Abstract
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 been advanced to explain the observed heating in such plasmas [I. Siemers et al., Phys. Rev. A38, 5121 (1988)]. This study investigates the nature of a onedimensional collisionless electron plasma that is confined by an rf field of the form , where is the space coordinate and is the rf frequency. Nonlinearly exact solutions are obtained. The distribution function and the plasma density are obtained in closed form and have constant shapes with time varying oscillations. These oscillations are at the rf frequency and its harmonics, modulated by a low frequency related to the electron bounce time. The linear limit of weak fields is recovered. Analytic expressions are obtained for the required external field to make it consistent with prescribed distribution functions. These solutions remain valid even in the presence of collisions. Solutions involving multiple species are also obtained, though only for collisionless traps. It is found that the ponderomotive force response needs to be corrected to account for the temperature fluctuations. No stochastic heating is observed in this field configuration.
I. INTRODUCTION
II. THE EQUATIONS OF MOTION
III. TIME EVOLUTION OF THE DISTRIBUTION FUNCTION AND DENSITY
IV. SELFCONSISTENT SOLUTIONS AND PLASMA CONFINEMENT
V. DISCUSSION
A. The ponderomotive potential
B. Relation to BGK theory
C. Collisional effects
D. Multispecies plasmas
E. Linear response theory
F. Experimental realizability
VI. CONCLUSIONS
Key Topics
 Plasma confinement
 24.0
 Electric fields
 23.0
 Particle distribution functions
 20.0
 Cumulative distribution functions
 15.0
 Nonlinear dynamics
 13.0
Figures
The solid line shows the plot of the path of the particle as per numerical integration of Eq. (5). The dashed line shows the path as predicted by the mathematical expression given in Eq. (11), considering only the first three terms. This was done for , , and the initial conditions are , . We can see that there is close agreement between mathematics and simulation and little discrepancy can be reduced further by considering higher order terms in the mathematical expression.
The solid line shows the plot of the path of the particle as per numerical integration of Eq. (5). The dashed line shows the path as predicted by the mathematical expression given in Eq. (11), considering only the first three terms. This was done for , , and the initial conditions are , . We can see that there is close agreement between mathematics and simulation and little discrepancy can be reduced further by considering higher order terms in the mathematical expression.
This is the phase space plot of the trajectory shown in Fig. 1. We can see that near and , the particle is undergoing high frequency oscillations. This is the region of phase space for which the ponderomotive theory holds.
This is the phase space plot of the trajectory shown in Fig. 1. We can see that near and , the particle is undergoing high frequency oscillations. This is the region of phase space for which the ponderomotive theory holds.
This shows the density plots with , at two different times and , where is such that is close to . The three plots are for different values of . (a) ; (b) , where is given in Eq. (27); (c) . In (a), it can be seen that the two curves are quite close to each other. This shows that, approximately, is the value of at which the density function does not 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 more accurate expression for the ponderomotive energy of a particle under the linearly varying oscillatory electric field. The overlap of the curves at and is so good that it is not visible in this graph. In (c) it can be clearly seen that for this value of , the density function has a strong dependence on .
This shows the density plots with , at two different times and , where is such that is close to . The three plots are for different values of . (a) ; (b) , where is given in Eq. (27); (c) . In (a), it can be seen that the two curves are quite close to each other. This shows that, approximately, is the value of at which the density function does not 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 more accurate expression for the ponderomotive energy of a particle under the linearly varying oscillatory electric field. The overlap of the curves at and is so good that it is not visible in this graph. In (c) it can be clearly seen that for this value of , the density function has a strong dependence on .
This is the contour plot of the distribution function of the plasma for the case with , . The two superimposed contour plots correspond to the two times of the curves shown in Fig. 3(c). This clearly shows that the drastic change in the distribution function is the reason for the huge change in the density function over the time scale.
This is the contour plot of the distribution function of the plasma for the case with , . The two superimposed contour plots correspond to the two times of the curves shown in Fig. 3(c). This clearly shows that the drastic change in the distribution function is the reason for the huge change in the density function over the time scale.
This shows the spatial variation of the magnitude of various frequency components present in the Fourier transform of as given by Eq. (32). This is for the case when is given by Eq. (27) and , . As can be clearly seen, leaving the dc component, the component at is dominating. And we also have small contributions from components at frequencies , ( in our normalization). The remaining harmonics are lower in magnitude than the ones shown. The component at is two orders of magnitude lower than that at . So, the field given by Eq. (32) is essentially a nonuniform monochromatic electric field for all practical purposes.
This shows the spatial variation of the magnitude of various frequency components present in the Fourier transform of as given by Eq. (32). This is for the case when is given by Eq. (27) and , . As can be clearly seen, leaving the dc component, the component at is dominating. And we also have small contributions from components at frequencies , ( in our normalization). The remaining harmonics are lower in magnitude than the ones shown. The component at is two orders of magnitude lower than that at . So, the field given by Eq. (32) is essentially a nonuniform monochromatic electric field for all practical purposes.
This shows the relative spatial variation of the fields corresponding to the rf solution considered in this paper and the fields in static equilibrium. If the timeaveraged density of the plasma goes like , then the “Effective static Field” corresponds to the electric field for which the potential goes like . The “Induced Field” is the field induced by the electron density in the absence of ions. The “Applied static Field” is the sum of these two fields, and is the external 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.
This shows the relative spatial variation of the fields corresponding to the rf solution considered in this paper and the fields in static equilibrium. If the timeaveraged density of the plasma goes like , then the “Effective static Field” corresponds to the electric field for which the potential goes like . The “Induced Field” is the field induced by the electron density in the absence of ions. The “Applied static Field” is the sum of these two fields, and is the external 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.
This figure shows the plot of the magnitude of the coefficients of and in the expression for as given in Eq. (38), normalized by . It can be clearly seen that magnitude of the response at lies below the response at . The solid thick line labeled serves to divide two prominent regions of plasma response. When , 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 . This also demarcates two regions in the space. As crosses this line, there is a visible change in the slope of the curves. The plots clearly show that as becomes large compared to , the first nonlinearity to set in is of the order of . Also, the curves are well in agreement with the expressions derived in Eq. (37).
This figure shows the plot of the magnitude of the coefficients of and in the expression for as given in Eq. (38), normalized by . It can be clearly seen that magnitude of the response at lies below the response at . The solid thick line labeled serves to divide two prominent regions of plasma response. When , 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 . This also demarcates two regions in the space. As crosses this line, there is a visible change in the slope of the curves. The plots clearly show that as becomes large compared to , the first nonlinearity to set in is of the order of . Also, the curves are well in agreement with the expressions derived in Eq. (37).
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