Cold Plasma Theory Research Articles

Backstreaming ions at a high Mach number interplanetary shock

Context.Solar Orbiter, a mission developed by the European Space Agency, explores in situ plasma across the inner heliosphere while providing remote-sensing observations of the Sun. The mission aims to study the solar wind, but also transient structures such as interplanetary coronal mass ejections and stream interaction regions. These structures often contain a leading shock wave that can differ from other plasma shock waves, such as those around planets. Importantly, the Mach number of these interplanetary shocks is typically low (1–3) compared to planetary bow shocks and most astrophysical shocks. However, our shock survey revealed that on 30 October 2021, Solar Orbiter measured a shock with an Alfvén Mach number above 6, which can be considered high in this context.Aims.Our study examines particle observations for the 30 October 2021 shock. The particles provide clear evidence of ion reflection up to several minutes upstream of the shock. Additionally, the magnetic and electric field observations contain complex electromagnetic structures near the shock, and we aim to investigate how they are connected to ion dynamics. The main goal of this study is to advance our understanding of the complex coupling between particles and the shock structure in high Mach number regimes of interplanetary shocks.Methods.We used observations of magnetic and electric fields, probe-spacecraft potential, and thermal and energetic particles to characterize the structure of the shock front and particle dynamics. Furthermore, ion velocity distribution functions were used to study reflected ions and their coupling to the shock. To determine shock parameters and study waves, we used several methods, including cold plasma theory, singular-value decomposition, minimum variance analysis, and shock Rankine-Hugoniot relations. To support the analysis and interpretation of the experimental data, test-particle analysis, and hybrid particle in-cell simulations were used.Results.The ion velocity distribution functions show clear evidence of particle reflection in the form of backstreaming ions several minutes upstream. The shock structure has complex features at the ramp and whistler precursors. The backstreaming ions may be modulated by the complex shock structure, and the whistler waves are likely driven by gyrating ions in the foot. Supra-thermal ions up to 20 keV were observed, but shock-accelerated particles with energies above this were not.

Resonance cones in cold plasma: Origin, singularities, and power flow

In magnetized tenuous plasma, typical at the plasma edge of fusion devices, a nearly electrostatic wave mode with relatively enhanced electric field can propagate along a specific angle with the magnetic field. For this characteristic, it is known as a “resonance cone.” For instance, these waves can be excited by radio frequency antennas in the ion-cyclotron and lower-hybrid range of frequencies. We consider the resonance cones emitted by idealized spatially extended sources. In 2D, we use a novel geometric construction which generalizes the d'Alembert solution to curved boundaries/moving sources, and show, for the first time, that singular electric fields arise under these conditions, thereby bringing the resonance cones in line with the other resonances of the cold plasma theory. Still in 2D, we give an expression for the amount of power radiated by resonance cones in terms of surface quantities on the source, which is finite despite the singular electric field. We generalize the conclusions regarding the presence and location of singular electric fields to the 3D electromagnetic case.

Relativistic analysis of upper hybrid wave propagation and trapping

We investigate the impact of relativistic effects on upper hybrid (UH) waves in plasmas with thermal electrons, particularly focusing on modifications of the conditions under which UH wave trapping and related low-threshold parametric decay instabilities (PDIs) may occur. A moderately relativistic (MR) dispersion relation for UH waves, valid for electron temperatures up to 25 keV and wave frequencies up to twice the electron cyclotron frequency, is obtained from previous results and shown to reduce to the warm non-relativistic result commonly used for PDI studies at low electron temperatures. The conditions under which MR UH waves propagate are then determined and compared with warm and cold plasma theory, showing a general increase in the electron density and background magnetic field strength at which the UH resonance occurs for finite electron temperatures. We next investigate the impact of the MR corrections on the possibility of UH wave trapping for X-mode electron cyclotron resonance heated (ECRH) plasmas at the ASDEX Upgrade tokamak and scaled versions of the ASDEX Upgrade parameters with core electron temperatures resembling those expected in ITER X-mode ECRH plasmas. The MR UH wave trapping conditions are virtually unchanged for ASDEX Upgrade relative to warm theory, due to the low electron temperatures, while potentially important differences between warm and MR theory exist for ITER-like core electron temperatures; cold theory is found to be insufficient in both cases. Finally, the MR dispersion relation is shown to qualitatively reproduce the PDI thresholds from warm theory for previously studied ASDEX Upgrade cases.

Quantifying the Sheath Impedance of the Electric Double Probe Instrument on the Van Allen Probes

AbstractSpherical double probe electric field sensors become electrically coupled to magnetospheric plasma during operation, leading to an instrument response that varies with the local plasma environment. Here, a method is developed for determining this variable coupling impedance for each measurement direction by using periods of favorable boom, wave, and magnetic field geometry. Comparing electric field complex amplitudes between 30 Hz and 10 kHz observed along each boom direction to those predicted from simultaneous magnetic field measurements and cold plasma theory allows for the amplitude and phase response of the instrument to be quantified over the full range of plasma densities encountered on‐orbit. A sheath model is developed to describe how the sheath resistance, sheath capacitance, and relative effective length vary as a function of plasma density. An additional empirical correction is also included to describe the phase response along the spin‐axis. The modeled sheath correction is subsequently tested for case studies of burst‐mode data and statistical analyses of survey‐mode data. It is demonstrated that the levels of agreement between observations and theoretical predictions based on Faraday's Law are substantially greater for the sheath corrected data than for uncorrected observations. Comparisons between observations with oppositely directed Poynting vector directions reveals that the sheath correction reconciles a bifurcated distribution in the uncorrected data to a single peak centered on agreement with Faraday's Law. A full sheath corrected EMFISIS L4 survey mode data set has been produced for final archive. Full details of the sheath correction are also provided for manual data correction.

A Statistical Examination of EMIC Wave‐Driven Electron Pitch Angle Scattering Signatures

AbstractElectromagnetic ion cyclotron (EMIC) waves have long been thought to contribute significantly to the variability of Earth's outer radiation belt because of their ability to scatter relativistic (>0.5 MeV) electrons into the loss cone on short timescales. We investigate statistically such pitch angle scattering signatures (“bite‐outs”) and their quantitative agreement with cold plasma theory. Our analysis is based on EMIC wave events and local plasma parameters observed simultaneously with energetic electrons by the Van Allen Probes between February and May 2017. We determine the energy and pitch angle ranges in which relativistic electron scattering is likely and find that, on average, EMIC wave‐driven scattering of core (E ≤ 2 MeV) electrons is possible in 28% of events. By comparing the expected ranges for scattering to electron flux measurements co‐located with the wave activity, we determine that scattering signatures are found in 46% of those. Additionally, we find that the scattering signatures are more likely to occur under increased geomagnetic activity and for waves with greater wave power. These results provide observational evidence that EMIC wave‐driven scattering of core relativistic electrons is common in the duskside outer radiation belt.

Electromagnetic Characteristics of Fast Magnetosonic Waves in the Inner Magnetosphere

AbstractIn evaluating the effects of fast magnetosonic (MS) waves on magnetospheric particles, their magnetic spectra are often obtained from satellite observations, while electric field components are usually derived under the cold plasma approximation. However, such an approximation has not been verified with in situ observations yet. In this study, we report the electromagnetic characteristic for MS waves in various plasma environments with observations of the Van Allen Probe A. It is found that a considerable number of observed MS waves consist of dominated electrostatic components, which also suggest the importance of inspecting the estimation algorithm for the electric field components. Moreover, the comparison between results from statistical and theoretical analysis shows that electromagnetic characteristics of the observed MS waves can be well predicted by cold plasma theory. Our result indicates the validation of cold plasma approximation to estimate the electric field components of MS waves from their magnetic counterparts in the inner magnetosphere.

Realistic Dispersion of Plasmaspheric Hiss in the Inner Magnetosphere and Its Effect on Wave-induced Electron Scattering Rates

Abstract The cold plasma approximation is a common treatment to study wave–particle interactions between plasmaspheric hiss and magnetospheric electrons, which, however, can become a challenge during periods of disturbed geomagnetic activity. To assess the validity of the cold plasma dispersion relation of plasmaspheric hiss, we adopt the cold plasma theory to calculate the wave magnetic field intensities from the electric field intensities observed by Van Allen Probe A from 2012 October 1 to 2018 February 28. Comparisons between the observed and converted hiss magnetic field intensities capture pronounced differences with the enhancement of substorm activity, and exhibit the largest discrepancies on the nightside MLT sector for low-frequency hiss waves. We also use both the hiss dispersion curves derived from the observations and the cold plasma dispersion relation to evaluate hiss-induced electron diffusion coefficients under different substorm activity conditions. The results indicate that the profiles of electron diffusion rates vary considerably for the two hiss dispersion relations and that the differences between them become increasingly distinct with substorm activity intensification. Our study therefore demonstrates that the cold plasma theory can become less reliable for plasmaspheric hiss waves under disturbed geomagnetic circumstances and that the realistic wave dispersion is essential to better quantify the electron scattering effect of hiss waves, which needs to be carefully incorporated into future global simulations of the generation and propagation of plasmaspheric hiss and associated dynamic variability of radiation belt electrons.

Examining Wave Vector and Minimum Cyclotron Resonant Electron Energy of EMIC Waves With Magnetospheric Multiscale Mission

AbstractWave number vectors k and minimum cyclotron resonant electron energies Emin of electromagnetic ion cyclotron (EMIC) waves are analyzed via the phase differencing technique by using Magnetospheric Multiscale Mission data. It is demonstrated that the phase differencing method provides an estimate of the dominant wave number when finite k spectrum broadenings occur. A case study is conducted for the EMIC event on 20 November 2015, showing remarkable agreements with spectral analysis in wave propagation directions. We find that obtained wave vectors, roughly agreeing with the validity of cold plasma theory, might significantly vary from wave packet to wave packet. Numerical calculations indicate that Emin can range from 0.5 to tens of MeV, suggesting that EMIC waves can effectively interact with those relativistic electrons. This study enriches our understanding of the applicability of phase differencing. It further supports that EMIC waves can be responsible for the loss of electrons with an extremely broad energy range in the magnetosphere.

Examining Coherency Scales, Substructure, and Propagation of Whistler Mode Chorus Elements With Magnetospheric Multiscale (MMS)

AbstractWhistler mode chorus waves are a naturally occurring electromagnetic emission observed in Earth's magnetosphere. Here, for the first time, data from NASA's Magnetospheric Multiscale (MMS) mission were used to analyze chorus waves in detail, including the calculation of chorus wave normal vectors, k. A case study was examined from a period of substorm activity around the time of a conjunction between the MMS constellation and NASA's Van Allen Probes mission on 07 April 2016. Chorus wave activity was simultaneously observed by all six spacecraft over a broad range of L shells (5.5 < L < 8.5), magnetic local time (06:00 < MLT < 09:00), and magnetic latitude (−32° < MLAT < −15°), implying a large chorus active region. Eight chorus elements and their substructure were analyzed in detail with MMS. These chorus elements were all lower band and rising tone emissions, right‐handed and nearly circularly polarized, and propagating away from the magnetic equator when they were observed at MMS (MLAT~−31°). Most of the elements had “hook”‐like signatures on their wave power spectra, characterized by enhanced wave power at flat or falling frequency following the peak, and all the elements exhibited complex and well‐organized substructure observed consistently at all four MMS spacecraft at separations up to 70 km (60 km perpendicular and 38 km parallel to the background magnetic field). The waveforms in field‐aligned coordinates also demonstrated that these waves were all phase coherent, allowing for the direct calculation of k. Error estimates on calculated k revealed that the plane wave approximation was valid for six of the eight elements and most of the subelements. The wave normal vectors were within 20–30° from the direction antiparallel to the background field for all elements and changed from subelement to subelement through at least two of the eight elements. The azimuthal angle of k in the perpendicular plane was oriented earthward and was oblique to that of the Poynting vector, which has implications for the validity of cold plasma theory.

Quasi‐linear diffusion coefficients for highly oblique whistler mode waves

AbstractQuasi‐linear diffusion coefficients are considered for highly oblique whistler mode waves, which exhibit a singular “resonance cone” in cold plasma theory. The refractive index becomes both very large and rapidly varying as a function of wave parameters, making the diffusion coefficients difficult to calculate and to characterize. Since such waves have been repeatedly observed both outside and inside the plasmasphere, this problem has received renewed attention. Here the diffusion equations are analytically treated in the limit of large refractive index μ. It is shown that a common approximation to the refractive index allows the associated “normalization integral” to be evaluated in closed form and that this can be exploited in the numerical evaluation of the exact expression. The overall diffusion coefficient formulas for large μ are then reduced to a very simple form, and the remaining integral and sum over resonances are approximated analytically. These formulas are typically written for a modeled distribution of wave magnetic field intensity, but this may not be appropriate for highly oblique whistlers, which become quasi‐electrostatic. Thus, the analysis is also presented in terms of wave electric field intensity. The final results depend strongly on the maximum μ (or μ∥) used to model the wave distribution, so realistic determination of these limiting values becomes paramount.

An improved sheath impedance model for the Van Allen Probes EFW instrument: Effects of the spin axis antenna

AbstractA technique to quantitatively determine the sheath impedance of the Van Allen Probes Electric Field and Waves (EFW) instrument is presented. This is achieved, for whistler mode waves, through a comparison between the total electric field wave power spectra calculated from magnetic field observations and cold plasma theory and the total electric field wave power measured by the EFW spherical double probes instrument. In a previous study, a simple density‐dependent sheath impedance model was developed in order to account for the differences between the observed and calculated wave electric field. The current study builds on this previous work by investigating the remaining discrepancies, identifying their cause, and developing an improved sheath impedance correction. Analysis reveals that anomalous gains are caused by the spin axis antennas measuring too much electric field at specific densities and frequencies. This is accounted for in an improved sheath impedance model by introducing a density‐dependent function describing the relative effective length of the probe separation, Leff, in addition to the sheath capacitance and resistance values previously calculated. Leff values vary between between 0.5 and 1.2, with values >1 accounting for the anomalous gains and values <1 accounting for the shorting effect at low densities. Applying this improved sheath impedance model results in a significant increase in the agreement level between observed and calculated electric field power spectra and wave powers over the previous model.

Analytical investigation of microwave resonances of a curling probe for low and high-pressure plasma diagnostics

The concept of ‘active plasma resonance spectroscopy’ (APRS) has attracted greater interest in recent years as an established plasma diagnostic technique. The APRS describes a class of related methods utilizing the intrinsic ability of plasma to resonate at or near the electron plasma frequency . The Curling probe (CP) as a novel realization of the APRS idea, is a miniaturized spiral slot embedded flatly in the chamber wall. Consequently, a plasma diagnostic technique with minimum disturbance and without metal contamination can be developed. To measure the plasma parameters the CP is fed with a weak frequency-swept signal from the exterior of the plasma chamber by a network analyzer which also records the response of the plasma versus the frequency. The resonance behavior is strongly dependent on the electron density and the gas pressure. The CP has also the advantage of resonating at a frequency greater than which is dependent on the spiral’s length. The double resonance characteristic gives the CP the ability to be applied in varying plasma regimes. Assuming that the spiralization does not have a considerable effect on the resonances, a ‘straightened’ infinite length CP has recently been investigated (Arshadi and Brinkmann 2016 Plasma Sources Sci. Technol. 25 045014) to obtain the surface wave resonances. This work generalizes the approach and models the CP by a rectangular slot-type resonator located between plasma and quartz. Cold plasma theory and Maxwell’s equations are utilized to compute the electromagnetic fields propagating into the plasma by the diffraction of an incident plane wave at the slot. A mathematical model is employed and both kinds of resonances are derived. The analytical study of this paper shows good agreement with the numerical results of the probe inventors.

Experimental determination of the dispersion relation of magnetosonic waves

AbstractMagnetosonic waves are commonly observed in the vicinity of the terrestrial magnetic equator. It has been proposed that within this region they may interact with radiation belt electrons, accelerating some to high energies. These wave‐particle interactions depend upon the characteristic properties of the wave mode. Hence, determination of the wave properties is a fundamental part of understanding these interaction processes. Using data collected during the Cluster Inner Magnetosphere Campaign, this paper identifies an occurrence of magnetosonic waves, discusses their generation and propagation properties from a theoretical perspective, and utilizes multispacecraft measurements to experimentally determine their dispersion relation. Their experimental dispersion is found to be in accordance with that based on cold plasma theory.

Effect of spatial density variation and O+ concentration on the growth and evolution of electromagnetic ion cyclotron waves

AbstractWe simulate electromagnetic ion cyclotron (EMIC) wave growth and evolution within three regions, the plasmasphere (or plasmaspheric plume), the plasmapause, and the low‐density plasmatrough outside the plasmapause. First, we use a ring current simulation with a plasmasphere model to model the particle populations that give rise to the instability for conditions observed on 9 June 2001. Then, using two different models for the cold ion composition, we do a full‐scale hybrid code simulation in dipole coordinates of the EMIC waves on a meridional plane at magnetic local time = 18 and at 1900 UT within a range of L shell from L=4.9 to 6.7. EMIC waves were observed during 9 June 2001 by Geostationary Operational Environmental Satellite (GOES) spacecraft. While an exact comparison between observed and simulated spectra is not possible here, we do find significant similarities between the two, at least at one location within the region of largest wave growth. We find that the plasmapause is not a preferred region for EMIC wave growth, though waves can grow in that region. The density gradient within the plasmapause does, however, affect the orientation of wavefronts and wave vector both within the plasmapause and in adjacent regions. There is a preference for EMIC waves to be driven in the He+ band (frequencies between the O+ and He+ gyrofrequencies) within the plasmasphere, although they can also grow in the plasmatrough. If present, H+ band waves are more likely to grow in the plasmatrough. This fact, plus L dependence of the frequency and possible time evolution toward lower frequency waves, can be explained by a simple model. Large O+ concentration limits the frequency range of or even totally quenches EMIC waves. This is more likely to occur in the plasmatrough at solar maximum. Such large O+ concentration significantly affects the H+ cutoff frequency and hence the width in frequency of the stop band above the He+ gyrofrequency. EMIC wave surfaces predicted by cold plasma theory are altered by the finite temperature of the ring current H+.

Effects of ion abundances on electromagnetic ion cyclotron wave growth rate in the vicinity of the plasmapause

Electromagnetic ion cyclotron (EMIC) waves in multi-ion species plasmas propagate in branches. Except for the branch corresponding to the heaviest ion species, which has only a resonance at its gyrofrequency, these branches are bounded below by a cutoff frequency and above by a resonant gyrofrequency. The condition for wave growth is determined by the thermal anisotropies of each ion species, j, which sets an upper bound, ωj∗, on the wave frequency below which that ion species contributes positively to the growth rate. It follows that the relative positions of the cutoffs and the critical frequencies ωj∗ play a crucial role in determining whether a particular wave branch will be unstable. The effect of the magnetospheric ion abundances on the growth rate of each branch of the EMIC instability in a model where all the ion species have kappa velocity distributions is investigated by appealing to the above ideas. Using the variation of the cutoff frequencies predicted by cold plasma theory as a guide, optimal ion abundances that maximise the EMIC instability growth rate are sought. When the ring current is comprised predominantly of H+ ions, all branches of the EMIC wave are destabilised, with the proton branch having the maximum growth rate. When the O+ ion abundance in the ring current is increased, a decrease in the growth rate of the proton branch and cyclotron damping of the helium branch are observed. The oxygen branch, on the other hand, experiences an increase in the maximum growth rate with an increase in the O+ ion abundance. When the ring current is comprised predominantly of He+ ions, only the helium and oxygen branches of the EMIC wave are destabilised, with the helium branch having the maximum growth rate.

Finite-beta effects on quasi-linear diffusion coefficients

[1] One of the most important theoretical tools for modeling the sources and losses of Earth's radiation belt electrons is that based on quasi-linear theory. However, one of the key assumptions made in such efforts is that the waves are described by cold plasma theory. In this paper, quasi-linear diffusion coefficients for hot/warm (or finite-beta) plasmas are formulated and computed on the basis of a recently developed empirical theory in which exact numerical hot/warm plasma dispersion relations are modeled by analytical functions. It is found that finite-beta effects associated with whistler mode waves lead to a reduction of the diffusion coefficients for relativistic electrons near 90° pitch angles, which tends to reduce the efficiency of electron acceleration. Finite-beta effects associated with electron scattering by electromagnetic ion cyclotron (EMIC) waves broaden the range of wave-particle interaction in pitch angle space, and hence increase the efficiency of pitch angle diffusion. The rate of pitch angle diffusion of protons by EMIC waves is also generally enhanced by finite-beta effects.

Experimental verification of relativistic finite temperature polarimetry effects at JET

Previously, the effect of high electron temperature on plasma polarimetry has been studied theoretically with specific relevance to its use as a tokamak plasma diagnostic. Approximations have been given in both the non-relativistic and weakly relativistic limits but neither has been verified experimentally. This paper compares over 23 000 measurements of induced ellipticity made by the JET polarimeter during 268 selected pulses with predictions based on both temperature corrections and on purely cold plasma wave theory. The cold plasma theory is shown to significantly underestimate while the non-relativistic warm model is shown to significantly over-estimate the induced ellipticity. Good agreement with the relativistic warm model is shown for plasmas in the range 3 keV < Te < 12 keV, which constitutes the first experimental verification of relativistic effects in plasma polarimetry.

Transversely localized shear Alfven waves in resonance cones

Using fully electromagnetic 2.5-D particle-in-cell (2DPIC) simulations, we study the scattering of shear Alfven waves (SAWs) having long perpendicular wavelengths from multiple localized density cavities. We show that the scattered fields are radiated in conical wave structures, which are the inertial Alfven wave (IAW) cones with apexes at the cavities and half-cone angle approximately determined by the cold plasma theory. The waves within the cones obey the basic dispersion relation of IAWs. The scattered fields have relatively short transverse length scale lengths and are confined within the cones, which are embedded in the large-scale incident SAW. The resulting wave structures and their Fourier spectral behaviour found in the simulations are compared with observations on transversely localized IAWs from Freja, FAST and Polar. We highlight the fine structures in the parallel electric fields within the cones and in the associated parallel and transverse currents. The current closure consisting of the parallel and transverse currents forming loops inside the cone is seen. We demonstrate parallel acceleration of the electrons forming energetic tails and bulk transverse heating of ions in the scattered fields. The scattering mechanism discussed here provides a linear mechanism for converting large-scale SAW into the solitary structures of small-scale IAWs.

Behaviour of a low frequency wave in a FRC plasma

A low frequency torsional Alfven wave (angular frequency ω ∼ 0.3ωci0, ωci0; ion gyro frequency in vacuum) is induced in an extremely high beta (beta = plasma pressure/pressure of confining magnetic field) plasma with the field reversed configuration (FRC) by an external antenna. When the wave is introduced from outside towards the magnetic axis of the FRC plasma, not only the Alfven velocity vA but also the local ion gyro frequency ωci decreases as the density increases and the magnetic field decreases. The spatial structure of the wave is examined in the region between the separatrix and the location near the O-point. Near the former location, the plasma is tenuous and the magnetic field has nearly external value so that ω ∼ 0.3ωci0, and near the latter location, the beta value is large and ω > 0.7ωci. The observed amplitude of the toroidal component of the magnetic disturbance has a broad peak around the radial location of the resonance position obtained from the cold plasma theory, although the cold plasma theory yields a sharper spatial structure. The polarization of the magnetic disturbance component perpendicular to the static magnetic field is left-handed outside the theoretical resonance position and right-handed deep in the plasma. The measured perpendicular wave number kr is small except for the region near the resonance position where the kinetic Alfven wave is expected to have a large wave number. This behaviour will be interpreted as, with the help of the warm plasma theory, the torsional Alfven wave evolves into the kinetic Alfven wave and is then converted to the compressional wave with a smaller wave number when the wave propagates inside the plasma.

Low-noise electromagnetic δf particle-in-cell simulation of electron Bernstein waves

The conversion of the extraordinary (X) mode to an electron Bernstein wave (EBW) is one way to get rf energy into an overdense plasma. Analysis of this is complex, as the EBW is a fully kinetic wave, and so its linear propagation is described by an intractable integro-differential equation. Nonlinear effects cannot be calculated within this rubric at all. Full particle-in-cell (PIC) simulations cannot be used for these analyses, as the noise levels for reasonable simulation parameters are much greater than the typical rf amplitudes. It is shown that the delta-f computations are effective for this analysis. In particular, the accuracy of those computations has been verified by comparison with full PIC, cold plasma theory, and small gyroradius theory. This computational method is then used to analyze mode conversion in different frequency regimes. In particular, reasonable agreement with the theoretical predictions of Ram and Schultz [Phys. Plasmas 7, 4084 (2000)] in the linear regime is found, where 100% X−B mode conversion has been obtained when the driving frequency is less than twice the electron gyrofrequency. The results show that cold-plasma theory well predicts the mode conversion efficiency, as is consistent with the phase-space picture of mode conversion. From this it can be shown that nearly 100% X−B mode conversion cannot be obtained when the frequency is higher than the electron second harmonic cyclotron frequency.