Lecture 2: Plasma particles with E and B fields

Lecture 2: Plasma particles with E and B fields

Today s Menu Magnetized plasma & Larmor radius Plasma s diamagnetism Charged particle in a multitude of EM fields: drift motion ExB drift, gradient drift, (later: curvature drift, polarization drift, ) Concept of a guiding center Magnetic moment Magnetic mirror Loss cone 2

Plasmas of interest Not only are the plasmas of our interest (space & fusion) weakly coupled, they are also magnetized Why? B ~ tens of μt B ~ several T Earth has its own magnetic field that, in the first approximation, can be considered a dipole field. In fusion energy research, the VERY hot plasma is kept away from the vessel walls by magnetic field. 3

Charged particles in magnetic field Consider a charge particle (q,m) in uniform magnetic field, BB = BB 0 zz. Lorentz force: mm ddvv dddd = qqvv BB mm ddvv xx dddd = qqvv yybb 0 mm ddvv yy dddd = qqvv xxbb 0 mm ddvv zz dddd = 0 Collect the constants into Ω qqbb 0 /mm, Larmor/cyclotron frequency HW vv xx = vv sin Ωtt with vv yy = vv cos Ωtt (or vice versa), vv zz = vv 4

Larmor motion Integrate in time (HW) xx = vv sin Ωtt & y = vv cos Ωtt Ω Ω charged particles are gyrating around the magnetic field line on a circle with the radius defined by their perpendicular velocity and magnetic field strength: Larmor radius: r L = mmvv qqqq Notice rightaway (effects one-by-one): Strong field stick close to field line Big charge number stick close to field line Large perpendicular velocity large gyro radius Large mass large excursions from the field line Larmor stuff: Ω is the highest frequency in a magnetized plasma r L = the shortest scale in a magnetized plasma 5

and diamagnetism Particles in plasma thus carry out circular motion around field lines. A charged particle on a circular path forms a current ring Ampere s law BB = μμ 0 jj recall your course in EM + additional magnetic field opposite to the background field A plasma is diamagnetic ( except in some special cases ), i.e., tends to reduce the imposed magnetic field 6

Concept of magnetized plasma A plasma is considered magnetized if the Larmor radius is much much smaller than the scale length L over which the magnetic field changes appreciably. r L << L Note: not exactly uniform B fields 7

Charged particle motion in simple or simplish fields 8

Add a uniform electric field, E = E 0 EE = EE 0 zz simply acceleration in the direction of B Take E perpendicular to B, e.g., EE = EE 0 xx Think what happens now during the gyration period Can this be true? Particle seems to move in direction perpendicular to both E and B fields!!! 9

Do the math Equations of motion: HW vv xx = vv sin Ωtt ddvv xx = Ωvv dddd yy + qqee 0 mm ddvv yy dddd = Ωvv xx vv yy = vv cos Ωtt + EE 0 BB 0 Indeed, the particle drifts perpendicular to both fields! Useful concept: the center of gyro motion, the guiding center, drifts. 10

The ExB drift This guiding-center drift is called the EE BB drift and it has a very important role especially in fusion plasma physics. General (vector) form: Things to notice: vv EEEEEE = EE BB BB 2 The drift does not depend on the particle everybody drifts in the same direction with the same velocity! This drift is not really specific to just electric field. Any external force, E F/q, would cause such a drift but this time depending on the charge! e.g., gravitational force 11

Charged particle motion in nonuniform magnetic field 12

Part I: BB BB = BB 0 Choose the axes so that B yy What happens now during one gyration period zz The particle moving (= drifting) in direction perpendicular to both the B field and its gradient!!! 13

Do the math Taylor expand the magnetic field remembering that r L << L BB zz = BB oo + yy BB zz + FF yy = qqvv xx BB zz yy qqvv (sin Ωtt) BB oo + rr LL (sin Ωtt) BB zz where we have also used the unperturbed orbit to evaluate the force. Recall: Ω gives the shortest time scale average over one gyro period < sin Ωtt > = 0, < (sin Ωtt) 2 > = 1 2 < FF yy > = ± 1 2 qqvv rr LL BB zz 14

The gradient drift So there is an effective net force on the particle obtain GC drift from the generalized ExB drift: vv GGGG = 1 qq FF BB BB 2 = 1 qq FF yy xx = ± 1 BB zz vv BB 0 2BB rr LL 0 The gradient drift ( -drift) in general vector form vv = ± 1 2 vv rr LL BB BB 2 This drift does depend on the charge, as indicated by the ± sign 15

Part II: B B = B 0 z For axial magnetic to have parallel gradient means that the field must have also a radial component. It can be obtained from BB = 0: Cylindrical symmetry cylindrical coordinates: Assume slowly varying magnetic field 1 rr rrbb rr + BB zz = 0 rr BB rrbb rr = 0 rr zz dddd 1 2 rr2 BB zz r=0 BB rr 1 rr BB zz 2 r=0 Non-uniform BB rr gradient drift in poloidal direction. No problem. (Radial drift would require non-uniformity in poloidal direction) 16

Full Lorentz force in cylindrical coordinates FF rr = qqvv θθ BB zz FF θθ = qq vv zz BB rr vv rr BB zz FF zz = qqvv θθ BB rr Gyro motion around the fieldline The 1st term in FF θθ causes a radial drift that forces the particle to follow the bending field lines The new physics is brought about by FF zz. For simplicity, study a particle on the axis, r = 0: FF zz = qqvv 1 2 rr LL BB zz r=0 17

Magnetic force along the field rr LL = mmmm /qqqq FF zz = 1 2 mmvv 2 mmvv 2 BB BB zz = μμ BB zz where μμ 1 is the so-called magnetic moment of the particle. 2 BB General (vector) form: FF = μμμμ BB Note: μ can be understood as the magnetic moment due to the current loop created by the gyrating particle (HW) The force causes a braking action when particle moves towards higher field 18

Now we have a bunch of drifts So what? 19

Magnetic mirrors and invariance of μ Magnetic bottle : first attempt to magnetic confinement Linear device BB BB z mm ddvv dddd = μμ Multiply by vv = dddd dddd zz s = distance along a field line BB mmmmmm BB mmmmmm mm 2 dd dddd vv 2 = μμ ss tt = μμ dddd dddd Note: B does not depend on time, but a particle sees it varying in time. 20

and invariance of μ dd dddd 1 mmvv 2 2 + μμμμ = BB dddd dddd Recall the definition: μμ 1 2 EE tttttt = 1 mmvv 2 2 + μμμμ mmvv 2 BB 1 2 mmvv 2 = μμμμ Total energy is conserved: ddee tttttt dddd dddd dddd = 0 = 0 The magnetic moment is an (adiabatic) invariant!!! 21

In the house of mirrors mmvv 2 μμ 1 = constant 2 BB So what happens if the particle moves to a region with increasing BB? Perpendicular energy must increase Total energy conserved vv must decrease BB mmmmmm high enough Larmor motion eats up all vv particle stops Now FF = μμμμ BB kicks in particle gets reflected particle gets trapped in the mirror = particle is confined! This was the idea behind the magnetic bottle. 22

Magnetic bottle is not plasma-tight But we do not get fusion electrons out of our electrical outlets. Why? There was an if above: if BB mmaaaa high enough What is high enough? Let vv,0 & vv,0 correspond to the mid-bottle, i.e., where BB = BB mmiiii At the (potential) turning point, BB = BB mmaaaa : vv = 0 & vv = vv μμ = constant vv 2,0 2 BB mmmmmm = vv BB mmaaaa Energy is conserved: vv 2,0 +vv 2 2,0 = vv Particle confined only if vv,0 is low enough (HW): vv,0 2 vv2 0 < 1 BB mmmmmm /BB mmmmmm 23

The concept of a loss cone It is common to denote vv 2 pitch of the particle vv 0 2 ξξ2, called the Correspondingly, θθ coss 1 ξξ is the pitch angle. The value of ξξ in the weak-field region defines the loss cone: ξξ 0 2 > 1 BBmmmmmm /BB mmmmmm It is clear that for BB mmmmmm <, the magnetic bottle leaks and not all the particles are confined. 24

Things to keep in mind Even though in uniform magnetic field particles are stuck with their field line, with additional fields and/or uniformities, the particles will start drifting from their mother-fieldline Many interesting plasmas have their mirrors and loss cones In a mirror field, particles with small ξξ bounce between the mirror points w/ bounce frequency ωω bb More drifts to come in the second period ;-) 25