D. B. Batchelor

Power deposition in high‐density inductively coupled plasma tools for semiconductor processing

A two‐dimensional, computationally efficient numerical model is developed to study power deposition in high‐density inductively coupled plasma sources. Calculations include both inductive coupling, caused by plasma response to external coil currents, and capacitive coupling, caused by plasma response to external voltages on the coils and wafer. Variation in current along the induction coil is determined self‐consistently from the integral constraint of charge conservation. Sheath phenomena are incorporated through previously published analytic models. The system behavior is analogous in some respects to that of a transmission line. Comparison with measurement suggests that this model provides a good description of self‐consistent coil response when the electric field exhibits less than a quarter wavelength per coil turn.

Physics design of the national spherical torus experiment

The mission of the National Spherical Torus Experiment (NSTX) is to prove the principles of spherical torus physics by producing high-beta toroidal plasmas that are non-inductively sustained, and whose current profiles are in steady-state. NSTX will be one of the first ultra low a[P(input) up to 11 MW] in order to produce high-beta toroidal (25 to 40%), low collisionality, high bootstrap fraction (less than or equal to 70%) discharges. Both radio-frequency (RF) and neutral-beam (NB) heating and current drive will be employed. Built into NSTX is sufficient configurational flexibility to study a range of operating space and the resulting dependences of the confinement, micro- and MHD stability, and particle and power handling properties. NSTX research will be carried out by a nationally based science team.

All-orders spectral calculation of radio-frequency heating in two-dimensional toroidal plasmas

Spectral calculations of radio-frequency (rf) heating in tokamak plasmas are extended to two dimensions (2-D) by taking advantage of new computational tools for distributed memory, parallel computers. The integral form of the wave equation is solved in 2-D without any assumption regarding the smallness of the ion Larmor radius (ρ) relative to the perpendicular wavelength (λ⊥). Results are therefore applicable to all orders in k⊥ρ, where k⊥=2π/λ⊥. Previous calculations of rf wave propagation and heating in 2-D magnetized plasmas have relied on finite Larmor radius expansions (k⊥ρ≪1) and are thus limited to relatively long wavelengths. In this paper, no such assumption is made, and we consider short wavelength processes such as the excitation and absorption of ion Bernstein waves in 2-D with k⊥ρ>1. Results show that this phenomenon is far more complex than simple one-dimensional plasma models would suggest. Other applications include fully self-consistent 2-D solutions for high-harmonic fast-wave heating in...

Conversion between cold plasma modes in an inhomogeneous plasma

For a particular angle of incidence, it is possible for an ordinary mode electromagnetic wave, incident on an inhomogeneous plasma slab, to be totally converted into an extraordinary mode. A previous analysis of this process has been extended to permit an arbitrary angle of incidence and to include magnetic field gradients. Although this process has recently been proposed as a method by which toroidal plasmas might be electron cyclotron heated from the outside (low magnetic field region), it was found that for plasma parameters characteristic of modern tokamaks, significant mode conversion occurs only for waves incident within a very narrow cone (δϑ∼1°) about the critical angle. Also addressed is the question of the location at which ordinary/extraordinary mode conversion occurs.

Advances in full-wave modeling of radio frequency heated, multidimensional plasmas

Previous full-wave models for rf heating in multidimensional plasmas have relied on either cold-plasma or finite Larmor radius approximations. These models assume that the perpendicular wavelength of the rf field is much larger than the ion Larmor radius, and they are therefore limited to relatively long wavelengths and low cyclotron harmonics. Recently, alternate full-wave models have been developed that eliminate these restrictions. These “all orders spectral algorithms” take advantage of new computational techniques for massively parallel computers to solve the integral form of the wave equation in multiple dimensions without any restriction on wavelength relative to orbit size, and with no limit on the number of cyclotron harmonics retained. These new models give high-resolution, two-dimensional solutions for mode conversion and high harmonic fast wave heating in tokamak geometry. In addition, they have been extended to give fully three-dimensional solutions of the integral wave equation for minority ...

Exact and approximate solutions to the finite temperature wave equation in a one-dimensional perpendicularly stratified plasma

The sixth order wave equation which results from a finite temperature expansion of the Vlasov equation is solved globally in the ion cyclotron range of frequencies. A perpendicularly stratified, onedimensional slab plasma is assumed. The diamagnetic drift and the associated anisotropy are included in the unperturbed distribution function to ensure a self-adjoint system. All x-dependence in the plasma pressure and magnetic field is retained along with the electric field parallel to . Thus, Landau damping of the ion Bernstein wave is included self-consistently. Because of the global nature of the solution, the evanescent short wavelength Bernstein waves do not grow exponentially as in shooting methods. Strong variations occur in the absorption and in the structure of the wave fields as resonance topology is varied. Solutions to the complete sixth order differential equation are compared to those from an approximate second order equation based on local dispersion theory.

Propagation and absorption of electromagnetic waves in fully relativistic plasmas

The propagation and absorption of electromagnetic waves in a relativistic Maxwellian plasma are investigated by solving the uniform plasma dispersion relation. Both the Hermitian and the anti‐Hermitian parts of the plasma conductivity tensor σ are calculated relativistically. The Bessel functions occurring in σ are not expanded, and many cyclotron harmonic terms are included at high temperatures. The dispersion relation is solved numerically for perpendicular propagation, k∥ =0, where the relativistic effects are maximum and are not masked by Doppler broadening, which has been more thoroughly investigated. It is found that relativistic broadening has a substantial effect on wave dispersion, shifting the extraordinary mode right‐hand cutoff and the upper‐hybrid resonance to higher magnetic field with increasing temperature. Above a critical temperature, the cutoff disappears entirely. There is a broad range of temperatures, 20 keV≤Te ≤500 keV, for which the wavenumber k⊥ differs significantly from both the...

Self-consistent full-wave and Fokker-Planck calculations for ion cyclotron heating in non-Maxwellian plasmas

Magnetically confined plasmas can contain significant concentrations of nonthermal plasma particles arising from fusion reactions, neutral beam injection, and wave-driven diffusion in velocity space. Initial studies in one-dimensional and experimental results show that nonthermal energetic ions can significantly affect wave propagation and heating in the ion cyclotron range of frequencies. In addition, these ions can absorb power at high harmonics of the cyclotron frequency where conventional two-dimensional global-wave models are not valid. In this work, the all-orders global-wave solver AORSA [E. F. Jaeger et al., Phys. Rev. Lett. 90, 195001 (2003)] is generalized to treat non-Maxwellian velocity distributions. Quasilinear diffusion coefficients are derived directly from the wave fields and used to calculate energetic ion velocity distributions with the CQL3D Fokker-Planck code [R. W. Harvey and M. G. McCoy, Proceedings of the IAEA Technical Committee Meeting on Simulation and Modeling of Thermonuclear ...

ICRF wave propagation and absorption in tokamak and mirror magnetic fields — A full-wave calculation

Global solutions for the ion cyclotron resonant frequency (ICRF) wave fields in a straight tokamak with rotational transform and a poloidally symmetric mirror are calculated in the cold plasma limit. The component of the wave electric field parallel to B is assumed zero. Symmetry in each problem allows Fourier decomposition in one ignorable coordinate, and the remaining set of two coupled, two-dimensional partial differential equations is solved by finite differencing. Energy absorption and antenna impedance are calculated using a simple collisional absorption model. When large gradients in ‖B‖ along B are present in either geometry, ICRF heating at the fundamental ion cyclotron resonance is observed. For the mirror, such gradients are always present. But for the tokamak, the rotational transform must be large enough that B·▿B ⪆ 0(1). For smaller transforms more typical of real tokamaks, only heating at the two-ion hybrid resonance is observed. This suggests that direct resonant absorption at the fundamental ion cyclotron resonance may be possible in stellarators where B·▿B ≈ 0(1) naturally.

Full-wave calculation of sheared poloidal flow driven by high-harmonic ion Bernstein waves in tokamak plasmas

A full-wave, one-dimensional spectral model is developed to study sheared poloidal flow driven by high-harmonic ion Bernstein waves (IBWs) in tokamak plasmas. The local plasma conductivity is corrected to lowest order in ρ/L where ρ is the ion Larmor radius and L is the equilibrium scale length. This correction takes into account gradients in equilibrium quantities and is necessary for conservation of energy. It is equivalent to the “odd-order derivative” terms in finite difference models. No assumption is made regarding the smallness of the ion Larmor radius relative to wavelength, and results are applicable to all orders in k⊥ρ where k⊥ is the perpendicular wave number. Previous numerical results for flow drive have relied on expansions in k⊥ρ, and are thus limited to cyclotron harmonics of two and below. In this article, we consider higher-harmonic cases corresponding to recent IBW flow drive experiments on the Tokamak Fusion Test Reactor [B. P. LeBlanc, R. E. Bell, S. Bernabei et al., Phys. Rev. Lett....