INTRODUCTION TO PLASMA PHYSICS Fulvio ZoncaThe Debye length is not a characteristic feature of a plasma but is common to any ionized system in thermal equilibrium. At thermal equilibrium

A.A. 2019-20 TECHNOLOGIES FOR NUCLEAR FUSION PART I – Physics Lecture 1 – 1

Lecture 1

INTRODUCTION TO PLASMA PHYSICS

Fulvio Zonca

http://www.afs.enea.it/zonca

ENEA C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C.

March 10.th, 2020

TECHNOLOGIES FOR NUCLEAR FUSIONPART I – Physics

INTRODUCTION TO PLASMA PHYSICS

Fulvio Zonca


SyllabusMain areas that will be explored are:

• Lecture 1

i) Classification of plasmas, Debye length

ii) Collisions between charged particles

iii) Collisional slowing down, plasma resistivity

iv) Fusion reactor scheme

v) Power balance, Lawson Criterion

vi) Ideal ignition temperature

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• Lecture 2

i) Solution of the power balance equation

ii) Main heating and loss terms

iii) Scaling laws for the energy confinement time

iv) Tokamak and stellarator

v) Brief introduction to inertial confinement fusion

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Classification of plasmas

✷ A plasma can be defined as a fully or partly ionized gas, which is generallyneutral in average and exhibits collective behaviors.

✷ Main properties of a plasma and its classification can be assigned by itsparticle density n and temperature T .

✷ In fact, characteristic plasma parameters depend on n and T

• Debye length λD (see later): characteristic length over which Coulombpotential is screened

• Mean inter-particle spacing ∼ n−1/3

• Distance of closest approach

✷ Plasmas are classified according to the ratio of average kinetic energy ofits particles with the potential energy due to interaction among plasmaparticles themselves

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✷ Plasma classification scheme: (note, nλ3

D = 1 ⇔ Ecoul = kT )

• weakly coupled plasmas (fusion interest): nλ3

D > 1

• strongly coupled plasmas: nλ3

D < 1

• degenerate plasmas: kT ≤ ǫF = (h2/2me)(3π2n)2/3

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Debye length

✷ The Debye length is not a characteristic feature of a plasma but is commonto any ionized system in thermal equilibrium.

✷ At thermal equilibrium the ionized (plasma) particles of charge q are de-scribed by the Gibbs distribution

n = n0 exp

(

−qΦ

kT

)

= n0

(

1−qΦ

kT+ ...

)

= n0 + n

✷ Assume that the system is neutral at equilibrium; that is there exists aneutralizing background (ions in a plasma). Then introduce a test chargeat r = r0.

✷ The scalar potential function satisfies the Poisson’s law

−∇2Φ = 4πq [n+ δ (r − r0)]

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✷ Taking the expression for n ≃ −n0(qΦ/kT ) and substituting into the Pois-son’s equations, we obtain

(

−∇2 +1

λ2

D

)

Φ = 4πqδ (r − r0) ; λD =

(

kT

4πn0q2

)1/2

✷ Without the ∝ λ−2

D term, the general solution of the above equation isthe potential of a point charge q located at r = r0; i.e., Φ = (q/r), withr = |r − r0|.

✷ The effect of the ∝ λ−2

D term is to introduce a screening:

Φ =q

rexp

(

−r

λD

)

=q

r+

q

r

[

exp

(

−r

λD

)

− 1

]

• the green term is the bare potential

• the blue term is the screened potential due to collective effects

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✷ Ratio of thermal energy over the Coulomb energy: assume the mean inter-particle spacing n−1/3

kT

qΦ∼

kT

n1/3q2exp(nλ3

D)1/3 ≃ 4π(nλ3

D)2/3 exp(nλ3

D)1/3

✷ The ratio of thermal energy over the Coulomb energy is larger or smallerthan unity depending on the value of Λ = nλ3

D.

✷ Fusion plasmas are characterized by Λ = nλ3

D ≫ 1 and are weakly coupledplasmas, dominated by collective effects.

✷ The distance of closest approach b is reached when Coulomb energy is equalto the kinetic energy

mv2

2≃

3

2kT = qΦ =

q2

b; ⇒ b =

2

3

q2

kT=

1

6π

1

nλ2

D

; ⇒b

λD

=1

6π

1

nλ3

D

✷ In a fusion plasma, the distance of closest approach is much smaller thanthe Debye length. The role of binary collisions with the bare potential ofcharged particles can be important.

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Collisions between charged particles

✷ Cross section σ: definition (n = number density of target particles;N = rate of events; v = particle speed.)

N = nvσ ;dN

dx= nσ .

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✷ Differential cross section:

✷ Information on collisions between charged particles (Coulomb scattering)is contained in the dependence of scattering angle on impact parameter:Rutherford cross section

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✷ Simple derivation of Rutherford cross section:

F =e1e2r2

er ; F∆P =e1e2r2

cosφ

∆P =

∫

e1e2r2

cosφdt = 2mv0 sin

(

θ

2

)

✷ Conservation of angular momentum (central force field):

L = mv0b = mr2dφ

dt; r2 =

v0b

dφ/dt

∆P =

∫

e1e2r2

cosφdt =e1e2v0b

∫

cosφdφ = 2mv0 sin

(

θ

2

)

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✷ Collecting terms:

e1e2v0b

(sinφf − sinφi) = 2mv0 sin

(

θ

2

)

φi = −(1/2)(π − θ) ; φf = (1/2)(π − θ)

sinφf − sinφi = 2 cos(θ/2)

b =e1e22E

cot(θ/2)

✷ Substituting into the general expression for σ, we obtain the Rutherfordcross section (watch the sign!!):

σ(θ) = −b

sin θ

db

dθ=

e21e22

16E2 sin4(θ/2)

✷ Note dependences on energy and scattering angle!

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✷ Large vs. small angle scattering:

• because of the dependence on scattering angle, large angle deflectionsare dominated by multiple collisions

• probability of scattering at θ ≥ 90o

σ(θ ≥ 90o) = πb2900

=πe2

1e22

4E2

• small angle deflection due to a single collision

∆θ =e1e2Eb

• mean square deflection of a particle traveling a distance L and collid-ing with plasma particles of density n2

∆θ2 = 2πn2L

∫

∆θmax

∆θmin

∆θ2σ(∆θ) sin(∆θ)d∆θ = 2πn2L

∫ bmax

bmin

∆θ2bdb

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✷ Collecting terms:

∆θ2 = 2πn2Le21e22

E2ln

(

bmax

bmin

)

• because of Debye screening, we can take bmax = λD withλD = (kT2)

1/2/(4πn2e22)1/2

• we can also take bmin = b900 = (e1e2)/(2E) ≃ (e1e2)/(3kT2)

• substituting back into the expression for ∆θ2 and introducing theCoulomb logarithm

lnΛ = ln

(

bmax

bmin

)

= ln

(

4π

3n2λ

3

D

e2e1

)

∆θ2 = 2πn2Le21e

22

E2ln Λ

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✷ Setting ∆θ2 = 1 allows estimating the length L90 to be traveled by a testparticle to undergo a large angle scattering (e.g., 90o) by multiple collisions:

L90 =E2

2πn2e21e22 ln Λ

• the cross section corresponding to such event is

σ90 =1

n2L90

=2πe2

1e22

E2ln Λ

• the last expression can be used to estimate the relative probabilityof a particle to undergo large angle scattering (e.g., 90o) by multiplecollisions rather by one single collision

σ90

σ(θ ≥ 90o)= 8 lnΛ

• In fusion plasmas, lnΛ ∼ 15÷ 20 typically. Collisions are dominatedby small angle scattering (q.e.d.)

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Collisional slowing down

✷ The present analysis of Rutherford cross section assumes that recoil of nu-cleus is negligible.

✷ As a matter of fact, the analysis is general in the center of mass. Labelingwith 1 the beam particle and 2 the target, we have

R =m1r1 +m2r2

m1 +m2

; r = r1 − r2

r1 = R +m2

m1 +m2

r ; r2 = R−m1

m1 +m2

r

✷ Noting conservation of energy and angular momentum, and denoting x thedirection of initial beam propagation and y the direction transverse to it

v1x =m1

m1 +m2

v0 +m2

m1 +m2

v0 cos θ

v1y =m2

m1 +m2

v0 sin θ

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✷ From kinematics, one can therefore obtain the relationship between scatter-ing angles in the laboratory and in the center of mass

cot θL =m1

m2

csc θ + cot θ

✷ Special cases of interest are

• scattering off heavy particles: m2 ≫ m1 ⇒ θL ≃ θ

• scattering of like particles: m2 = m1 ⇒ θL = θ/2

• scattering off light particles: m2 ≪ m1 ⇒ θL ≃ (m2/m1) sin θ

✷ Characteristic time for a particle to be deflected at 900 in the center of masssystem (with m = m1m2/(m1 +m2))

τ90 =L90

v=

(m/2)1/2E3/2

2πn2e21e22 ln Λ

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✷ Thus:

• for e − e and e− i collisions, the deflection time at 900 in the centerof mass system is a good representation for the deflection time in thelaboratory frame

• for i− i collisions, the deflection time is longer by ∼ (mi/me)1/2

• for i − e collisions, ∆θL ≃ (me/mi)∆θ, so particles must travel a∼ (mi/me) longer distance to be deflected by 900 in the laboratoryframe

τ e−e90

∼ τ e−i90

∼ (me/mi)1/2τ i−i

90∼ (me/mi)τ

i−e90

✷ In the laboratory frame, the change of momentum in x direction and ofenergy are

∆p1xp10

= −2m2

m1 +m2

sin2(θ/2) ;∆E1

E0

= −4m1m2

(m1 +m2)2sin2(θ/2)

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✷ This shows that the slowing down rates (momentum transfer) due to colli-sions satisfy

τ e−e90 ∼ τ e−e

S ∼ τ e−iS ∼ (me/mi)

1/2τ i−iS ∼ (me/mi)τ

i−eS

✷ Furthermore, the rates for energy transfer due to collisions satisfy

τ e−e90

∼ τ e−eE ∼ (me/mi)

1/2τ i−iE ∼ (me/mi)τ

e−iE ∼ (me/mi)τ

i−eE

✷ Thus, in a thermonuclear plasma:

• electrons equilibrate and reach thermal equilibrium on τ e−e90

• ions equilibrate among themselves on a ∼ (mi/me)1/2 scale

• ions and electrons reach mutual equilibrium on the longest scale∼ (mi/me)τ

e−e90

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Plasma resistivity

✷ The equation of motion for plasma electrons is

medvedt

= −eE −meve

τ e−i90

✷ Defining the current density J = −eneve, the equation of motion can bewritten as

me

nee2dJ

dt= E −

me

nee2τe−i90

J = E − ηJ

✷ The plasma resistivity is defined as

η =me

nee2τe−i90

=8πm

1/2e Ze2 ln Λ

(3kT )3/2

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✷ Importance of ion-ion collisions vs. fu-sion reactions in fusion plasmas:

✷ The figure illustrates the relative im-portance of ion-ion collisions by com-paring the fusion and Coulomb scat-tering cross sections.

✷ An ion suffers a large number of colli-sions before it undergoes fusion.

✷ Effects of collisions are quite impor-tant and need to be taken into ac-count.

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Fusion reactor scheme

✷ Fusion reactions of primary interest

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✷ Fusion reaction cross sections (reaction rate 〈σv〉)

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✷ The need of fusion fuel: D exists at 0.0156% of all H in nature.

✷ On the contrary, T is radioactive with a half time of 12.32yr; thus it mustbe produced.

✷ The n produced in D − T fusion reactions allows to close the fuel cyclethrough

n+6Li → T +4He ; (σ ∼ 950b)

n+7Li → T +4He+ n′ ; (σ<∼ 1b)

6Li occurs at about 7.6% of natural lithium.

✷ Need for a breeding blanket.

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✷ Magnetic confinement

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✷ Magnetic confinement in toroidal geometry

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✷ Schematic view of a fusion reactor

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✷ Artistic view of a fusion reactor

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Power balance

✷ The global power balance of a fusion reactor at steady state can be writtenas:

Pα + Padd = PL + PR

• Pα is the (volume averaged) nuclear heating power from alpha parti-cles; for a 50% D-T mixture

Pα =n2i

4〈σv〉Eα

• Padd is the (volume averaged) supplemental or additional heating thatneeds to be provided externally in order to keep the reactor in steadystate

• PL is the power loss due to plasma transport (lecture 2)

• PR is the power loss due to plasma radiation (lecture 2)

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Lawson Criterion

✷ It is possible to globally take into account the power losses from a fusionplasma through the energy confinement time:

PL + PR ≡(3/2)(neTe + niTi)

τE; τE ≡

(3/2)(neTe + niTi)

PL + PR

≃3nT

PL + PR

✷ The energy produced in a D − T fusion reaction consists of the neutronenergy, En = 14.06 MeV, and of the α-particle energy, Eα = 3.52 MeV.Note that the energy produced in a fusion reaction, Ef = (En+Eα) ≃ 5Eα.

✷ Breakeven and Ignition conditions are defined comparing Pf and Pα withPR + PL.

• Breakeven condition: Pf = PR + PL

• Ignition condition: Pα = PR + PL

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✷ Generally, only a fraction of the Pf can be reused to balance the losses:

PL + PR =3nT

τE< ηPf = η

n2

4〈σv〉Ef ; nτE >

12T

η 〈σv〉 (T )Ef

✷ It is common to introduce the power gain factor, Q = Pf/Padd = Pf/(PR +PL − Pα). Thus, Q = 1/(η − 0.2) and η = (Q+ 5)/(5Q).

✷ These values are also related to the fraction of Pα over the total plasmaheating Pα + Padd: Pα/(Pα + Padd) = Q/(Q+ 5) = 1/(5η).Typical values of η:

• Breakeven condition: η = 1, Q = 1.25 (sometimes Q = 1)

• Ignition condition: η = 0.2, Q → ∞

• Lawson criterion: η ≃ 0.3, Q ≃ 10

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✷ Lawson criterion and breakeven and ignition conditions

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✷ Relative progress of triple product nTτE as figure of merit

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Ideal ignition temperature

✷ The ignition condition, again, is Pα = PR +PL. It is instructive to comparethe radiation loss PR (mostly bremsstrahlung; see lecture 2) with Pα.

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✷ At low temperature, radiation lossesovercome Pα, so ignition cannot befeasible. Thus, there exists an idealignition temperature defined as thevalue of T at which Pα can balanceradiation losses (to bremsstrahlung).

✷ Since radiation losses depend on thecomposition of the fusion fuel (con-centration of impurities), it is possibleto define a maximum impurity con-centration, NZ/NDT , as a function ofthe atomic number Z above which ig-nition becomes impossible (solid anddashed curves refer to actual and ap-proximated expressions of radiationloss by impurities).

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Exercises

✷ Consider the Poisson’s equation

(

−∇2 +1

λ2

D

)

Φ = 4πqδ (r − r0) .

Demonstrate that the solution is Φ = (q/r) exp (−r/λD), with r = |r− r0|.

✷ Repeat the derivation of the Rutherford cross section (see pp. 12-13) usingthe formulation in the center of mass frame.

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Documents

INTRODUCTION TO PLASMA PHYSICS Fulvio ZoncaThe Debye length is not a characteristic feature of a plasma but is common to any ionized system in thermal equilibrium. At thermal equilibrium