# ED3220 Motion of Charged Particles, Collision Processes and Basis of Transport Theory II8.0 credits

This course provides the theoretical basis for motion of charged particles in static and time-dependent electromagnetic fields as well as effects of interactions with a background plasma. The course provides the basis for understanding transport theory from random walk of single particles in magnetic bottles. This course is aimed for students studying fusion research. It is similar to FED3210 except that it treats drift orbits in toroidal plasmas in more detail.

Third cycle

P, F

## Information for research students about course offerings

The course is given when there is sufficient demand. Please contact the examiner if you are interested in taking the course.

## Intended learning outcomes

When completing the course, the student should be able to:

Describe particle motion in terms of drift motion of guiding centre,

Understand the concept of adiabatic invariants, knowledge of particle orbits in toroidal geometries in presence of static electric fields. Stochastisation of orbits by asymmetries.

Know how to use Poincaré plots and standard mapping for analysing regular and stochastic orbits, Chirikov criterion for determining stochastisation, and KAM surfaces. Understand how Coulomb collisions affect the motion of single particles and how relaxation towards isotropic thermal plasmas takes place.

Be familiar with the concept of stochastic differential equations and how to use it for solving diffusion equations. The most important collision processes in plasma including nuclear reactions.

Understand the basis of curvilinear coordinate system: covariant and contra variant representation, differentiation in curvilinear coordinate system, flux coordinate system, Clebsch representation of magnetic field and coordinate system suitable for analysing guiding centre motion.

Understand the basis of classical mechanics: Lagrange equation, Hamilton equation, canonical transformation, cyclical coordinates, action-angle variables, Lagrange and Hamiltonian equations of motion of charged particles.

Perturbation theory of charged particle motion and adiabatic invariants. Drift motion of guiding centre. Particle motion in toroidal geometry in presence of static radial and toroidal electric fields. Stochastisation of the orbits by asymmetries, Poincaré plots, Chirikov criterion. Collision processes. Relaxation processes by Coulomb collisions with a background plasma. Interactions by time dependent fields (including wave-particle interactions; resonance interactions) superadiabatic oscillations, collisionless absorption and stimulated emission processes. Brownian motions – Monte Carlo methods for describing particle motion. Curvilinear coordinate system with application to charged particle motion. Basis of analytic mechanics with application to charged particle motion in curvilinear coordinate system.

## Disposition

Seminars or discussion meetings

## Literature

Lecture notes. Research articles refered to in the lecture notes for understanding the notes.

Parts of the following, or similar, litterature:

P. Helander and D.J. Sigmar Collisional Transport in magnetized Plasma Introduction 1-13, Collision operator 22-58, 99-136, adiabatic invariants 99-117 with Lagrange and Hamiltonians.

D.J. Rose and McClark, Plasma and Controlled Fusion, M.I.T. and Wiley, New York-London 1961, p. 13-53, 228-256.

L. Spitzer, Physics of Fully Ionized Gases, second rev. Ed., New York, 1962, 120-154.

P. Rutherford and Goldstone Poincaré plot and Chirikov criterion. The subject is also included in Chap. 11 in Classical mechanics, Goldstein (Third Edition 2001).

Curvilinear coordinate system. R. B White, Ch. 1.,

E. Madelung Die Mathematischen Hilfsmittel des Physikers, p 212-220 or corresponding content in any other classic text book,

Classical mechanics, Goldstein Chapters corresponding to the lecture notes. Ch 8-10 in Third Edition, Addison Wesley, San Fransico 2002.

Stochastic differential equations Numerical Solutions of Stochastic Differential Equations, P. E. Klueden and E. Platen, Springer-Verlag, Berlin 1992, Introductory chapter XX-XXXV.

## Examination

• EXA1 - Examination, 8.0, grading scale: P, F

Final writen and oral exam.

## Offered by

EECS/Fusion Plasma Physics

Thomas Jonsson

## Examiner

Thomas Jonsson <johnso@kth.se>

## Version

Course syllabus valid from: Spring 2012.
Examination information valid from: Spring 2019.