Boltzmann Relation Research Articles

Improved erosion estimates for the STEP divertor

Net tungsten erosion and deposition profiles are simulated for the outer and inner lower divertor of STEP, a spherical prototype fusion reactor targeting ∼ 2040. In this contribution, previous modelling work [1] has been extended by studying the influence of various input parameters. Main aim is to analyse the influence of several modelling improvements on the erosion/deposition characteristics at the divertor targets. This comprises the consideration of an electron density decay according to the Boltzmann relation within the sheath region as well as a certain tungsten influx to the divertor originating from main wall erosion. Also, for the inner divertor improved background plasma parameters are applied taking into account a gradient along the flux surfaces compared to formerly constant electron temperature and density and ion temperature along the flux surfaces. ERO simulations have been performed for one selected plasma case with Ar seeding, both for the outer (with peak target Te ∼ 25 eV) and inner lower divertor (with peak target Te ∼ 3 eV).The simulated tungsten erosion and redeposition profiles do not significantly change for the cases studied if the Boltzmann-related decay of electron density within the sheath is considered. The assumption of tungsten within the background plasma can notably alter the overall erosion/redeposition behaviour at a background tungsten concentration of 1/10 with respect to the Ar divertor concentration. The inner divertor target plate shows net deposition everywhere, while at the outer one the net erosion zone becomes smaller and the maximum net erosion peak reduces by a factor of two. Lastly, the improved background plasma for the inner divertor has moderate effects, for instance the overall tungsten redeposition at the target plate increases from 88.7% to 94.4%.

Addendum: Thermodynamic quasi-equilibria in high power magnetron discharges: a generalized Poisson–Boltzmann relation (2023 Plasma Sources Sci. Technol. 32 055012)

A recent publication by Köhn et al (2023 Plasma Sources Sci. Technol. 32 055012) studied the quasi-equilibria of high power magnetron discharges through thermodynamic principles. A generalized, magnetic-field aware Poisson–Boltzmann relation for the electric potential and the electron density was established using a non-standard (multi-objective) variational principle. This addendum demonstrates that, assuming slow or quasistatic evolution, the same result can be realized via a standard (single-objective) variational principle, thereby streamlining the theoretical framework while preserving the robustness of the finding.

One-dimensional, multi-fluid model of the plasma-wall transition. II. Negative ions

The plasma-wall transition is investigated by a one-dimensional steady-state multifluid model, which was presented in detail in Part I [T. Gyergyek et al., AIP Adv. 14, 045201 (2024)]. In this work, the plasma-wall transition is analyzed for the case where the plasma consists of singly charged positive ions, electrons, and singly charged negative ions. When the temperature and initial density of the negative ions are varied, a transition between two types of solutions of the model is observed. We call them the low and high solution, with respect to the absolute value of the potential drop. When the density and temperature of the negative ions are above a critical value, the low solution is observed. As the mass of the positive ions increases, these critical values also increase, but only until the ion mass is below about 1000 electron masses. With larger ion masses, the critical density of the negative ions and the temperature no longer change. In the low solution, the potential drop in front of the sheath is determined by the negative ions and is smaller in absolute terms than in the case of the high solution, where the potential drop in front of the sheath is determined by the electrons. If the problem is analyzed on the pre-sheath scale, the transition between the low and high solution is very sharp. However, when the neutrality condition is replaced by the Poisson equation, this transition becomes blurred and the solutions of the model equations exhibit oscillations. The role of the smallness parameter is highlighted. It is shown how the initial electric field is determined. Deviation of the negative ion density profile from the Boltzmann relation is discussed.

Formation of singularities in plasma ion dynamics

We study the formation of singularity for the Euler–Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for C 1 blow-up of the Euler–Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the pressureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave C 1 class in a finite time when the gradients of the Riemann functions are initially large.

Adiabatic collapse of non-homogeneous self-gravitating gas cloud

In this letter, we find the critical mass of a self-gravitating, spherically symmetric gas cloud, above which the fluid, within the bubble, collapses. Our analysis departs from a non-homogeneous equilibrium density, satisfying the Boltzmann relation. A time scale is defined in terms of the adiabatic index of the gas. Subsequently, a sinusoidal perturbation around equilibrium is regarded, thereby leading to a dispersion relation of frequency with wavelength, which does not depend on geometrical curvature effects. Such a formulation clearly justifies that the collapse occurs much faster than predicted by the well-known Jeans approach. The equilibrium profiles of the density, gravitational field, and potential are obtained as functions of the spherical radius coordinate at marginal instability. Since our theory captures the essential physics of gravitational collapse, it can be used as the starting point for several advancements in galactic dynamics.

A Boltzmann Electron Drift Diffusion Model for Atmospheric Pressure Non-Thermal Plasma Simulations

We introduce a fluid computational model for the numerical simulation of atmospheric pressure dielectric barrier discharge plasmas. Ion and neutral species are treated with an explicit drift diffusion approach. The Boltzmann relation is used to compute the spatial distribution of electrons as a function of the electrostatic potential and the ionic charge density. This technique, widely used to speed up particle and fluid models for low-pressure conditions, poses several numerical challenges for high-pressure conditions and large electric field values typical of applications involving atmospheric-pressure plasmas. We develop a robust algorithm to solve the non-linear electrostatic Poisson problem arising from the Boltzmann electron approach under AC electric fields based on a charge-conserving iterative computation of the reference electric potential and electron density. We simulate a volumetric reactor in dry air, comparing the results yielded by the proposed method with those obtained when the drift diffusion approach is used for all charged species, including electrons. We show that the proposed methodology retains most of the physical information provided by the reference modeling approach while granting a substantial advantage in terms of computation time.

A finite element procedure for time-dependent radio-frequency sheaths based on a two-dimensional microscale fluid model

In this paper, we present a finite element scheme for the analysis of time-dependent radio-frequency (RF) sheath behavior in a plasma-filled domain bounded by periodically curved plates. This numerical scheme is based on a two-dimensional (2D) microscale model in which the time-dependent cold-ion fluid equations, the Maxwell–Boltzmann relation for electrons, and Poisson's equation are solved subject to periodic boundary conditions (BCs) and conducting-wall BCs. The continuity of the total current is employed in localized regions to provide a constraint on the reference sheath potential. The primary purpose of this work is to understand 2D dynamic sheath behavior in order to improve predictive capabilities for RF wave interactions in magnetic fusion experiments. In particular, this work treats cases where the local radius of curvature of the wall surface is comparable to the non-neutral sheath width. Using the developed numerical code, the dependences of the ion, electron, and displacement admittances on the wall bump height, ion magnetization, ion mobility, and the magnetic field angle are investigated. It is shown that the ion and electron admittances are nearly unchanged over wide ranges of the bump height and ion magnetization. In addition, the 2D sheath effects are assessed through the spatial distributions of various quantities such as the electron density, electrostatic potential, ion velocity, and surface wall current.

Spectroscopic investigations of detachment on the MAST Upgrade Super-X divertor

We present the first analysis of the atomic and molecular processes at play during detachment in the MAST-U Super-X divertor using divertor spectroscopy data. Our analysis indicates detachment in the MAST-U Super-X divertor can be separated into four sequential phases: first, the ionisation region detaches from the target at detachment onset leaving a region of increased molecular densities downstream. The plasma interacts with these molecules, resulting in molecular ions ( and/or ) that further react with the plasma leading to molecular activated recombination and dissociation (MAR and MAD), which results in excited atoms and significant Balmer line emission. Second, the MAR region detaches from the target leaving a sub-eV temperature region downstream. Third, an onset of strong emission from electron–ion recombination (EIR) ensues. Finally, the electron density decays near the target, resulting in the bulk of the electron density moving upstream. The analysis in this paper indicates that plasma–molecule interactions have a larger impact than previously reported and play a critical role in the intensity and interpretation of hydrogen atomic line emission characteristics on MAST-U. Furthermore, we find that the Fulcher band emission profile in the divertor can be used as a proxy for the ionisation region and may also be employed as a plasma temperature diagnostic for improving the separation of hydrogenic emission arising from electron-impact excitation and that from plasma–molecular interactions. We provide evidences for the presence of low electron temperatures (≪0.5 eV) during detachment phases III–IV based on quantitative spectroscopy analysis, a Boltzmann relation of the high-n Balmer line transitions together with an analysis of the brightness of high-n Balmer lines.

Simulating ion flux to 3D parts in vacuum arc coating: Investigating effect of part size using novel particle-based model

This work presents a novel particle-based computational model capable of predicting coating distribution and composition on real parts in ion-based physical vapor deposition (PVD) processes. The model treats ions as particles (using test particle Monte Carlo method) and electrons as a fluid (using Boltzmann's relation). This combination makes it possible to simulate a realistic plasma sheath around a coated part and locally enhanced electric field around the part's edges. This simulation tool is instrumental especially in predicting the so-called “antenna-effect” - i.e. changes in coating thickness and composition near sharp edges of coated parts. It can also be used for quantifying the effect of coater loading or fixture design, as illustrated by simulating the effect of a top-plate on the coating distribution. The model is validated by two experimental data sets - a milling tool and a drill bit, both coated by hard nitride coatings in a vacuum arc process.

Boltzmann relation reliability in optical temperature sensing based on upconversion studies of Er3+ /Yb3+ co-doped PZT ceramics.

The fluorescence intensity ratio (FIR) of two thermally coupled levels with temperature follows the Boltzmann equation and shows an exponential nature to the temperature that is purely dependent on the energy difference between the levels. Despite the identical energy difference between the thermally coupled levels, researchers have observed varying sensitivities for various samples. In this article, the FIR and sensitivities were calculated using the Boltzmann equation by changing various parameters such as energy difference (ΔE) and the value of the constant C. The results were compared with various reports for Er3+ /Yb3+ ions. After analysis, a new polynomial fit equation was used to determine the temperature sensitivities for the Er3+ /Yb3+ co-doped PbZrTiO3 phosphor in lieu of the conventional Boltzmann equation. The polynomial fit equation eliminated the dependency of the sensitivity on the inverse of the FIR factor and a flat sensitivity curve was obtained with temperature.

Isentropic plasma sheath model for improved fidelity

A model is developed for a collisionless plasma sheath assuming isentropic electrons in contrast to the standard isothermal electron assumption. This approach is enabled by the approximation of a Maxwellian electron velocity distribution function across the sheath, which is justified by near wall measurements. The conservation of entropy leads to a modified Boltzmann relation and a modified Bohm criterion. The predicted floating sheath potential is in excellent agreement with experimental data. Takamura's model for a space-charge limited plasma sheath near an emissive surface is also modified for isentropic electrons and with that modification agrees well with numerical results from a full fluid plasma model.

Plasma structure and electron cross-field transport induced by azimuthal manipulation of the radial magnetic field in a Hall thruster E × B discharge

Plasma structure and electron cross field in the z–θ plane of a Hall thruster E×B plasma under an azimuthally inhomogeneous magnetic field are studied by both experimental and numerical approaches. The work is intended to identify a primary role of electron dynamics on the structure formation by manipulating only the strongly magnetized electrons. The plasma potential distribution shows an axial–azimuthal variation; a low magnetic field region results in spatial potential saturation further downstream. The plasma density structure shows a 1D-like azimuthal variation with less axial deformation. A dense region is observed near the location of ∇B>0, where electrons are expected to undergo the ∇B and curvature drift toward the anode where neutrals are introduced. The potential structure is in close correlation to the Hall parameter distribution, indicating that electron dynamics plays a primary role in plasma structure formation, and via multiple consecutive stepwise physical steps, it eventually affects the density structure formation. In the z–θ space, the cross-field transport by E×B and diamagnetic drifts dominantly determines the electron flow and increases the overall axial electron mobility due to the azimuthal inhomogeneity. It is shown that most of the current is carried by the largest structure, but as the macroscopic structure fades out downstream, small structures grow and share the current. By considering the conservation laws, we show that a relation between azimuthal distributions of physical properties is formed to conserve the axial flux by a balance of specific forces, a balance between the resistive force and the magnetic force in the near-anode region and a balance between the electric/pressure force and the magnetic force in the acceleration and plume region, which differs from the Boltzmann relation satisfied in the radial dimension. Based on this principle, with a simplified test case having a uniform plasma density distribution, we show an analytic relation between azimuthal distributions of the magnetic field and the plasma potential and confirm the relation by a 2D hybrid simulation.

Fundamental stellar parameters of benchmark stars from CHARA interferometry

Context. Stellar models applied to large stellar surveys of the Milky Way need to be properly tested against a sample of stars with highly reliable fundamental stellar parameters. We have established a programme aiming to deliver such a sample of stars. Aims. Here we present new fundamental stellar parameters of nine dwarf stars that will be used as benchmark stars for large stellar surveys. One of these stars is the solar-twin 18 Sco, which is also one of the Gaia-ESO benchmarks. The goal is to reach a precision of 1% in effective temperature (Teff). This precision is important for accurate determinations of the full set of fundamental parameters and abundances of stars observed by the surveys. Methods. We observed HD 131156 (ξ Boo), HD 146233 (18 Sco), HD 152391, HD 173701, HD 185395 (θ Cyg), HD 186408 (16 Cyg A), HD 186427 (16 Cyg B), HD 190360, and HD 207978 (15 Peg) using the high angular resolution optical interferometric instrument PAVO at the CHARA Array. We derived limb-darkening corrections from 3D model atmospheres and determined Teff directly from the Stefan–Boltzmann relation, with an iterative procedure to interpolate over tables of bolometric corrections. Surface gravities were estimated from comparisons to Dartmouth stellar evolution model tracks. We collected spectroscopic observations from the ELODIE spectrograph and estimated metallicities ([Fe/H]) from a 1D non-local thermodynamic equilibrium (NLTE) abundance analysis of unblended lines of neutral and singly ionised iron. Results. For eight of the nine stars we measure the Teff ⪅ 1%, and for one star better than 2%. We determined the median uncertainties in log g and [Fe/H] as 0.015 dex and 0.05 dex, respectively. Conclusions. This study presents updated fundamental stellar parameters of nine dwarf stars that can be used as a new set of benchmarks. All the fundamental stellar parameters were based on consistently combining interferometric observations, 3D limb-darkening modelling, and spectroscopic analysis. The next paper in this series will extend our sample to giants in the metal-rich range.

Generalizing Boltzmann Configurational Entropy to Surfaces, Point Patterns and Landscape Mosaics.

Several methods have been recently proposed to calculate configurational entropy, based on Boltzmann entropy. Some of these methods appear to be fully thermodynamically consistent in their application to landscape patch mosaics, but none have been shown to be fully generalizable to all kinds of landscape patterns, such as point patterns, surfaces, and patch mosaics. The goal of this paper is to evaluate if the direct application of the Boltzmann relation is fully generalizable to surfaces, point patterns, and landscape mosaics. I simulated surfaces and point patterns with a fractal neutral model to control their degree of aggregation. I used spatial permutation analysis to produce distributions of microstates and fit functions to predict the distributions of microstates and the shape of the entropy function. The results confirmed that the direct application of the Boltzmann relation is generalizable across surfaces, point patterns, and landscape mosaics, providing a useful general approach to calculating landscape entropy.

Thermodynamic Consistency of the Cushman Method of Computing the Configurational Entropy of a Landscape Lattice.

There has been a recent surge of interest in theory and methods for calculating the entropy of landscape patterns, but relatively little is known about the thermodynamic consistency of these approaches. I posit that for any of these methods to be fully thermodynamically consistent, they must meet three conditions. First, the computed entropies must lie along the theoretical distribution of entropies as a function of total edge length, which Cushman showed was a parabolic function following from the fact that there is a normal distribution of permuted edge lengths, the entropy is the logarithm of the number of microstates in a macrostate, and the logarithm of a normal distribution is a parabolic function. Second, the entropy must increase over time through the period of the random mixing simulation, following the expectation that entropy increases in a closed system. Third, at full mixing, the entropy will fluctuate randomly around the maximum theoretical value, associated with a perfectly random arrangement of the lattice. I evaluated these criteria in a test condition involving a binary, two-class landscape using the Cushman method of directly applying the Boltzmann relation (s = klogW) to permuted landscape configurations and measuring the distribution of total edge length. The results show that the Cushman method directly applying the classical Boltzmann relation is fully consistent with these criteria and therefore fully thermodynamically consistent. I suggest that this method, which is a direct application of the classical and iconic formulation of Boltzmann, has advantages given its direct interpretability, theoretical elegance, and thermodynamic consistency.

Predictive estimation of vacuum ultraviolet emission intensity in a low-pressure inductively coupled hydrogen plasma based on the branching ratio technique

Vacuum ultraviolet (VUV) emission has recently attracted attention in low pressure processing plasmas because of the possibility of high-energy photon damages on the substrates. To quantify the VUV induced damages during the plasma processes, it is need to use of a VUV spectrometer equipped with vacuum systems not readily available in industries. In this work, therefore, we report a simple method to estimate the VUV emission intensity of hydrogen plasmas utilizing a conventional visible spectrometer widely used in plasma processes. From the measurement of hydrogen emission spectra in the visible wavelength region, the VUV emission line (Lyman-β) was calculated using the branching ratio technique and enabled the estimation of Lyman-α emission intensity based on the Boltzmann relation with given plasma parameters. In addition, it was found that the method could also predict the VUV emission intensity for high density hydrogen plasma cases by considering the self-absorption effect by hydrogen atoms. • VUV emission intensity measurement and predictive estimation. • Calculation of VUV Lyman-α (121 nm) emission based on Branching ratio technique. • Correlation between predicted and measured VUV Lyman-α (121 nm) emission intensity. • Self-absorption effect of hydrogen atoms on VUV emission intensity estimation.

A Simple Gas-Kinetic Model for Dilute and Weakly Charged Plasma Micro-Jet Flows

This paper presents a simple model for slightly charged gas expanding into a vacuum from a planar exit. The number density, bulk velocity, temperature, and potential at the exit are given. The electric field force is assumed weaker than the convection term and is neglected in the analysis. As such, the quasi-neutral condition is naturally adopted and the potential field is computed with the Boltzmann relation. At far field, the exit degenerates as a point source, and simplified analytical formulas for flow and electric fields are obtained. The results are generic and offer insights on many existing models in the literature. They can be used to quickly approximate the flowfield and potential distributions without numerical simulations. They can also be used to initialize a simulation. Based on these results, more advanced models may be further developed.

Three-fluid model of the plasma–sheath region for a planar probe immersed in an active oxygen discharge. Validity of the Boltzmann relation

This paper studies the effect of the generation-loss processes in the plasma–sheath region close to a probe immersed in an oxygen discharge considering a three-fluid model for the charged species. A significant deviation from the Boltzmann relationship is obtained for the negative ion density when different values of the electron temperature, the negative ion temperature, the negative ion concentration, and the collision mean free path are considered.

Generalized Boltzmann relations in semiconductors including band tails

Boltzmann relations are widely used in semiconductor physics to express the charge-carrier densities as a function of the Fermi level and temperature. However, these simple exponential relations only apply to sharp band edges of the conduction and valence bands. In this article, we present a generalization of the Boltzmann relations accounting for exponential band tails. To this end, the required Fermi–Dirac integral is first recast as a Gauss hypergeometric function followed by a suitable transformation of that special function and a zeroth-order series expansion using the hypergeometric series. This results in simple relations for the electron and hole densities that each involve two exponentials. One exponential depends on the temperature and the other one on the band-tail parameter. The proposed relations tend to the Boltzmann relations if the band-tail parameters tend to zero. This work is timely for the modeling of semiconductor devices at cryogenic temperatures for large-scale quantum computing.

Gravitational instability with dust charge gradient and ion drag forces in unmagnetized dusty plasma

The influence of dust charge gradient force and ion drag force on the fragmentation of unmagnetized, self-gravitating dust cloud has been studied. The thermal electrons satisfy the Boltzmann relation, while inertialess ions are affected by the ion-neutral collisions. The dynamics of dusty fluid are modified by ion drag, charge gradient, and gravitational forces. The onset criterion of pinching instability and gravitational instability is derived. The pinching instability depends upon the critical ion drag coefficient and dust charge variation parameter. In the laboratory complex plasma, with finite dust charge variations, the ion drag coefficient larger than the critical value causes pinching instability. This results in the fragmentation of the dusty cloud, which is affected due to the dust charge variations. The ion drag coefficient has destabilizing, while the dust charge variation parameter has stabilizing influence on the growth rate of the linear gravitational instability. The results have been discussed to understand the dust cloud collapse in the astrophysical system.