arXiv subjects

P. -A. Gourdain

Publications and source records attributed to P. -A. Gourdain.

16 recordsLinked to original sources

arXiv2025

The Klein-Gordon equation with relativistic mass: a relativistic Schrödinger equation

The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.

quant-ph↗

arXiv2023

Faraday rotation signal amplification using high-power lasers

Magnetic fields play an important role in plasma dynamics, yet it is a quantity difficult to measure accurately with physical probes, whose presence disturbs the very field they measure. The Faraday rotation of a polarized beam of light provides a mechanism to measure the magnetic field without disturbing the dynamics, and has been used with great success in astrophysics and high energy density plasma science, where physical probes cannot be used. However, the rotation is typically small, which degrades the accuracy of the measurement. Paradoxically, the main source of error is the probe beam itself. Since polarization cannot be measured directly, detectors rely on a polarizer to measure a small change in beam intensity instead. In this work, we show how suppress the beam intensity that is not part of the Faraday rotation signal by taking forming an optical derivative. Since the rotation measurement is now strictly proportional to the beam intensity, the system allows to amplify the rotation measurement simply by increasing the laser power.

physics.optics↗

arXiv2022

True optical spacial derivatives for plasma density measurements

This paper shows analytically and numerically that a vortex plate coupled to a neutral density filter can deliver a true optical derivative when placed at the focal plane of a $2f$ lens pair. This technique turns spatial variations in intensity into an intensity, which square root is the spatial derivative of the initial intensity variation. More surprisingly, it also turns any spatial variations in phase into an intensity, which square root is the spatial derivative of the initial phase variation. Since the optical derivative drops the DC component of the signal, it is possible to measure the full electron plasma turbulence spectrum optically, without using any interferometer.

physics.optics↗

arXiv2022

A fully implicit method using nodal radial basis functions to solve the linear advection equation

Radial basis functions are typically used when discretization sche-mes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrix-matrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to other solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is naturally conservative, keeping the error virtually constant throughout the computation.

math.NA↗

arXiv2018

The conceptual design of 100-kA pulsed magnetic field generator for magnetized high-energy-density plasma experiments

This paper presents the conceptual design of a high-voltage pulser intended to generate 30-T magnetic fields for magneto-inertial fusion experiments at the OMEGA facility. The pulser uses a custom capacitor bank and two externally triggered spark gaps to drive a multi-turn coil. This new high-voltage pulser is capable of storing 10 times more energy than the previous system, using a higher charge voltage (from 20 to 30 kV) and a larger capacitance (from 1 μF to 5 μF). Circuit simulations show that this pulser can deliver 100 kA into a 60-nH, 14-mΩ coil with a rise time of 1 μs. For a coil with 2 turns with an average coil diameter of 7.8 mm, this current translates into a 32-T peak magnetic field at coil center. This is a factor of three increase in the peak magnetic field compared to the present generator magnetic field capabilities.

physics.plasm-ph↗

arXiv2018

High voltage charging system for pulsed power generators

A robust and portable power supply has been developed specifically for charging linear transformer drivers, a modern incarnation of fast pulsed power generators. It is capable of generator +100 kV and -100 kV at 1 mA, while withstanding the large voltage spikes generated when the pulsed-power generator is triggered. The three-stage design combines a zero-voltage switching circuit, a step-up transformer using ferrite cores, and a dual Cockcroft-Walton voltage multiplier. The zero-voltage switching circuit drives the primary of the transformer in parallel with a capacitor. With this driver, the tank circuit naturally remain in its resonant state, allowing for maximum energy coupling between the zero-voltage switching circuit and the Cockcroft-Walton voltage multiplier across a wide range of loading conditions.

physics.ins-det↗

arXiv2018

Mega-ampère to mega-gauss: the generation of intense magnetic fields using fast pulsed-power drivers

Intense magnetic fields modify the properties of extremely dense matter via complex processes that call for precise measurements in very harsh conditions. This endeavor becomes even more challenging because the generation of mega-gauss fields in a laboratory is far from trivial. This paper presents a unique and compact approach to generate fields above 2 mega-gauss in less than 150 ns, inside a volume close to half a cubic centimeter. Magnetic insulation, keeping plasma ablation close to the wire surface, and mechanical inertia, limiting coil motion throughout the current discharge, enable the generation of intense magnetic fields where the shape of the conductor controls the field topology with exquisite precision and versatility, limiting the need for mapping exactly magnetic fields.

physics.plasm-ph↗

arXiv2018

The generation of warm dense matter using a magnetic anvil cell

Warm dense matter is present at the heart of gas giants and large exo-planets. Yet, its most basic properties are unknown and limit our understanding of planetary formation and evolution. In this state, where pressure climbs above 1 Mbar, matter is strongly coupled and quantum degenerate. This combination invalidates most theories capable of predicting the equation of state, the viscosity or heat conductivity of the material. When such properties are missing, understanding planetary evolution becomes an arduous endeavor. Henceforth, research in this field is actively growing, using high power laser or heavy ion beams to produce samples dense enough to overcome the 1 Mbar limit. These samples are not actively confined and tend to expand rapidly, precluding the existence of any thermodynamically stable equilibrium. However, a mega-ampere-class pulsed-power generator can produce confined matter in the Mbar range, providing two conditions are being met. First, the sample needs to be compressed cylindrically, to maximize magnetic pressure (compared to slab compression). Second, a damper must be used to preclude the formation of a corona around the sample. This corona robs the main sample from valuable current and limits the homogeneity of the compression. According to numerical simulations, the setup proposed here, and called a magnetic anvil cell, can reach pressure on the order of 1 Mbar using a mega-ampere pulsed power driver. These samples span several millimeters in length. Unlike diamond anvil cell, which pressure is limited below 1 Mbar due to materials strength, the magnetic anvil cell has virtually no pressure limit. Further, the current heats the sample to several eV, a temperature well beyond diamond anvil cell capabilities.

physics.plasm-ph↗

arXiv2017

The stability of unity beta equilibria in tokamaks

The plasma beta, ratio of kinetic to magnetic pressure, inside a tokamak should stay below the Troyon limit to avoid major plasma instabilities. However, this paper argues that Troyon limit occurs only when current profiles cannot sustain high beta equilibria. This paper shows that plasmas with peak beta on the order of unity are stable to ideal MHD modes. While the theoretical (ideal-MHD) stability of unity beta equilibrium has been resolved, the experimental existence of high beta equilibria in tokamaks remains an unsolved problem of plasma physics.

physics.plasm-ph↗

arXiv2017

Axial magnetic field injection into thick, imploding liners

MagLIF is a fairly new fusion concept using a puled-power generator as the main driver. This concept uses a Z-pinch configuration where the implosion is driven by the Z-machine using 27 MA of electrical current in 100 ns. Since the implosion time is long compared to the heat diffusion time of a non-magnetized plasma, MagLIF requires an initial axial magnetic of 30T to reduce heat losses to the liner wall. Since the field needs to penetrate the transmission lines of the pulsed-power generator, as well as the liner itself, the rise time must exceed tens of microseconds. Any coil capable of producing such field on that long a pulse-length is inevitably bulky. The space required to house the coil near the liner increase the inductance of the load, which becomes problematic since the voltage at the load cannot exceed what the driver can already provide. Yet, the enormous amount of current that the Z-machine can provide could be used to produce the required 30 T by tilting the current posts surrounding the liner. However, the field penetration is limited by the skin effect of the liner wall. This paper shows that when current densities are large enough, the material generate resistivity gradients which forces the current to diffusive across the liner wall much fast than the skin time. As a result, the 30T coil can be eliminated and replaced by return current posts with minimal helicity.

physics.plasm-ph↗

arXiv2017

HADES: a High Amperage Driver for Extreme States

Linear transformer drivers (LTD) allow to greatly reduce the size of pulsed-power drivers while increasing their efficiency and repetition rate. However, limitations on the capacitors voltage and current exist, mostly driven by technological imperatives. As a result, LTD required to be connected in series and in parallel to form a practical pulsed-power generator. The High Amperage Driver for Extreme States (HADES) proposed here can generate 250 GW of power, i.e. a 1 MA on current with a rise time of 150 ns into a 20nH load. The key feature of the driver is its size. The machine footprint is less than 5m2 and its height is less than 1.6m. This reduction in footprint, compared to other 1 MA LTD based-systems, comes from a compact transmission line geometry which allow to connect LTDs is series and in parallel efficiently. This paper summarizes this design and describes HADES operation and maintenance.

physics.ins-det↗

arXiv2017

The impact of the Hall term on tokamak plasmas

The Hall term has often been neglected in MHD codes as it is difficult to compute. Nevertheless setting it aside for numerical reasons led to ignoring it altogether. This is especially problematic when dealing with tokamak physics as the Hall term cannot be neglected as this paper shows.

physics.plasm-ph↗

arXiv2017

The impact of three dimensional MHD instabilities on the generation of warm dense matter using a MA-class linear transformer driver

Warm dense matter is difficult to generate since it corresponds to a state of matter which pressure is order of magnitude larger than can be handled by natural materials. A diamond anvil can be used to pressurize matter up to one Gbar, this matter is at high density but at room temperature. High power lasers and heavy ion beams can generate warm dense matter but they cannot confine it long enough to allow measurements of quasi-static transport coefficients such as viscosity or heat conduction. We present here a third method to generate warm dense matter. It uses a pulsed-power driver which current rise time is substantially shortened by using a plasma opening switch, limiting the development of electrothermal instabilities. The switch relies on the implosion of a gas puff Z-pinch which carries most of the discharge current until the pinch reaches the sample. After that, the sample is compressed until it reaches the warm dense matter regime. Three-dimensional magnetohydrodynamics computations show that if the density of the gas is low enough no detectable instabilities (e.g. kinks and sausages modes) impede the remainder of the implosion.

physics.plasm-ph↗

arXiv2017

A physics-based solver to improve the illumination of cylindrical targets using spherically-distributed high power laser systems

In recent years, our understanding of high energy density plasmas has played an important role in improving inertial fusion confinement and in emerging new fields of physics, such as laboratory astrophysics. Every new idea required developing innovative experimental platforms at high power laser facilities, such as OMEGA or NIF. These facilities, designed to focus all their beams onto spherical targets or hohlraum windows, are now required to shine them on more complex targets. While the pointing on planar geometries is relatively straightforward, it becomes problematic for cylindrical targets, or target with more complex geometries. This publication describes how the distribution of laser beams on a cylindrical target can be done simply by using a set of physical laws as a pointing procedure. The advantage of the method is threefold. First, it is straightforward, requiring no mathematical enterprise besides solving ordinary differential equations. Second, it will converge if a local optimum exists. Finally, it is computationally inexpensive. Experimental results show that this approach produces a geometrical beam distribution that yields cylindrically symmetric implosions.

physics.plasm-ph↗

arXiv2013

The impact of the Hall effect on high energy density plasma jets

Using a 1-MA, 100ns-rise-time pulsed power generator, radial foil configurations can produce strongly collimated plasma jets. The resulting jets have electron densities on the order of 10^20 cm^-3, temperatures above 50 eV and plasma velocities on the order of 100 km/s, giving Reynolds numbers of the order of 10^3, magnetic Reynolds and Péclet numbers on the order of 1. While Hall physics does not dominate jet dynamics due to the large particle density and flow inside, it strongly impacts flows in the jet periphery where plasma density is low. As a result, Hall physics affects indirectly the geometrical shape of the jet and its density profile. The comparison between experiments and numerical simulations demonstrates that the Hall term enhances the jet density when the plasma current flows away from the jet compared to the case where the plasma current flows towards it.

physics.plasm-ph↗

arXiv2008

Dual equilibrium in a finite aspect ratio tokamak

A new approach to high pressure magnetically-confined plasmas is necessary to design efficient fusion devices. This paper presents an equilibrium combining two solutions of the Grad-Shafranov equation, which describes the magnetohydrodynamic equilibrium in toroidal geometry. The outer equilibrium is paramagnetic and confines the inner equilibrium, whose strong diamagnetism permits to balance large pressure gradients. The existence of both equilibria in the same volume yields a dual equilibrium structure. Their combination also improves free-boundary mode stability.

physics.plasm-ph↗