Debye Length

Adrian Down

January 22, 2007

1 Particle number density

1.1 Review

Previously, we used Boltzmann’s law to express the number density n of aspecies of particle in thermal equilibrium subject to a potential φ,

n(r) = n0e− qφ(r)

kT

In the quasi-nuetral situtaion, the number of ions ni and electrons ne areapproximately equal, ni ≈ ne ≈ n0. However, the distribution of thesedifferent kinds of particles are not necessarily unifrom in space in the quasi-neutral situtation, and so charge gradients and fields can still exist withinthe plasma.

1.2 Example: Gaussian potential

Consider a Gaussian potential φ acting on a population of electrons. In thiscase, the charge is q = −e. The distribution of electrons is thus,

ne(r) = n0e+ eφ

kT

The sign of the exponent indicates that there is an enhancement of the num-ber of electrons in the region of the potential. This can be seen physicallyfrom the electric field, which is the gradient of the potential. In all space,the electric field points outwards from the peak of the potential, so the elec-trons, having negative charge, are drawn towards the potential, opposite ofthe electric field.

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2 Derivation and definition of the Debye length

2.1 Poisson’s equation

The previous example is not very physical. As soon as the distribution startsto evolve, the change in the density n changes the potential φ. To find thepotential, we need Poisson’s equation,

φ · E =ρ

ε0

E = −∇φ⇒ −∇2φ =ρ

ε0

ρ is the charge density, which must take into account all species of particle.Assuming the charge of the electrons and ions to be equal and opposite,

ρ =∑

s

nsqs = nie− nee

If ne and ni are equal and uniform, then there is complete cancellation. Ingeneral though, this is not the case.

Note. In an inhomogenious medium, a tensor must be used to represent theproperties of the dielectric,

∇ · ←→ε · E︸ ︷︷ ︸D

=ρ

ε0

2.2 Introduction of a test charge

We can consider a test change s placed in the plasma. Using the expressionsfor the number density derived previously to express the number density,

ρ = nie− nee + s

= en0

(e− eφ

kTi − eeφ

kTe

)+ s

Note. We allow for the case that Te 6= Ti.

In the one-dimensional case, Poisson’s equation with the inclusion of thetest charge is,

−∂2φ(x)

∂x2=

n0e

ε0

[e− eφ(x)

kTi − eeφ(x)kTe

]+ s

2

2.3 Linearization

Finding an analytic solution to this equation is nontrivial. The physicalsolution is to make an approximation and solve the equation in the resultinglimit. To linearize the equation, make the approximation that eφ

kT� 1, so

that the exponents can be expanded. This approximation corresponds tokT � eφ, meaning that most particles in the plasma are “free-streaming”,unaffected by the potential φ.

For small x, the exponential can be Taylor expanded,

ex ≈ 1 + x x� 1

⇒ e− eφ

kTi ≈ 1− eφ

kTi

eeφ

kTe ≈ 1 +eφ

kTe

The linearize form of Poisson’s equation is,

−∂2φ

∂x2=

n0e

ε0

[1− eφ

kTi

− 1− eφ

kTe

]+ s

=n0e

2

ε0

(1

kTi

+1

kTi

)φ + s

The coefficient of the φ term can be considered a constant. By comparingwith the left side, it must have units of length squared. The Debye length isdefined as the square root of this constant. We can simplify the expressionfurther by defining an effective temperature,

1

Teff

=1

Te

+1

Ti

The Debye length is then,

Definition (Debye length λD).

λD =

√ε0kTeff

n0e2

Note. In convenient units, the Debye length can be written,

λD = 7.4

√T (eV)

n (cm−3)m

3

3 Interpretation of λD: shielding

3.1 Introduction

Shielding in a plasma is often due primarily to the electrons in the plasma.Consider electrons impinging on a surface. The flux of these electrons isφ = nvth. The thermal velocity can be approximated by kinetic theory,modulo a constant factor,

kT ∼ m⟨v2

⟩⇒ vth ∼

√kT

m

Because of the dependance on the mass, the thermal velocity of the heav-ier ions is often much less than that of the electrons. Hence if there is a wallin a container, the flux of electrons onto that wall will often be much higherthan that of ions. The accumulation of negative charge will form a sheath.This effect is known as sheath theory, and it is presented in chapter 8 of thetext.

3.2 Differential equations

3.2.1 Possion’s equation

Poisson’s equation governs the formation and evolution of such a sheath ofcharge,

∂2φ

∂x2=

1

λ2D

φ + s

The presence of the constant factor s does not change that the solution tothis equation is exponential,

φ(x) = Ae|x|λD + Be

−|x|λD

where A and B are normalization constants. As a boundary condition, werequire that φ→ 0 as x→∞, so A = 0.

3.2.2 Electric field

We can obtain an expression for the potential φ from the electric field E. Todo so, we assume that the test charge is actually a sheet that is infinite in

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two demensions, so that s = σs(x), where σ is a charge density. The electricfield accross this sheet of charge can then be obtained by integrating over aGaussian pillbox. ∫

E · dA =Qenc

ε0

E(O+)− E(O−) =σ

ε0

⇒ 2E =σ

ε0

The constant B in the expression for φ is then fixed by the relationshipbetween E and φ. Taking x = 0, which corresponds to the location of thesheet of charge,

E = −∂φ

∂x

⇒ σ

2ε0

=B

λD

e0

⇒ B =σλD

2ε0

⇒ φ(x) =σλD

2ε0

e− |x|

λD

In position space, φ(x) decays exponentially away from the sheet ofcharge. The scale length of the exponential decay is λD. Hence the De-bye length is the scale length to which a charge in the plasma is shielded.

Note. • λD is inversely proportional to n0, and hence the shielding lengthdecreases as the density increases. This results because there are moreelectrons to cancel an existing change distribution, and hence a shorterlength scale to reach neutrality in the plasma, as the density of theplasma increases.

• It is left as an exercise to derive the full three-dimensional case. Theanswer is,

φ(r) =q

re− r

λD

This is often called the Yukawa potential in nuclear physics.

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3.3 Collective behavior

Shielding is the first example we have seen of collective behavior in a plasma.If something is introduced to the plasma system to perturb it, the particles ofthe plasma respond and rearrange themselves to maintain charge neutrality.Recall that exhibitng collective behavior was one of the characteristics listedin the definition of a plasma.

All of the previous work was based on the assumption that the Boltzmanndistribution is applicable. The system cannot exhibit collective behaviorif the number of particles in the system is too small and the Boltzmannstatistics break down. There are a few order of magnitude estimates that canbe used to determine when a plasma can be expected to exhibit collectivebehavior.

•

ND = n0

(4

3πλ3

D

)� 1

ND is representative of the number of particles in a sphere with radiusequal to the Debye length.

•

λD � L

where L is the characteristic scale of the system.

Note. The Debye length is the smallest natural scale in the plasma.This is because every particle in the plasma is effectively shieldingevery other plasma on the Debye scale.

•

ωτ � 1

ω is the characteristic frequency of the system, and τ is the time be-tween collisions. The meaning of this will become more clear later inthe course.

Some examples of plasma parameters are in table 1.

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n (cm−3) T (eV) B (G)Interstellar space 1 1 10−5

Solar wind 10 5 10−4

Ionosphere 105 0.1 0.5Corona 106 100 5Fusion 1011 104 104

Stellar interior 1027 103 unknown

Table 1: Examples of plasma parameters

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