Electromagnetic four-potential

An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector. As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is conventionally taken to be the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant. Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units. The electromagnetic four-potential can be defined as: {| class="wikitable" |- ! SI units ! Gaussian units |- | || |} in which φ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m−1 in SI, and Mx·cm−1 in Gaussian-cgs. The electric and magnetic fields associated with these four-potentials are: {| class="wikitable" |- ! SI units ! Gaussian units |- | || |- | || |} In special relativity, the electric and magnetic fields transform under Lorentz transformations. This can be written in the form of a tensor - the electromagnetic tensor. This is written in terms of the electromagnetic four-potential and the four-gradient as: assuming that the signature of the Minkowski metric is (+ − − −). If the said signature is instead (− + + +) then: This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

Monte Carlo Modeling of Crystal Channeling at High Energies

Philippe Jean Schoofs

Charged particles entering a crystal close to some preferred direction can be trapped in the electromagnetic potential well existing between consecutive planes or strings of atoms. This channeling effect can be used to extract beam particles if the crystal is bent beforehand. Crystal channeling is becoming a reliable and efficient technique for collimating beams and removing halo particles. At the European Organization for Nuclear Research (CERN), the installation of silicon crystals in the Large Hadron Collider (LHC) is under scrutiny by the UA9 collaboration with the goal of investigating if they are a viable option for the collimation system upgrade. This thesis describes a new Monte Carlo model of planar channeling which has been developed from scratch in order to be implemented in the FLUKA code simulating particle transport and interactions. Crystal channels are described through the concept of continuous potential taking into account thermal motion of the lattice atoms and using Moliere screening function. The energy of the particle transverse motion determines whether or not it is trapped between the crystal planes while single Coulomb scattering on lattice atoms can lead to dechanneling. The volume capture and reflection applying to quasi-channeled particles are also modeled. Analogously to dechanneling, single scattering is used to determine the occurrence of volume capture. The parameters of the crystals, such as torsion or miscut, are described as well. For channeled particles, the suppression of electromagnetic and nuclear collisions is implemented. It is stronger for particles oscillating close to the center of the channel and is crucial for a correct evaluation of the rate of dechanneling without having recourse to the use of a macroscopic dechanneling length. The UA9-H8 experiment conducted at CERN aims at investigating new crystal physics as well as characterizing crystals that can be of interest in view of the implementation of crystal collimation at CERN, including in the LHC. This experiment uses silicon strip detectors situated on both sides of the crystal. Putting together upstream and downstream tracks in coincidence and matching at an identical fitted location on the crystal, it yields information about the deflections given to the beam population. Several runs from the UA9-H8 experiment are analyzed and compared to the model results. Channeling and dechanneling rates, as well as angular distributions at crystal exit are shown to be in a very encouraging agreement both for strip and quasi-mosaic crystals.

EPFL2014

Low frequency electromagnetic wave propagation in 3D plasma configurations

Pavel Popovitch

We investigate low-frequency electromagnetic wave propagation and absorption properties in 2D and 3D plasma configurations. For these purposes, we have developed a new full-wave 3D code LEMan that determines a global solution of the wave equation in bounded stellarator plasmas excited with an external antenna. No assumption on the wavelength compared to the plasma size is made, all the effects of the 3D geometry and finite plasma extent are included. The equation is formulated in terms of electromagnetic potentials in order to avoid numerical pollution effects. The code utilises linear and Hermite cubic finite element discretisation in the radial direction and Fourier series in the poloidal and toroidal variables. The full cold plasma model including finite electron inertia and, thus, mode conversion effects is implemented. The code uses Boozer magnetic coordinates and has an interface to the TERPSICHORE code. Special care is taken to treat the magnetic axis and to ensure the unicity of the numerical solution. The discretisation, interpolation and numerical derivation methods specifically adapted for our problem avoid the energy sink in the origin and provide a very good local and global energy conservation. A special algorithm has been developed to analytically expand the wave equation coefficients in the full 3D stellarator geometry. The code has been specifically optimised for vector computing platform, reaching close to maximum average performances on the NEC SX5 machine. The code has been applied in 1D, 2D, and 3D geometries. No unphysical solutions have been observed. LEMan successfully recovers all the fundamental properties of the Alfvén spectrum (gaps, eigenmodes). Benchmarks have been made against the 2D LION code and JET experimental measurements, showing a good agreement between the results.

EPFL2004

Monte Carlo Modeling of Crystal Channeling at High Energies

Philippe Jean Schoofs

EPFL2014