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A general formalism is developed to construct and solve a system of linearized moment equations for parallel and perpendicular closures in high-collisionality plasmas. It is applicable for multiple ion species with arbitrary masses, temperatures, charges, and densities. The convergence of closure coefficients is evaluated by increasing the number of moments from 2 to 32 for scalar, vector, and rank-2 tensor moments. As an example, the complete set of closure coefficients for a deuteriumcarbon plasma over the entire Hall parameter range is presented. The closure coefficients at various temperature ratios show that one-temperature closure coefficients can differ significantly from two

A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field. Expanding a distribution function in general moments a set of ordinary differential equations are obtained. Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the density, temperature, and flow velocity perturbations in terms of radial gradients of equilibrium pressure and temperature. Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity, are also obtained by solving the non-Maxwellian moment equations. The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations. The method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field., Comment: 25 pages, 9 figures

A generalized formula for wave instability is developed for an anisotropic nonuniform plasma with finite flows and temperatures. Six-moment fluid equations are solved to give the analytic expression for wave instability in arbitrarily nonuniform plasmas. The analytic formula explicitly states the dependence of wave instability on the nonuniformities of number density, flow velocity, and anisotropic or isotropic pressure. The accuracy of the formalism is verified by a numerical calculation of implicit dispersion relations in complex Fourier space. The analysis shows that nonuniformity plays a critical role in plasma instability, while the flow velocity and anisotropic pressures determine the growth rate of the instability. The instability diagram and associated instability criterion for anisotropy-driven instability are introduced as applications of the formalism.

In a time-dependent flow, nonlinear harmonics can be excited by coupling between linear waves and flow-induced harmonic waves. Examining the dispersion relations and selection rules for the coupling, we investigate nonlinearly coupled harmonics for waves propagating along the magnetic field line in a magnetized plasma, as well as waves in an unmagnetized plasma. The coupled harmonics in a plasma flow are described by analytic dispersion relations and selection rules. This nonlinear coupling is corroborated by the particle-in-cell simulation. The coupled-harmonics model describes a mechanism for the excitation of nonlinear harmonics from linear waves in a time-dependent flow. The spectral analysis of the dispersion relation provides a useful way to evaluate the spatiotemporal behavior of a plasma flow.

A novel numerical method is employed to compute the integral form of the axi-symmetric Trubnikov-Rosenbluth potentials. Two methods for quadrature in pitch-angle are described and their convergence properties are studied. Careful attention is given to quadrature over a singular Green's function. It is shown that an infinite series representation of the Green's function can be used more efficiently than its closed form involving complete elliptic integrals. Then a collocation method in speed, with its associated quadrature scheme, is laid out and its convergence properties are studied. Using the proposed scheme, accurate low-order moments of the field collision operator are obtained using relatively few velocity space degrees of freedom. The scheme is showcased by solving for the equilibrium, axi-symmetric bootstrap current in tokamaks. A C 0 Gauss-Lobatto-Legendre finite element pitch-angle basis with vertex nodes at the trapped/passing boundary is shown, in the context of the integral methods used, to be much more efficient than the more common Legendre polynomial expansion.

Exact moments of the Boltzmann collision operator are calculated in the irreducible Hermitian moment expansion written in terms of the random-velocity variable of each species. The formulas are presented in closed, algebraic form and can be straightforwardly implemented in computer algebra systems. They are valid for two arbitrary masses, temperatures, and flow velocities, and hence include all other existing results derived for distribution functions expanded with respect to reference states of one temperature and flow velocity. In comparison, the Landau collisional moments are good approximations for large Coulomb logarithm and small relative flow velocity, but they fail to predict the correct behavior of most collisional moments for large relative flow even for weakly coupled plasmas.

We show that the energy, force, and torque between two spherically symmetric multipole density distributions are identical to those between two point multipoles, and apply this point-sphere equivalence to coated spherical dipole magnets. We also show that the potential and field of such a distribution are equivalent to those due to point multipoles located at the center of the distribution. We expand the inverse-distance potential in terms of harmonic (Hermite irreducible) tensors, whose properties enable us to express the potential energy, force, and torque for two arbitrary source distributions in a series of point-multipole interactions. This work generalizes recent work on interactions between uniformly magnetized dipole spheres [B. F. Edwards, D. M. Riffe, J.-Y. Ji, and W. A. Booth, Am. J. Phys. 85, 130 (2017)] to interactions between spherically-symmetric multipole spheres.

A mechanism is presented whereby relativistic electron beams localized in phase space are deterministically scattered by coherent circularly polarized electromagnetic waves without stochastic processes. It is shown via an exact single-particle analysis that the condition for maximal scattering is an off-resonant condition, contrary to previous kinetic analyses that predict maximal diffusion or interaction at exact resonance or its harmonics. The mechanism, verified by single-particle simulations, enables a fast, nonlinear redistribution of the beam particles. A possible application of this mechanism to runaway electron suppression is presented.

We use simple symmetry arguments suitable for undergraduate students to demonstrate that the magnetic energy, forces, and torques between two uniformly magnetized spheres are identical to those between two point magnetic dipoles. These arguments exploit the equivalence of the field outside of a uniformly magnetized sphere with that of a point magnetic dipole, and pertain to spheres of arbitrary sizes, positions, and magnetizations. The point dipole/sphere equivalence for magnetic interactions may be useful in teaching and research, where dipolar approximations for uniformly magnetized spheres can now be considered to be exact. The work was originally motivated by interest in the interactions between collections of small neodymium magnetic spheres used as desk toys.

Ion parallel closures are obtained for arbitrary atomic weights and charge numbers. For arbitrary collisionality, the heat flow and viscosity are expressed as kernel-weighted integrals of the temperature and flow-velocity gradients. Simple, fitted kernel functions are obtained from the 1600 parallel moment solution and the asymptotic behavior in the collisionless limit. The fitted kernel parameters are tabulated for various temperature ratios of ions to electrons. The closures can be used conveniently without solving the kinetic equation or higher order moment equations in closing ion fluid equations., Comment: 2 figures

The magnetized resistivity and electrothermal tensors when substituted into the induction equation lead to electrothermal magnetic field generation, resistive magnetic diffusion, and magnetic field advection due to resistivity gradients, temperature gradients, and currents. The advection terms driven by the temperature gradient and current have cross field components (perpendicular to both the magnetic field and the driving term) that depend on significantly modified versions of Braginskii's transport coefficients [S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205]. The improved fits to Braginskii's coefficients given by Epperlein and Haines [Phys. Fluids 29, 1029 (1986)] and Ji and Held [Phys. Plasmas 13, 042114 (2013)] give physically incorrect results for cross field advection at small Hall parameters (product of cyclotron frequency and collision time). The errors in Epperlein and Haines' fits are particularly severe, giving increasing advection velocities below a Hall parameter of one when they should decrease linearly to zero. Epperlein and Haines' fits can also give erroneous advection terms due to variations in the effective atomic number. The only serious error in Braginskii's fits is an overestimate in advection due to perpendicular resistivity. New fits for the cross field advection terms are obtained from a direct numerical solution of the Fokker–Planck equation and Ji and Held's higher order expansion approach that are continuous functions of the effective atomic number.

Nonlinear coupling of cold and hot waves in a flowing magnetized plasma is analyzed with the Vlasov equation. An analytical solution is obtained for cold waves of a small amplitude (weak flow) and a long wavelength. The distribution function is obtained by integrating the kinetic equation along a perturbed phase-space trajectory for a time-varying plasma flow. The kinetic description presents a generalized dispersion relation that involves resonances depending on cold and hot wave dispersions. Coherent fluid motion leads to radiation peaks in addition to the cyclotron harmonics, where the wavenumber of the cold wave determines the peak frequencies. The peaks appear narrow when the wave propagates perpendicular to the time-averaged flow while they become broad due to the Doppler effect when the wave propagates parallel to the flow. Fully kinetic particle-in-cell simulations corroborate the theoretical predictions. The dispersion relation and resulting wave spectra provide information about plasma parameters and flow properties.

The Maxwell–Boltzmann statistics of the quantum ideal gas is studied through the canonical partition function by exactly counting discrete quantum states without the continuum approximation. Analytic expressions for energy, pressure, entropy, and heat capacity are expressed in terms of Jacobi theta functions and complete elliptic integrals. The results show typical effects of discrete energy levels in the low temperature limit while they reproduce thermodynamics of the classical ideal gas in the high temperature limit.

Linear closures are obtained for arbitrary collisionality for the 3 þ 1 fluid model which includes the evolution of density, flow velocity, and pressure both parallel and perpendicular to a preferred direction, usually a magnetic field. A large set of 6400 moment equations is solved to provide closures that are accurate in the collisional regime and well into the collisionless regime. The closures in the collisionless limit are determined by solving the kinetic equation with a model collision operator. Simple fits for the kernel functions that define the closures are obtained for arbitrary collisionality in wave number space. The results are linearly accurate to within 3% across the entire range of collisionality.

Integral (nonlocal) closures [J.-Y. Ji and E. D. Held, Phys. Plasmas 21, 122116 (2014)] are combined with the momentum balance equation to derive electron parallel transport relations. For a single harmonic fluctuation, the relations take the same form as the classical Spitzer theory (with possible additional terms): The electric current and heat flux densities are connected to the modified electric field and temperature gradient by transport coefficients. In contrast to the classical theory, the dimensionless coefficients depend on the collisionality quantified by a Knudsen number, the ratio of the collision length to the angular wavelength. The key difference comes from the proper treatment of the viscosity and friction terms in the momentum balance equation, accurately reflecting the free streaming and collision terms in the kinetic equation. For an arbitrary fluctuation, the transport relations may be expressed by a Fourier series or transform. For low collisionality, the electric resistivity can be s...

In the moment approach, a parallel heat flux and a viscous stress are derived for arbitrary collisionality, which is becoming increasingly important in emerging concept devices. This derivation improves upon previous derivations by using the full linearized collision operators instead of the pitch-angle scattering operator and also by including the ion-electron collision operator. The parallel viscous stress can be computed by integrating thermodynamic drives along a magnetic field line weighted by kernel functions that are simple linear combinations of exponential functions. The convergence of the ion viscous stress is verified for sinusoidally varying drives with increasing number of moments.

PH3 adsorption on Si(1 1 1)-7 × 7 was studied after various exposures between 0.3 and 60 L at room temperature by means of scanning tunneling microscopy (STM). PH3-, PH2-, H-reacted, and unreacted adatoms can be identified by analyzing empty-state STM images at different sample biases. PHx-reacted rest-atoms can be observed in empty-state STM images if neighboring adatoms are hydrogen terminated. Most of the PH3 adsorbs dissociatively on the surface, generating H- and PH2-adsorbed rest-atom and adatom sites. Dangling-bonds at rest-atom sites are more reactive than adatom sites and the faulted half of the 7 × 7 unit cell is more reactive than the unfaulted half. Center adatoms are overwhelmingly preferred over corner adatoms for PH2 adsorption. The saturation P coverage is ∼0.18 ML. Annealing of PH3-reacted 7 × 7 surfaces at 900 K generates disordered, partially P-covered surfaces, but dosing PH3 at 900 K forms P/Si(1 1 1)- 6 3 surfaces. Si deposition at 510 K leaves disordered clusters on the surface, which cannot be reordered by annealing up to 800 K. However, annealing above 900 K recreates P/Si(1 1 1)- 6 3 surfaces. Surface morphologies formed by sequential rapid thermal annealing are also presented.

The vacuum excitation and squeezing of two harmonic oscillators with delta-kicked interactions for four types of elementary coupling are studied. The exact quantum motion for the Heisenberg operators and the explicit form of squeezing operators are found. The variances are calculated to study the squeezing properties for position, momentum and generalized quadrature operators.

A general quantization scheme of the electromagnetic (EM) fields, including the near-field, of a dielectric material is presented. By expanding the classical EM fields in terms of orthonormal modes, the Lagrangian is expressed as a sum of the Lagrangians of the independent harmonic oscillators. For a dielectric sphere, as an example, two possible modes satisfying the boundary conditions, as well as the orthonormality conditions, are explicitly obtained: spherical modes and Mie modes. Using these modes, the EM fields are quantized and the transformation between two one-photon states having different modes is also discussed. We apply our results to calculate the expectation value of the quantum momentum-density operator for a one-photon state and show that it is equivalent to the classical Poynting vector. Our rigorous results may be useful in the study of the quantum optical properties of the EM near-fields of a dielectric microsphere of sub-wavelength size.

The signal generated by a weak impulse propagates in an oscillatory way and dispersively in a gravitationally compacted granular chain. For the power-law type contact force, we show analytically that the type of dispersion follows power-laws in depth. The power-law for grain displacement signal is given by $h^{-1/4(1-1/p)}$ where $h$ and $p$ denote depth and the exponent of contact force, and the power-law for the grain velocity is $h^{-1/4({1/3}+1/p)}$. Other depth-dependent power-laws for oscillation frequency, wavelength, and period are given by combining above two and the phase velocity power-law $h^{1/2(1-1/p)}$. We verify above power-laws by comparing with the data obtained by numerical simulations., Comment: 12 pages, 3 figures; Changed content

For $N$-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained., Comment: 8 pages, revTeX

We consider classical and quantum electromagnetic fields in a three-dimensional (3D) cavity and in a waveguide with oscillating boundaries of the frequency $\Omega $. The photons created by the parametric resonance are distributed in the wave number space around $\Omega/2 $ along the axis of the oscillation. When classical waves propagate along the waveguide in the one direction, we observe the amplification of the original waves and another wave generation in the opposite direction by the oscillation of side walls. This can be understood as the classical counterpart of the photon production. In the case of two opposite walls oscillating with the same frequency but with a phase difference, the interferences are shown to occur due to the phase difference in the photon numbers and in the intensity of the generated waves.

We study the photon production in a 1D cavity whose left and right walls oscillate with the frequency $\Omega_{L} $ and $\Omega_{R} $, respectively. For $\Omega_{L} \neq \Omega_{R}, $ the number of generated photons by the parametric resonance is the sum of the photon numbers produced when the left and the right wall oscillates separately. But for $\Omega_{L} = \Omega_{R} $, the interference term proportional to $\cos \phi $ is found additionally, where $\phi $ is the phase difference between two oscillations of the walls., Comment: 7 pages, RevTeX, no figures, a sign error corrected

We present the Heisenberg-picture approach to the quantum evolution of the scalar fields in an expanding FRW universe which incorporates relatively simply the initial quantum conditions such as the vacuum state, the thermal equilibrium state, and the coherent state. We calculate the Wightman function, two-point function, and correlation function of a massive scalar field. We find the quantum evolution of fluctuations of a self-interacting field perturbatively and discuss the renormalization of field equations., Comment: 15 pages, RevTeX, no figures

We study a cyclic initial state (CIS) and corresponding (nonadiabatic) Berry phase for a τ-periodic harmonic oscillator which alternates between two frequencies. The existence of CISs depends on two frequencies and their duration intervals. We find a new type of CIS which is nτ -periodic ( n = 1,2,…).

Exact Schr\"odinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\ensuremath{\tau}$-periodic Hamiltonian. There may exist $N\ensuremath{\tau}$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.

Electron parallel closures for the ion charge number $Z=1$ [J.-Y. Ji and E. D. Held, Phys. Plasmas \textbf{21}, 122116 (2014)] are extended for $1\le Z\le10$. Parameters are computed for various $Z$ with the same form of the $Z=1$ kernels adopted. The parameters are smoothly varying in $Z$ and hence can be used to interpolate parameters and closures for noninteger, effective ion charge numbers.

Radial heat transport induced by magnetic field line fluctuations is obtained from the integral parallel heat flow closure for arbitrary collisionality. The parallel heat flow and its radial component are computed for a single harmonic sinusoidal field line perturbation. In the collisional and collisionless limits, averaging the heat flow over an unperturbed surface yields Rechester–Rosenbluth like formulae with quantitative factors. The single harmonic result is generalized to multiple harmonics given a spectrum of small magnetic perturbations. In the collisionless limit, the heat and particle transport relations are also derived.

Electron parallel closures for heat flow, viscosity, and friction force are expressed as kernel-weighted integrals of thermodynamic drives, the temperature gradient, relative electron-ion flow velocity, and flow-velocity gradient. Simple, fitted kernel functions are obtained for arbitrary collisionality from the 6400 moment solution and the asymptotic behavior in the collisionless limit. The fitted kernels circumvent having to solve higher order moment equations in order to close the electron fluid equations. For this reason, the electron parallel closures provide a useful and general tool for theoretical and computational models of astrophysical and laboratory plasmas.

Phosphorous δ-doping layers were fabricated in silicon by PH3 deposition at room temperature, followed by low-temperature Si epitaxy. Scanning tunneling microscope images indicate large H coverage, and regions of c(2×2) structure. Hall data imply full carrier activation with mobility

The kinetic equation for a single-component uniform plasma in a magnetic field is analytically solved by the moment method. The linear system of ordinary differential equations for the moments is decomposed into subsystems of lower dimensions by a geometric method. The eigensystem of each subsystem shows that parallel moments decay monotonically, but perpendicular lth harmonic moments decay while oscillating with the l,l-2,… ,-th harmonics of gyrofrequency. A generalization to a multicomponent plasma is discussed.

According to analytical calculations of the ion collision operator, the ion-electron collision terms could be larger than the ion-ion collision terms. In the previous work [J.-Y. Ji and E. D. Held, Phys. Plasmas 20, 042114 (2013)], the ion-electron collision effects are diminished by the ion temperature change terms introduced from unlikely assumptions. In this work, the high-collisionality closures for ions are calculated without the temperature change terms. The ion-electron collision terms significantly modify existing closure coefficients.

Verification of continuum solutions to the electron and ion drift kinetic equations (DKEs) in NIMROD [C. R. Sovinec et al., J. Comp. Phys. 195, 355 (2004)] is demonstrated through comparison with several neoclassical transport codes, most notably NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 54, 015015 (2012)]. The DKE solutions use NIMROD's spatial representation, 2D finite-elements in the poloidal plane and a 1D Fourier expansion in toroidal angle. For 2D velocity space, a novel 1D expansion in finite elements is applied for the pitch angle dependence and a collocation grid is used for the normalized speed coordinate. The full, linearized Coulomb collision operator is kept and shown to be important for obtaining quantitative results. Bootstrap currents, parallel ion flows, and radial particle and heat fluxes show quantitative agreement between NIMROD and NEO for a variety of tokamak equilibria. In addition, velocity space distribution function contours for ions and electrons show nearly identical detailed structure and agree quantitatively. A Θ-centered, implicit time discretization and a block-preconditioned, iterative linear algebra solver provide efficient electron and ion DKE solutions that ultimately will be used to obtain closures for NIMROD's evolving fluid model.

Electron parallel closures for heat flow, viscosity, and friction force are expressed as kernel-weighted integrals of thermodynamic drives, the temperature gradient, relative electron-ion flow velocity, and flow-velocity gradient. Simple, fitted kernel functions are obtained for arbitrary collisionality from the 6400 moment solution and the asymptotic behavior in the collisionless limit. The fitted kernels circumvent having to solve higher order moment equations in order to close the electron fluid equations. For this reason, the electron parallel closures provide a useful and general tool for theoretical and computational models of astrophysical and laboratory plasmas.