtain the ponderomotive drift. However, if b is non-negligible, higher-order terms may yield a significant contribution. Thesehigher-order contributions to the ponderomotive drift are particularly import-ant if the frequency of the applied electromagnetic field is a sub-harmonic ofthe cyclotron frequency. The nonlinearity of the equation of motion leads, inthis case, to oscillatory motion at the cyclotron frequency, resulting in aresonant ponderomotive contribution. Thus if (14) is approximately satisfied,the oscillation-centre approximation will yield an inaccurate calculation of thefull nonlinear ponderomotive force.4. ConclusionsBy analysing in detail a particular example, the motion of a single particleunder the influence of an electromagnetic field has been investigated. It has beenshown that the particle motion is strongly modified by a change in theelectromagnetic field amplitude (if sufficiently large) or the wave frequency (ifin the vicinity of the cyclotron frequency).Comparison has been made of the ponderomotive drift obtained using theoscillation-centre approximation and from numerical integration of the equa-tion of motion. Within the regime of validity of such a perturbation approach,the oscillation-centre approximation provides a good description of the exactsolution. However, it has been shown that even for a modest field amplitude,the particle motion is unstable for frequencies in the vicinity of twice thecyclotron frequency and its sub-harmonics. In addition, the ponderomotivedrift velocity calculated from numerical integration diverges not only if theapplied frequency equals the cyclotron frequency, but also at the sub-harmonicsof the cyclotron frequency. In the vicinity of these frequencies, the applicationof the oscillation-centre approximation is therefore clearly inappropriate.The results of the present study reinforce the assertion of Sawley & Vaclavik(1986) that the single-particle and fluid (collective) ponderomotive forces arefundamentally different. I t has been shown that the oscillation-centre approxi-mation is complete (within the limits of its application) in describing single-particle behaviour under the influence of the perpendicular ponderomotiveforce. It is not possible to obtain the fluid result from single-particle theory(whether oscillation-centre approximation or exact solution).use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:25:29, subject to the Cambridge Core terms of506 M. C. Festeau-Barrioz, M. L. Sawley and J. VdclavikThis work was partially supported by the Fonds National Suisse de laRecherche Scientifique.REFERENCESFADER, W. J., JONG, R. A., STUFFLEBEAM, J. H. & SZIKLAS, E. A. 1981 Phys. Rev. Lett. 46,999.FERRON, J. R., HEKSHKOWITZ, N., BREUN, R. A., GOLOVATO, S. N. & GOULDING, R.1983 Phys. Rev. Lett. 51, 1955.KLIMA, R. 1968 Czech. J. Phys. 518, 1280.MCLACHLAN, N. W. 1964 Theory and Applications of Mathieu Functions. Dover.MOTZ, H. & WATSON, C. J. H. 1967 Adv. Electron. Electron Phys. 23, 153.SAWLEY, M. L. & VACLAVIK, J. 1986 Comments Plasma Phys. Contr. Fusion, 10, 19.STATHAM, G. & TER HAAR, D. 1983 Plasma Phys. 25, 681.VACLAVIK, J., SAWLEY, M. L. & ANDEREGG, F. 1986 Phys. Fluids, 29, 2034.WATARI, T. et al. 1978 Phys. Fluids, 21, 2076.YASAKA, Y. & ITATANI, R. 1984 Nucl. Fusion, 24, 445.use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0022377800012332Downloaded from https:/www.cambridge.org/core. 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Single-particle motion under the influence of the perpendicular ponderomotive force

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