Lecture3: Plasma Physics 1

APPH E6101x Columbia University

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Homework #1 (1 of 3)

Problems 43

• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)

Γ = q2

4πε0a2W S kBT

.

Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.

Problems

2.1 Prove that the electron Debye length can be written as

λDe = 69 m!

T (K)

ne(m−3)

"1/2

2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).

2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?

2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).

2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.

2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from

1T

= ∂S∂λ

∂λ

∂U

and use#

ni = 1.

2

Homework #1 (2 of 3)

Problems 43

• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)

Γ = q2

4πε0a2W S kBT

.

Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.

Problems

2.1 Prove that the electron Debye length can be written as

λDe = 69 m!

T (K)

ne(m−3)

"1/2

2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).

2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?

2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).

2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.

2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from

1T

= ∂S∂λ

∂λ

∂U

and use#

ni = 1.3

Homework #1 (3 of 3)

Problems 43

• The coupling parameter Γ determines the state of each plasma component(electrons, ions, dust)

Γ = q2

4πε0a2W S kBT

.

Γ may be different for the components, depending on the individual tem-peratures and densities. A gaseous phase is found for Γ ≪ 1, the liquidstate for 1 < Γ < 180 and the solid phase for Γ > 180.

Problems

2.1 Prove that the electron Debye length can be written as

λDe = 69 m!

T (K)

ne(m−3)

"1/2

2.2 Calculate the electron and ion Debye length(a) for the ionospheric plasma (Te = Ti = 3000 K, n = 1012 m−3).(b) for a neon gas discharge (Te = 3 eV, Ti = 300 K, n = 1016 m−3).

2.3 Consider an infinitely large homogeneous plasma with ne = ni = 1016 m−3.From this plasma, all electrons are removed from a slab of thickness d = 0.01 mextending from x = −d to x = 0 and redeposited in the neighboring slab from x =0 to x = d. (a) Calculate the electric potential in this double slab using Poisson’sequation. What are the boundary conditions at x = ±d? (b) Draw a sketch of spacecharge, electric field and potential for this situation. What is the potential differencebetween x = −d and x = d?

2.4 Show that the equation for the shielding contribution (2.24) results from (2.21)and (2.23).

2.5 Derive the relationship between the coupling parameter for ion-ion interactionΓ Eqs. (2.15) and ND (2.33) under the assumption that Te = Ti.

2.6 Show that the second Lagrange multiplier in Eq. (2.6) is λ = (kBT )−1.Hint: Start from

1T

= ∂S∂λ

∂λ

∂U

and use#

ni = 1.

4

Outline

• Charged Particle Drifts

• Adiabatic Invariants

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Charged particle motion in inhomogeneous, static and slowly-varying electric and magnetic fields

Part 1: Gyromotion in Uniform Field

48 3 Single Particle Motion in Electric and Magnetic Fields

The magnetic induction B(r) of a dipole field is given by the expression

B(r) = µ0

4π3r(r · M) − r2M

r5 . (3.6)

Here r is a vector pointing from the magnetic dipole to the field point. At the author’slocation (54.3◦ N.; 10.1◦ E), the Earth magnetic field has a horizontal componentBh = 17, 700 nT and a vertical component Bv = −46, 150 nT.

3.1.4 E×B Drift

When we now allow for a stationary and homogeneous electric field, we can choosethe orientation of our coordinate system, without loss of generality, to have electricand magnetic field in the x-y plane, E = (Ex , 0, Ez) and B = (0, 0, Bz). In thiscase, Newton’s equation of motion reads

vx = qm

!Ex + vy Bz

"

vy = qm

(−vx Bz)

vz = qm

Ez . (3.7)

The motion along the magnetic field is now accelerated but independent of themotion in the x-y plane. According to the principle of superposition of motions,we can consider both effects separately. For the x-y plane, the motion can again bedecomposed

vx = −ω2cvx

vy = −ω2c (vy + Ex/Bz) . (3.8)

Again, we find a harmonic oscillation in x-direction, but the motion in y-directionis more complex. In a moving frame of reference, vy = vy + Ex/Bz , which movesat a constant velocity −Ex/Bz in negative y-direction, we obtain a simple harmonicmotion

¨vy = −ω2c vy . (3.9)

Thus the solution for the velocities is the superposition of a circular orbit and aconstant motion in the same plane. This constant motion is called the E×B-drift.For a particle that is initially at rest, the solution reads

vx = Ex

Bzsinωct

vy = Ex

Bz[cosωct − 1] . (3.10)

48 3 Single Particle Motion in Electric and Magnetic Fields

The magnetic induction B(r) of a dipole field is given by the expression

B(r) = µ0

4π3r(r · M) − r2M

r5 . (3.6)

Here r is a vector pointing from the magnetic dipole to the field point. At the author’slocation (54.3◦ N.; 10.1◦ E), the Earth magnetic field has a horizontal componentBh = 17, 700 nT and a vertical component Bv = −46, 150 nT.

3.1.4 E×B Drift

When we now allow for a stationary and homogeneous electric field, we can choosethe orientation of our coordinate system, without loss of generality, to have electricand magnetic field in the x-y plane, E = (Ex , 0, Ez) and B = (0, 0, Bz). In thiscase, Newton’s equation of motion reads

vx = qm

!Ex + vy Bz

"

vy = qm

(−vx Bz)

vz = qm

Ez . (3.7)

The motion along the magnetic field is now accelerated but independent of themotion in the x-y plane. According to the principle of superposition of motions,we can consider both effects separately. For the x-y plane, the motion can again bedecomposed

vx = −ω2cvx

vy = −ω2c (vy + Ex/Bz) . (3.8)

Again, we find a harmonic oscillation in x-direction, but the motion in y-directionis more complex. In a moving frame of reference, vy = vy + Ex/Bz , which movesat a constant velocity −Ex/Bz in negative y-direction, we obtain a simple harmonicmotion

¨vy = −ω2c vy . (3.9)

Thus the solution for the velocities is the superposition of a circular orbit and aconstant motion in the same plane. This constant motion is called the E×B-drift.For a particle that is initially at rest, the solution reads

vx = Ex

Bzsinωct

vy = Ex

Bz[cosωct − 1] . (3.10)

What drifts result from a static force that has the same sign for ions and electrons (like gravity)?

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Key Dynamics

• Separate descriptions of perpendicular and parallel motion

• Fast gyration around B

• Slow perpendicular drift of gyro center

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Guiding Center Motion in a Strong Magnetic Field

Part 2: Guiding Center Motion

Reference: I.B. Bernstein, “The Motion of a Charged Particle in a Strong Magnetic Field,” Advances in Plasma Physics 4,

311 (1971).

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Some Definitions & Cases

• 3 equations, 2nd order, 6 initial conditions

• Constant E and B

• Slowly varying E (with uniform B)

• Let E → 0, with B slowly varying in space

• …

Use Bernstein’s notation…

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Just Mechanics

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Simple Note 1

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Simple Note 2

so we can decouple parallel and perpendicular motion

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Initial Value Case 1

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Initial Value Case 2

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Initial Value Case 3

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Initial Value Case 4

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Defining Gyro-Averaging

A physical solution to the E.O.M. in terms of gyration and drift

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Separating the Vector E.O.M.

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Fast Gyromotion(Separately)

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Slow Drift (Separately)

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a(t) slowly…

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a(t) slowly…

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Polarization Drift

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B(x) changing slowly…

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B(x) changing slowly…

(1) (2) (3)

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Drifts (separately)

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Drifts (separately)

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Drifts (separately)

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Drifts (separately)

!!

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Gradient-B Drift

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(Parallel to B) Mirroring…

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(Perpendicular to B) Curvature Drift

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Curvature Drift

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Summary

(but Ω changes…)

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Simple Adiabatic Invariant

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Next Week

• Guiding Center Hamiltonian (Littlejohn & Boozer)

• Tokamaks and bananas

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