Doppler-shifted cyclotron resonance of fast ions with circularly polarized shear Alfvén waves

The Doppler-shifted cyclotron resonance between fast ions and shear Alfvén waves (cid:2) SAWs (cid:3) has been experimentally investigated with a test-particle fast-ion (cid:2) Li + (cid:3) beam launched in the helium plasma of the Large Plasma Device (cid:4) Gekelman et al. , Rev. Sci. Instrum. 62 , 2875 (cid:2) 1991 (cid:3)(cid:5) . Left- or right-hand circularly polarized SAWs are launched by an antenna with four current channels. A collimated fast-ion energy analyzer characterizes the resonance by measuring the nonclassical spreading of the averaged beam signal. Left-hand circularly polarized SAWs resonate with the fast ions but right-hand circularly polarized SAWs do not. The measured fast-ion proﬁles are compared with simulations by a Monte Carlo Lorentz code that uses the measured wave ﬁeld data. © 2009 American Institute of Physics . (cid:4) DOI: 10.1063/1.3103813 (cid:5)


I. INTRODUCTION
Fast ions and Alfvén waves are pervasive in both natural 1,2 and laboratory plasmas. 3 The interaction of fast ions with Alfvén waves and instabilities is challenging to study experimentally because of difficulties in diagnosing the fast-ion distribution function and the wave fields accurately, in either fusion devices or space plasmas. In a hot fusion device, emitted particles are used to infer the distribution function in the core. In the solar-terrestrial plasma environments, where Alfvén waves constitute the dominant components of the EM wave spectra, expensive spacecraft measurements near the earth can cover but a fraction of the daunting space.
The fast-ion campaign at the Large Plasma Device ͑LAPD͒ ͑Ref. 4͒ at the University of California, Los Angeles has been focused on fast-ion transport and Alfvén activities in this linear device starting from the year 2002. So far, several experiments have been conducted in the device and reported, including a study of classical fast-ion transport, 5 the observation of a gap in the shear Alfvén wave ͑SAW͒ spectrum induced by an array of magnetic mirrors, 6 and a study of the Doppler-shifted resonance of fast ions with linearly polarized SAWs. 7 The unique LAPD provides a probeaccessible plasma that features dimensions comparable to magnetic fusion research devices, which can accommodate both large Alfvén wavelengths and fast-ion gyro-orbits.
The approach of the work reported here is to launch test-particle fast-ion beams with a narrow initial distribution function in phase space using plasma-immersible fast-ion sources. 8,9 Here the test-particle assumption is made since the energy transfer between the particles and waves is much smaller than the wave energy. The fast-ion beam is readily detected by a collimated fast-ion analyzer. When resonance overlap of fast ions and SAWs occurs, resonant beam transport in addition to the well calibrated classical transport 5 is analyzed with good phase-space resolution.
An earlier paper 7 described measurements in the LAPD of the Doppler-shifted cyclotron resonance of fast ions with SAWs. In that work, the waves were essentially linearly polarized and the main interaction region was only about onethird of the cyclotron orbit. The observed nonclassical spreading caused by the resonant interaction is in good qualitative and quantitative agreement with theoretical predictions. In particular, the maximum broadening occurs at the predicted wave frequency, and, as the frequency is varied, subsidiary peaks and nulls are observed at the expected locations. The work reported here extends this earlier study in several ways. With the exception of the SAW antenna, the hardware and experimental technique are identical to the previous work.
The organization of this paper is as follows: In Sec. II, the analytical theory for this work is summarized with the results from a Monte Carlo Lorentz code simulation. The experimental setup and fast ion and SAW diagnostics are introduced in Sec. III and the new experimental results are reported in Sec. IV. Conclusions are drawn in Sec. V along with planned future fast-ion transport studies.

A. Resonance models
A charged particle is in cyclotron resonance with a transverse electromagnetic ͑EM͒ wave if the oscillation of wave electric field Ẽ matches the Doppler-shifted cyclotron motion of the particle. 10 In this case the following Doppler-shifted cyclotron resonance condition will be satisfied, where is the wave frequency, k ʈ and v ʈ are the wave number and velocity components parallel to the equilibrium magnetic field B 0 , and ⍀ f = q f B 0 / m f is the cyclotron frequency of the fast ions. The Ϯ sign in Eq. ͑1͒ corresponds to the Doppler and anomalous Doppler resonances, where the resonant fast ions move slower or faster than the parallel wave phase speed / k ʈ , respectively. Equation ͑1͒ is however not a sufficient condition for effective energy exchange between waves and fast ions; the wave field itself is required to have the correct polarization. For example, ions gyrate in a left handed sense around magnetic field lines and so the most effective wave-particle interaction will occur for an EM wave satisfying Eq. ͑1͒ with an electric field that is circularly polarized in the same direction. In contrast with the previous work, 7 which investigated linearly polarized regions of the waves, this paper investigates the Doppler resonance of circularly polarized SAWs with fast ions. If not otherwise specified, the default polarization of the waves in this work is left handed.
Here the goodness of the resonance can be defined to be In the presence of an EM wave the fast ion motion is given by the Lorentz force law where Ẽ and B are the perturbed fields from the EM wave. In general, the rate of change of fast ion energy parallel and perpendicular to the background magnetic field can be represented as 11 where the energy ͑W͒ change for each component is a combination of the work done by the wave electric field and a transfer of energy from the other component via the magnetic field. By following a gyrating fast ion's trajectory, a plane wave can be represented as a sum of helical waves moving coaxially with the fast ion as 11 where = k Ќ f , f is the fast-ion Larmor radius, and is the gyroangle. By insisting that the fast ion must experience the wave at a constant phase in order to achieve effective waveparticle interaction, the resonance condition in Eq. ͑1͒ is obtained for m = Ϯ 1. For the m =1 ͑the Doppler case͒ component of the wave, if the plasma is assumed to be sufficiently cold to avoid finite Larmor radius effects, then the time averaged energy change for the fast ions in the wave field takes the approximate form 11 where Ẽ + is the component of the perpendicular electric field that rotates in the same direction as the gyrating fast ions. From this it can be seen that unless a circularly polarized wave has the correct handedness, then no energy will be exchanged on average between the fast ions and the wave. Efficient energy transfer can also occur via the parallel component of the wave electric field. For this the m = 0 part of the wave in Eq. ͑5͒ is required. The constant phase criterion then leads to the Landau resonance condition In this case the time averaged energy change for the fast ions becomes 11 Experimentally, the most obvious change in beam-ion profile during interaction with the SAW is a change in gyroradius, suggesting that the energy change caused by the perpendicular electric field ͓Eq. ͑6͔͒ dominates over the energy change caused by the parallel electric field ͓Eq. ͑8͔͒. As predicted by Eq. ͑6͒ ͓but not by Eq. ͑8͔͒, significant energy exchange between the particles and the waves only occurs for left-handed polarization of the SAW ͑Sec. IV͒.

B. SAW dispersion relation
There are two regimes of plasma parameters for SAW propagation: the kinetic Alfvén wave ͑KAW͒ for plasma electrons having a Boltzmann distribution in the presence of the Alfvén wave fields and the inertial Alfvén wave ͑IAW͒ for electrons responding inertially to the wave. The KAW is more relevant to the physics of the interior regions of tokamak plasmas and the IAW to the edge and limiter regions. In this experiment, KAWs launched during the discharge of the LAPD are investigated.
A dimensionless parameter, ␤ e ϵ v te 2 / v A 2 , is a quantitative measure of how inertial or kinetic a plasma is, where v te = ͱ 2T e / m e is the thermal electron speed with T e as the electron temperature. For example, if ␤ e ӷ 1, as in the discharge plasma of the LAPD, it is kinetic; if ␤ e Ӷ 1, as in the afterglow plasma, the wave is inertial. For the KAW, the dispersion relation is where k ʈ is the component of the wave vector parallel to the background magnetic field, k Ќ is the perpendicular wave number, and s is the ion sound gyroradius In a typical case, k Ќ 2 s 2 is estimated to be 0.12. The intensity of the parallel wave electric field is 12 A typical ratio of ͉Ẽ ʈ / Ẽ Ќ ͉ is 0.01, which means that the nonvanishing parallel electric field will slightly modify fast ion v ʈ in the resonance experiment. The perpendicular electric field can be calculated from Ampère's law and the dispersion relation, given measured wave magnetic field data ͑Sec. IV͒. To indicate the relative strength of different SAWs, B max is defined in this article as the maximum amplitude of B x in a specific z plane.

C. Monte Carlo simulation
A Monte Carlo Lorentz code 7,13 is used to simulate the SAW-induced fast-ion transport, with the classical transport caused by thermal ions and electrons included. In the LAPD discharge plasma, the electrons dominate Coulomb slowing down, while thermal ions dominate the pitch-angle scattering rate. A number of fast ions are launched numerically according to the initial beam divergence in phase space ͑Ϯ5°in pitch angle and ϳ5 eV in energy͒. Due to the difference in time scales of the transport mechanisms shown in Table I, the classical transport effects can be readily decoupled from the SAW influence in the code.
During the simulation of each fast-ion orbit, the pitch angle scattering event is considered at a randomly selected time ͑t = t 1 ͒, by which each gyroperiod is divided into two partitions ͑ Cyclotron = t 1 + t 2 ͒. The single-particle Lorentz code is carried on throughout t 1 and t 2 for SAW perturbation. At the end of each time partition, the fast-ion energy is updated by the Coulomb slowing-down effect, At t = t 1 , a Monte Carlo collision operator 13 developed from the Coulomb scattering theory 14 is loaded to scatter the direction of the fast-ion velocity without changing its magnitude, 15   The simulated beam spatial profile shows substantial widening when the Doppler resonance condition is met. 7 To compare the simulation results with experimental data collected by the fast-ion detector ͑Sec. III͒, the computed beam signals are simulated with a Lorentz code that calculates the fast-ion orbits inside the detector. The internal geometric and electric configurations of the detector are included in the code. Electric field lines are modeled by the COMSOL software ͑COMSOL, Inc.͒. When a fast ion enters the detector, its orbit is calculated by the Lorentz force law with the dc electric and magnetic fields inside the detector ͑SAW influence is negligible here͒. The number of fast ions collected at each position in the x-y plane is proportional to the collected signal strength ͑Sec. III͒. The simulated fast-ion beam profile can then be plotted by scanning a spatial grid near the orbit, similar to the motion of the probe in the experiment.
Contour plots generated by simulation are compared with experimental beam profiles in Fig. 1. During this experiment, to efficiently utilize the fast-ion source life time, 7 only the upper half of the beam spot is collected as shown in Figs. 1͑a͒ and 1͑b͒. Using the same color scale, the contour plot shows obvious widening in both the gyroradial ͑r͒ and gyrophase ͑ ͒ direction when the SAW satisfies the resonance condition. Similarly, the simulated profiles show the same physics in Figs. 1͑c͒ and 1͑d͒.

A. Overview
This experiment is performed in the upgraded LAPD, which has a 17.56-m-long, 1-m-diameter cylindrical main vacuum chamber. Pulsed plasmas ͑70 cm diameter, ϳ10 ms in duration, 1 Hz repetition rate, and Ϯ10% spatial unifor-mity͒ are created by a discharge between a barium oxide coated cathode and a gridded molybdenum anode. 16 The cathode and anode are separated by 50 cm and both located at the south end of the machine. The working gas is helium at a partial pressure of ϳ3 ϫ 10 −5 Torr with less than 3% of impurities. Typical average plasma parameters for this experiment are n i ϳ 2.5ϫ 10 12 cm −3 , T e ϳ 6 eV, and T i ϳ 1.0Ϯ 0.5 eV. The fast-ion beam density ͑5.0ϫ 10 8 cm −3 ͒ used in this experiment is typically three to four orders of magnitude smaller than the plasma density, which ensures the test-particle assumption.
SAWs have been studied extensively in the LAPD. The waves are launched with inserted disk or loop antennas that drive propagating waves. Because of the large physical size, the plasma can accommodate up to ten axial wavelengths and the transverse wave structure is not determined by the radial dimensions of the plasma. The waves are partially reflected at the anode. Estimates indicate that ϳ10% of the measured wave amplitude may be associated with the reflected wave. 7 The hardware configuration is illustrated in Fig. 2͑a͒ for this experiment. The origin of the z axis is defined as the location of the lithium fast-ion source ͑Sec. III B͒. The beam-wave interaction region is ϳ10 m downstream of the cathode to ensure radially uniform background plasma properties. The SAW antenna ͑Sec. III C͒ is located at z = −3.52 m to avoid the near field effect ͑typical SAW wavelength parallel to B 0 is ϳ4 m͒. The fast-ion analyzer ͑Sec. III B͒ scans fast-ion signals in the x-y plane 0.32-0.96   m away from the source to vary the interaction time between wave and particle. Figure 2͑b͒ shows the actual LAPD port locations of all the instruments relative to the ambient magnetic field profile. The distance between two adjacent ports is 0.32 m. The axial magnetic field profiles are calculated from the actual current distribution for all the magnets in the machine. 17 The estimated axial field ripple amplitude at the radius of the machine is less than 2% for a constant field configuration.
In Table II, fast-ion orbit parameters, SAW frequency, and plasma conditions for the Doppler-shifted resonance studies in this work are listed.

B. Fast-ion sources and diagnostics
Lithium ion sources 8 with lithium aluminosilicate as thermionic emitters of different sizes ͑0.6 and 0.25 in. diameter, Heat Wave Inc. 18 ͒ were designed, constructed, and characterized at the University of California, Irvine. For this transport study, the 0.6 in. emitter version with isotopically purified Li-7 ͓99.99% grade, experimentally determined to be greater than 90% ͑Ref. 19͔͒ is operated with a 5 mm diameter circular aperture to achieve a narrow, high uniformity beam.
After the beam leaves the source and travels in the plasma and wave fields, a collimated fast-ion analyzer 7 is employed to produce a good fast-ion signal-to-noise in the strong rf noise environment during the LAPD discharge. The analyzer contains a 3.2-mm-diameter collector, an identical dummy for differential amplification, and three molybdenum grids for variable electrical barriers. 8 Measurements of the beam profile and wave field utilize a two-dimensional ͑2D͒ probe drive system ͑Ϯ0.5 mm precision͒ and take advantage of the highly reproducible plasma of the LAPD. At each spatial location, multiple time traces are collected with ten plasma shots repeated. The probe tip is then moved to the next spatial position according to the preselected grid and the process is repeated.
For this transport study, fast-ion parameters are always selected to complete integer numbers of gyrocycles that coincide with the location of diagnostic ports since the slightly divergent beam is only refocused at integer numbers of gyroorbits, as described by the "geometrical effect" in Ref. 5. The beam originates from the circular aperture, experiences classical cross-field diffusion, and passes the collection x-y plane with a pitch-angle close to the initial value.

C. SAW antennae and diagnostics
In order to generate circularly polarized SAWs with selectable handedness, this work uses a newly designed quadruple channel antenna ͑Fig. 3͒, which is equivalent to two identical rectangular loop antennae 20 as used in the previous work. 7 Each current loop is 30 cm along B 0 and generates a linearly polarized wave field pattern near the center of the loop in the x-y plane. The wave propagates along B 0 without significant changes in amplitude and pattern. The second current loop generates another linearly polarized wave field driven at Ϯ / 2 phase difference from the first one. The combined wave field becomes left-or right-hand circularly polarized as described in Eq. ͑5͒. The fast-ion orbit is aligned with the wave field to maximize the resonant interaction.
The wave magnetic field is measured by a set of identically designed b-dot probes featuring three orthogonal, differentially wound induction coil pairs, which are sensitive to the time derivative of the wave magnetic field. 21 Each coil pair is connected to a differential amplifier to select the mag-

D. System synchronization and data collection
Synchronization of the fast-ion source, the SAW antenna and the LAPD plasma is critical to a successful resonance experiment. It is arranged that the ϳ20 ms fast-ion pulse is turned on and off every two plasma discharges, with an adjustable delay and duration to overlap in time with the ϳ10 ms LAPD discharge. The background signal is taken when the source is disabled for one shot right after the previous beam-on shot. Even numbers of shots are repeated at one spatial location before the probe drive moves on. The signals with the source off are subtracted from the signals with the source on. The fast-ion signals shown in this work are net signals with the background subtracted.
Close to the end of each discharge, when the plasma parameters reach the best uniformity in time and space, the quadruple channel antenna drives an adjustable rf current for 1.0 ms, launching circularly polarized SAWs at the specified frequency. The data collection time window opens for ϳ2 ms with ϳ0.8 ms before the SAW duration. In this article, the SAW-influenced beam signal at each collection point ͑in the x-y plane͒ is averaged from 0.7 to 1 ms after the onset of the antenna current. The data before the SAWs are averaged to get the controls for no-wave situations. The time-averaged beam signals taken at a selected spatial grid are then combined to show the 2D fast-ion beam profile.

A. Fast-ion SAW induced transport is proportional to wave amplitude
The fast-ion beam profile shows significant widening as the SAW satisfies Doppler-shifted resonance condition ͑Fig. 1, experimental fast-ion orbits are different from machine center͒. The amplitude of the wave is adjusted at several different levels in order to observe the wave amplitude dependency. Here Fig. 4 shows the shade surfaces of the fastion beam signal profiles, where signal strength is the z axis. As the wave amplitude increases, the fast-ion profile becomes flatter and broader in both the r and directions.
The changes in the full width at half maximum ͑FWHM͒ and the peak intensity are both good indicators 7 of the resonance effect and they are plotted against B max in Fig. 5. The four sets of data points are acquired from the averaged fastion beam radial profiles at four different amplitudes of the waves. Linear dependence is not expected here since the resonance effect is an integration of the wave field influence along several gyrocycles and the experimental wave amplitude is not spatially uniform.

B. Selective fast-ion resonance with left-hand circularly polarized SAWs
The Doppler-shifted fast-ion resonance only happens when left-hand polarized SAWs are present to accelerate/ decelerate the fast-ion at the correct phases. The new quadruple channel antenna can select SAWs with different polarization directions relative to B 0 . Resonance experiments are conducted with left-and right-hand polarized SAWs at the same frequency and amplitude. Here Fig. 6 compares the statistically averaged radial beam profiles for both polariza-  tions with simulation results that use the measured wave field data. Both the experiment and the simulation show significant widening only for left-hand polarized waves. This confirms that the profile broadening is caused by changes in the fast-ion orbits rather than the field-line displacement produced by the SAW.

V. CONCLUSIONS AND FUTURE WORK
In this experimental work, the Doppler-shifted cyclotron resonance effect is directly measured from the beam spatial nonclassical spreading caused by circularly polarized SAWs. The resonance effect is observed to be dependent on the amplitude of the waves. Only the left-hand polarized SAWs resonate with the fast ions at the resonance condition.
Several extensions of this experimental study are possible. Simulation predicts that the fast-ion beam energy is changed up to ϳ23 eV at the Doppler resonance condition with 1.6 G wave magnetic field amplitude. Direct measurements of energy changes could be obtained by biasing the fast-ion collector. A higher bandwidth fast-ion analyzer would permit detection of coherent oscillations in the collector current at the wave frequency. Observation over longer axial distances is also desirable.
More generally, the successful observation of fast-ion interaction with a single coherent Alfvén wave lays the groundwork for studies of more complicated wave-particle interactions, including interaction with Alfvén waves in the nonlinear regime and studies of fast-ion diffusion by driftwave turbulence.