Plasma Physics Concepts in Beams

Pat Colestock

Los Alamos National Lab

[Goals for these lectures]{.underline}:

Familiarize you with the concepts and terminology of plasma physics

Describe specific plasma applications in accelerator science

Develop some new accelerator concepts based on similar phenomena in
plasmas

Course emphasis will be [conceptual]{.underline}

[Some prerequisites:]{.underline}

Basic understanding of electromagnetics

Familiarity with basic concepts of circular and linear accelerators

Mathematical dexterity

Syllabus of the Course

Lecture #1

Basic physical parameters

Debye shielding

Plasma oscillations

Sheaths

Plasma sources

DC discharges

Lecture #2

Distribution Function

Vlasov/Boltzmann equation

Solution methods

Moments

Application

Beam distributions

Lecture #3

Diffusion

Collisions

Dielectric function

Beam transport

Space charge forces

Lecture #4

Wave propagation fundamentals

Linear plasma waves

Analogues in storage rings

Analogues in linacs

Lecture #5

Three-wave coupling

Quasilinear diffusion

Application to storage rings

Lecture #6

Nonlinear Landau damping

Solitons

KDV and NLS equations

Experiments

Lecture #7

Echoes

Lecture #8

Intense beam transport

Halo generation

Plasma acceleration

Lecture #9

Thermal fluctuations

Weak turbulence

Strong turbulence

What is a plasma ?

A plasma is essentially a gas of charged particles. Typically it is a
gas where the interparticle electric potential energy is smaller than
the average particle kinetic energy.

Plasmas are commonly electrically neutral, but not always. Particle
beams are examples of non-neutral plasmas.

A key feature of plasmas is that the interparticle interaction exhibits
a long-range collective behavior that is very different from that
observed in uncharged gases. The term plasma physics is synonymous with
the study of this collective behavior, both linear and nonlinear.

![](media/image1.jpeg){width="5.990972222222222in"
height="6.868055555555555in"}

Ref: Rutherford and Goldston

A plasma is in essence an ensemble of particles exhibiting [collective
behavior]{.underline}.

A fundamental concept in describing such ensembles is the [distribution
function]{.underline}, used because it is usually impossible to describe
the behavior of individual particles in this many-body system.

f(x,v) = probability density that particles exist between
[x]{.underline}, [v]{.underline} and [x]{.underline}+d[x]{.underline},
[v]{.underline}+d[v]{.underline} in *six-dimensional phase space*

Maxwellian distribution  most probable distribution

$f(v) = n_{o}\left( \frac{m}{2\pi\text{kT}} \right)^{\frac{1}{2}}\text{exp}\left( - \frac{\text{mv}^{2}}{2\text{kT}} \right)$
(in one dimension)

Where does this come from?

We start with the thermodynamic [definition]{.underline} of temperature,
defined for systems in equilibrium

> $$\frac{1}{T} = \frac{\text{kd}\text{ln}\Omega}{\text{dW}}$$

where *k* is Boltzmann\'s constant, *k* ln Ω = S, entropy, and where Ω
is the number of microstates available for a system with total energy W.

Now we pose the question what is the probability that a given particle
has an energy W~i~ ? In a system in equilibrium, this particle obtains a
given energy by removing this amount from the rest of the system.

Thus, the total system energy becomes W~tot~ - W~i~ and we can expand
Ω  as follows

$\text{ln}\Omega_{W_{\text{total}} - W_{i}} \simeq \text{ln}\Omega_{\text{total}} - \frac{\partial\text{ln}\Omega}{\partial W}W_{i}$

Taking the exponential of this expression gives us the number of
microstates available for a particle of energy W~I~ Thus the probability
for a particle to be found in this state is

> $$\frac{\Omega_{\text{tot} - W_{i}}}{\Omega_{\text{tot}}} = \text{exp}\left\{ - \frac{W_{i}}{\text{kT}} \right\}$$

We ignore, for now, any potential energy the particle may have and write
the probability that a particle can be found in a velocity space volume
$\text{dv}_{x}\text{dv}_{y}\text{dv}_{z}$ as

$P_{o}\text{exp}\left\{ - \frac{m\left( v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \right)}{2\text{kT}} \right\}\text{dv}_{x}\text{dv}_{y}\text{dv}_{z}$

Note: Implicit in the above result is the fact that we have assumed the
plasma to be spatially uniform, and all particles to be
indistinguishable. Not true, of course, in general.

We further adopt the normalization that

$n_{o} = \int_{}^{}{\text{fdv}_{x}\text{dv}_{y}\text{dv}_{z}}$

which leads to the so-called [Maxwell-Boltzmann]{.underline}
distribution (in 3-D)

$f_{\text{MB}} = \frac{n_{o}}{\left( 2\pi\frac{\text{kT}}{m} \right)^{\frac{3}{2}}}\text{exp}\left\{ - \frac{m\left( v_{x}^{2} + v_{y}^{2} + v_{z}^{2} \right)}{2\text{kT}} \right\}$

This distribution is the fundamental distribution function for a plasma
in equilibrium, i.e. at a minimum of [free energy]{.underline}. In these
lectures, we will be concerned with small (or sometimes large)
deviations of a plasma from this equilibrium state.

Also in the future, we will relax the condition that the plasma is
spatially uniform, and use a distribution function that is a function of
both velocity and configuration space.

$f = f\left( x,y,z,v_{x},v_{y},v_{z} \right)$

This situation is frequently encountered in beams where *f* is a
function on [phase space]{.underline}. This implies the equilibrium
discussed above can be evaluated in a local sense.

We also note that a high-energy beam also typically has a Gaussian (or
Maxwell-Boltzmann) distribution in the particles\' rest frame. This is
only true when the potential energy part of W is small compared to the
kinetic part. In addition, it requires that the [thermalization
time]{.underline}, sometimes referred to as the [filamentation
time]{.underline}, is short with respect to the beam lifetime.
Thermalization can occur as a result of true random process, like
collisional scattering or radiation, as well as due to a number of
nonlinear collective

processes.

Debye Shielding

A plasma in equilibrium is electrically neutral and isothermal. Now
consider the perturbation of this equilibrium that occurs when a test
charge is introduced, forming a potential φ. Thus the particle energy is
given by

$W_{i} = \frac{1}{2}\text{mv}_{i}^{2} + q\varphi$

and integrating the distribution function over all velocity space to
find the density of both ions and electrons gives

$n_{e} = \int_{}^{}{f_{e}d^{3}v = n_{e0}\text{exp}\left\{ \frac{e\varphi}{\text{kT}_{e}} \right\}}$

$n_{i} = \int_{}^{}{f_{i}d^{3}v = \frac{1}{Z}n_{e0}\text{exp}\left\{ - \frac{e\varphi Z}{\text{kT}_{i}} \right\}}$

This implies that electrons will tend to gather near positive charges,
and vice versa in equilibrium, which will lead to shielding of the
original test charge.

For purposes of illustration, we assume a one-dimensional plasma and
solve for the potential distribution using Poisson\'s equation

$$$\frac{d^{2}\varphi}{\text{dx}^{2}} = \frac{e}{\varepsilon_{o}}\left( n_{e} - \text{Zn}_{i} \right) = \frac{\text{en}_{e0}}{\varepsilon_{o}}\left( \text{exp}\left\{ \frac{e\varphi}{\text{kT}_{e}} \right\} - \text{exp}\left\{ - \frac{e\varphi}{\text{kT}_{i}} \right\} \right)$

We now invoke the assumption that the potential energy is much smaller
than the kinetic energy

$e\varphi\text{<<}\text{kT}_{e}$

and take T~i~ \~ 0, which allows us to expand the exponentials

$\frac{d^{2}\varphi}{\text{dx}^{2}} \simeq \frac{\text{en}_{e0}}{\varepsilon_{o}}\left( \frac{e\varphi}{\text{kT}_{e}} \right)$

and this equation has the solution

$\varphi = \varphi_{o}\text{exp}\left\{ - \frac{x}{\lambda_{D}} \right\}$

where

$\lambda_{D} = \left( \frac{\varepsilon_{o}\text{kT}_{e}}{n_{\text{eo}}e^{2}} \right)^{\frac{1}{2}}$

$\lambda_{D}$is the Debye length and represents the characteristic scale
over which the potential from a test charge can exist in the plasma.

Another implicit assumption in the preceding derivation is that the
density in the region of nonzero potential is a smooth function of
space. This means there must be a sufficient number of charges within a
[Debye sphere]{.underline} so that the \'graininess\' of the potential
is a negligible effect. This is a typical feature of a neutral plasma,
and is part of the definition of the plasma state.

However, in a beam, or nonneutral plasma, this is not always true,
though usually it is.

The number of particles in a Debye sphere is given approximately by

$\Gamma = n_{e}\lambda_{D}^{3}$

and $\frac{1}{\Gamma}$is known as the [plasma parameter]{.underline}. We
typically want Γ \>\> 1.

For a typical laboratory plasma, T~e~ \~ 3 eV,

n~e~ = 10^19^ m^-3^,mwith the number of particles in a Debye sphere \~
10^3^, adequate for our purposes.

Plasma Oscillations

To understand the oscillatory behavior of a plasma, we consider a slab
of uniform, neutral plasma composed of equal numbers of electrons and
singly-charged ions

> ![](media/image2.jpeg){width="2.297222222222222in"
> height="1.8430555555555554in"}

For the purposes of this discussion, the ions are taken to be immobile
due to their mass, and we displace (somehow) all the electrons by a
distance ∆x.

Using Gauss\' Law, we may write the resulting 1-D electric field as

$\int_{}^{}{\text{EdA} = \text{EA} = \frac{\rho A\Delta x}{\varepsilon_{o}}} = \frac{\text{qn}_{e}}{\varepsilon_{o}}A\Delta x$

 $E = \frac{\text{qn}_{e}}{\varepsilon_{o}}\Delta x$

The force on an individual electron is -qE and the electron\'s motion
follows from Newton\'s Law

$m_{e}\Delta\overset{¨}{x} = - \frac{q^{2}n_{e}}{\varepsilon_{o}}\Delta x$

The resulting oscillatory behavior occurs at the [plasma
frequency]{.underline} given by

$\omega_{\text{pe}}^{2} = \frac{q^{2}n_{e}}{m_{e}\varepsilon_{o}}$

$\omega_{\text{pe}}^{2} = 3\text{.}\text{174} \times \text{10}^{9}n_{e}\left( \text{cm}^{- 3} \right)\left( \text{Hz}^{2} \right)$

$f_{\text{pe}} = 8\text{.}\text{97} \times \text{10}^{3}\sqrt{n_{e}\left( \text{cm}^{- 3} \right)}\left( \text{Hz} \right)$

This frequency is at the heart of the collective behavior of a plasma,
and forms the basis for a rich variety of wave phenomena, linear and
nonlinear.

We note that the Debye length can be expressed in terms of the plasma
frequency as

$\lambda_{D} = \frac{v_{\text{te}}}{\omega_{\text{pe}}}$

where v~te~ is the mean thermal speed
$\left( \frac{\text{kT}_{e}}{m_{e}} \right)^{\frac{1}{2}}$.

This suggests the rather intuitive picture that the Debye length is the
distance over which an average particle can travel before the plasma can
respond with shielding.

In a beam, a similar frequency can be defined, however there is some
ambiguity due to relativistic effects, and the exact definitions depend
on the particular application. If we define all quantities in the lab
frame, the effective density in the moving frame is lower by a factor
$\frac{1}{\gamma}$. However, the electron\'s mass is also lower
by$\frac{1}{\gamma}$ resulting in an apparent cancellation, i.e. the
plasma frequency appears to be the same in both the moving frame and the
lab frame. But the analysis must be applied differently to plasma
oscillations parallel or perpendicular to the motion.

The appropriate formulation is to transform the time, length scale and
fields according to the Lorentz transformation from one frame to
another, resulting in two different expressions for parallel and
perpendicular directions.

For transverse motion we have

$\overset{¨}{x} = \omega_{\text{pe}}^{2}x = \frac{F}{\gamma m}$

The force term is made up of a Coulomb repulsion that is reduced by
magnetic attraction thus

$F = \text{qE}\left( 1 - \frac{1}{\beta^{2}} \right) = \frac{\text{qE}}{\gamma^{2}}$

which yields

$\omega_{p}^{2} = \frac{q^{2}n}{{m\varepsilon}_{o}\gamma^{3}}$

The same result may be obtained by transforming the plasma frequency
from the beam frame to the lab frame.

Now for the longitudinal motion, the reduction of the Coulomb force does
not occur; however, the longitudinal motion involves a different
effective mass.

This can be seen by

$\frac{dp}{\text{dt}} = m\frac{d}{\text{dt}}(\gamma v) = F$

$\frac{dv}{\text{dt}} = \frac{1}{\gamma m}\left( F - \lbrack F \cdot \beta\rbrack\beta \right)$
; $\beta = \frac{v}{c}$

The effective longitudinal mass is seen to be ${m\gamma}^{3}$.

We may define a temperature using the invariance of the normalized
transverse emittance (we will see this later)

$\varepsilon_{N} = \gamma\frac{x_{\text{RMS}}v_{\text{RMS}}}{c} = \text{const}\text{.}$

Let a transverse temperature be *defined* by
$v_{\text{RMS}} = \left( \frac{\text{kT}_{m}}{m} \right)^{\frac{1}{2}}$

We then find the temperature scales as

$T_{\text{lab}} = \frac{T_{\text{beam}}}{\gamma}$

Assuming $v_{}\text{<<}c$, i.e. the transverse motion is
non-relativistic, we may construct the effective Debye length

$\lambda_{D} = \frac{v_{}}{\omega_{p}} = \left( \frac{\varepsilon_{o}\gamma^{3}v_{}^{2}}{q^{2}n}m \right)^{\frac{1}{2}} = \left( \frac{\varepsilon_{o}\gamma^{2}\text{kT}_{\text{lab}}}{q^{2}n} \right)^{\frac{1}{2}} = \left( \frac{\varepsilon_{o}\gamma\text{kT}_{\text{beam}}}{q^{2}n} \right)^{\frac{1}{2}}$

We note that a relativistic beam is typically very \'non-Maxwellian\',
meaning very different temperatures exist in the perpendicular and
parallel directions. And the Debye lengths for relativistic beams are
usually much greater than the beams themselves. So do we still have a
plasma? Yes, but not a conventional one. More on this topic later.

Plasma Sheaths

A consequence of Debye shielding is the formation of a boundary layer,
called a sheath, at the interface between the plasma and any surface,
conducting or non-conducting.

![](media/image3.jpeg){width="4.997916666666667in"
height="2.3847222222222224in"}

At any boundary, the electrons typically leave the plasma faster than
the ions, leaving a net positive charge behind. This builds up a
retarding field which tends to confine electrons and expel ions. The
field builds up until an equilibrium in the particle fluxes is reached.

An important consequence of a sheath is the fact that the energy of ions
leaving a plasma is typically characteristic of the mean electron energy
in the plasma.

![](media/image4.jpeg){width="3.8493055555555555in"
height="2.964583333333333in"}

> The potential variation across a sheath.

We wish to solve for the self-consistent drift velocity that the ions
must acquire. By energy conservation, we have

$\frac{1}{2}m_{i}v_{i}^{2} = \frac{1}{2}m_{i}v_{o}^{2} - e\varphi(x)$

where v~o~ is the ion drift velocity in the plasma we wish to find.

We now assume that the electrons are in [local thermodynamic
equilibrium]{.underline}. Remember this means that

$n_{e} = n_{o}\text{exp}\left\{ \frac{e\varphi}{\text{kT}_{e}} \right\}$

Now the continuity relation gives us an expression for the local ion
density

$n_{o}v_{o} = n_{i}v_{i}$

Poisson\'s equation then becomes

$\frac{d^{2}}{\text{dx}^{2}}\varphi = \frac{e}{\varepsilon_{o}}\left( n_{e} - n_{i} \right) = \text{en}_{o}\left( \text{exp}\left\{ \frac{e\varphi}{\text{kT}_{e}} \right\} - \left\lbrack 1 - \frac{2e\varphi}{m_{i}v_{o}^{2}} \right\rbrack^{\frac{1}{2}} \right)$

This is difficult to solve directly, but a consistency relation can be
found by multiplying by $\frac{d\varphi}{\text{dx}}$ and integrating
once

$\varphi^{} - \varphi_{o}^{} = \text{en}_{o}\left( \frac{\text{exp}\left\lbrack \frac{e\varphi}{\text{kT}_{e}} \right\rbrack - 1}{\frac{e}{\text{kT}_{e}}} + \frac{\sqrt{1 - \frac{2e\varphi}{m_{i}v_{o}^{2}}} - 1}{\frac{e}{m_{i}v_{o}^{2}}} \right)$

Now assuming that $\frac{e\varphi}{\text{kT}_{e}}\text{<<}1$ and
expanding the right hand side in a Taylor series leads to the
consistency relation known as the [Bohm sheath criterion]{.underline}

$m_{i}v_{o}^{2} > \text{kT}_{e}$

The implication of this criterion is that ions must drift into the
sheath region faster than the ion sound speed.

And whenever particles drift faster than the speed of some wave in a
medium, this can lead to instability, or collective oscillations. Hence,
sheaths can be a source of instabilities

Child-Langmuir Law

Another aspect of the sheath is the retardation of electron flow due to
the build-up of space charge.

An expression for this effect can be found by considering the motion in
1-D

$\frac{m_{e}}{2}v_{e}^{2} = e\varphi(x)$

$J_{x} = n_{e}v_{e} = \text{const}\text{.}$

along with Poisson\'s equation for the potential

$\frac{d^{2}\varphi}{\text{dx}^{2}} = - \frac{n_{e}e}{\varepsilon_{o}}$

Substituting for the density n~e~

From the continuity relation, we find

$\frac{d^{2}\varphi}{\text{dx}^{2}} = \frac{J_{x}}{e\left( \frac{2e}{m_{e}} \right)^{\frac{1}{2}}}\frac{1}{\varphi^{\frac{1}{2}}}$

This equation can be integrated once to give

$\left( \frac{d\varphi}{\text{dx}} \right)^{2} = \frac{4J_{x}}{\varepsilon_{o}\left( \frac{2e}{m_{e}} \right)^{\frac{1}{2}}} + \text{const}$

Now using the boundary conditions that

$\varphi = 0$ at x = 0

$\frac{d\varphi}{\text{dx}} = 0$ at x = 0

this equation can be integrated again to give

$\varphi^{\frac{3}{4}} = \frac{3}{2}\left( \frac{J_{x}}{\varepsilon_{o}} \right)^{\frac{1}{2}}\left( \frac{2e}{m_{e}} \right)^{\frac{- 1}{4}}x$

For an applied voltage V~o~ this relation implies the potential
$\varphi$ can be replaced by the expression

$\varphi(x) = V_{o}\left( \frac{x}{d} \right)^{\frac{4}{3}}$

Substituting this expression for $\varphi$ leads to the final result
known as the [Child-Langmuir law]{.underline}.

It describes the flow of electron current across a sheath, and is the
primary scaling law at work in vacuum tubes, among many other
situations.

$J = \frac{4}{9}\varepsilon_{o}\left( \frac{2e}{m_{e}} \right)^{\frac{1}{2}}\frac{V_{o}^{\frac{3}{2}}}{d^{2}}$

Example: Langmuir Probes

![](media/image5.jpeg){width="3.4493055555555556in"
height="2.6131944444444444in"}

If an external bias is applied to a wire conductor inserted into a
plasma, either ion or electrons can be collected, depending on the
polarity.

![](media/image6.jpeg){width="4.25in" height="2.71875in"}

If the plasma is largely unperturbed by the probe (i.e. it doesn\'t
collect too much current), then each particle species has a particle
flux given by

$\Gamma_{\alpha} = \frac{1}{4}n_{\alpha}\left( \frac{\text{kT}_{\alpha}}{m_{\alpha}} \right)^{\frac{1}{2}}$

so that for a probe of area A, the collected current is

$I = - \text{eA}(\frac{1}{4}n_{i}\left\lbrack \frac{\text{kT}_{i}}{m_{i}} \right\rbrack^{\frac{1}{2}} - \frac{1}{4}n_{e}\left\lbrack \frac{\text{kT}_{e}}{m_{e}} \right\rbrack^{\frac{1}{2}})$

However, due to Debye shielding, the plasma is strongly perturbed in the
vicinity of the probe and particles must fall across a sheath potential
in the manner we have just studied.

The electron density varies according to the local equilibrium condition

$n_{e} = n_{e0}\text{exp}\left\{ \frac{\text{eV}}{\text{kT}_{e}} \right\}$

Moreover, the ion flux is an invariant across any surface (in this 1-D
problem) so we can calculate the ion flux to the probe at any convenient
place, and we choose to do so at the edge of the sheath.

At this point the plasma is quasi-neutral,

$n_{e}|_{\text{sheath}} \simeq n_{i}|_{\text{sheath}}$

$I = \text{eAn}_{i}v_{i} = \text{aAn}_{e0}\text{exp}\left\{ \frac{\text{eV}}{\text{kT}_{e}} \right\} v_{i}$

But for typically low temperature ion plasmas, the ion velocity is given
by only that which it acquires as it falls across the sheath,

$v_{i} = \left( \frac{- 2\text{eV}}{m_{i}} \right)^{\frac{1}{2}}$

Using the Bohm sheath criterion (at equality) we have

$V_{\text{sheath}} = - \frac{\text{kT}_{e}}{2e}$

![](media/image7.jpeg){width="3.3534722222222224in"
height="2.2909722222222224in"}

So finally, we get for the collected current

$I_{i} = e^{\frac{- 1}{2}}\text{An}_{e0}\left( \frac{\text{kT}_{e}}{m_{i}} \right)^{\frac{1}{2}}$

which is called the [ion saturation current]{.underline}, namely the
current collected at large negative voltages when all the electrons are
repelled.

Note that the ion current depends on the electron temperature, which is
due to the collective character of the ion and electron flows. The above
expression can be used to deduce the electron density, if the electron
temperature is known.

The total current to the probe is given by

$I = n_{e0}\text{eA}\left( \frac{\text{kT}_{e}}{m_{i}} \right)^{\frac{1}{2}}\left( \left\lbrack \frac{m_{i}}{m_{e}} \right\rbrack^{\frac{1}{2}}\text{exp}\left\{ \frac{\text{eV}}{\text{kT}_{e}} \right\} - 1 \right)$

We note that there is also a geometric correction due to the fact that
the sheath thickness changes slightly with applied voltage. For a
cylindrical probe, for instance, this increase is given by

$A = A_{o}\left( 1 + \frac{x_{s}}{a} \right)$

$\frac{x_{s}}{\lambda_{D}} = \frac{2}{3}(2e)^{\frac{1}{4}}\left( \left\lbrack - \frac{\text{eV}}{\text{kT}_{e}} \right\rbrack^{\frac{1}{2}} - \frac{\sqrt{2}}{2} \right)^{\frac{1}{2}}\left( \left\lbrack - \frac{\text{eV}}{\text{kT}_{e}} \right\rbrack^{\frac{1}{2}} + \frac{\sqrt{2}}{2} \right)^{\frac{1}{2}}$

and is related to the Child-Langmuir law in cylindrical geometry, and in
this case, the ions are the charge carriers.

Plotting the V-I characteristic of the probe, the logarithmic slope can
be used to find T~e~, and once found, then the ion saturation current
gives n~e~.

There is no easy way to find T~I~ since the ion velocities are
determined mainly by the electron supported sheath. Hence the ion
temperature drops out of the equation. This is not the cast when the ion
temperatures are comparable to the electron temperature, but this case
is rare experimentally, and the equations must then be solved
numerically.

How do we make a plasma?

![](media/image8.jpeg){width="4.649305555555555in"
height="2.0506944444444444in"}

Particle balance:

$\frac{\text{dn}_{\alpha}}{\text{dt}} = S_{i} + \frac{n_{\alpha}}{\tau_{\text{wall}}} - \frac{n_{\alpha}}{\tau_{\text{convection}}} - \frac{n_{\alpha}}{\tau_{\text{diffusion}}}$

Energy balance:

$\frac{\text{dn}_{\alpha}\text{kT}_{\alpha}}{\text{dt}} = P_{\alpha}^{\text{in}} - P_{\alpha}^{\text{ionization}} - P_{\alpha}^{\text{radiation}} - P_{\alpha}^{\text{convection}} - P_{\alpha}^{\text{diffusion}}$

We will now take a look at some of the mechanisms that can produce or
destroy particles in the bulk plasma and at the plasma surface.

Ionization:

Electron impact

$e^{-} + A \rightarrow 2e^{-} + A^{+}$

Ion Ionization potential

H 13.6 eV

He 24.6

Ne 21.6

Ar 15.8

![](media/image9.jpeg){width="4.999305555555556in"
height="3.0388888888888888in"}

$\left\langle \sigma v \right\rangle_{\text{ionization}} \simeq \frac{2\text{.}5 \times \text{10}^{- 6}}{u^{\frac{3}{2}}}\frac{\eta\left( \frac{T}{u} \right)^{\frac{1}{2}}}{1 + \frac{T}{u}}e^{- \frac{u}{T}}\left( \frac{\text{cm}^{3}}{\text{sec}} \right)$

η = \# of equivalent electrons

u = ionization potential

T = electron temperature (eV)

Dissociative ionization

$e^{-} + \text{AB} \rightarrow 2e^{-} + A^{+} + B$

$e^{-} + \text{AB} \rightarrow e^{-} + A + B$

$e^{-} + \text{AB} \rightarrow e^{-} + \text{AB}^{} \rightarrow e^{-} + A + B + h\nu$

$\sigma \leq \text{10}^{- \text{17}}\text{cm}^{2}$ at 100 eV

Photoionization

$h\nu + A \rightarrow A^{+} + A$

> $\sigma \approx \text{10}^{- \text{17}}\text{cm}^{2}$ at 1000
> $$$\overset{0}{A}$

Charge exchange

$A + A^{+} \rightarrow A^{+} + A$

This does not produce an change in the ion density, so why does it
matter? \-\--\> Energy balance

Example:

Plasma creation with particle beams

A plasma can be readily created in and around a particle beam
propagating through a neutral gas.

The plasma density increases as

$\frac{\text{dn}_{\alpha}}{\text{dt}} = n_{b}n_{g}\sigma_{\text{ionization}}v_{o}$

where n~b~ is the beam density, n~g~ is the gas density,
$\sigma_{\text{ionization}}$is the cross-section for ionization and v~o~
is the beam velocity.

As the gas becomes ionized, like-signed particles are repelled and
opposite-signed particles become trapped in the beam\'s potential well.
The characteristic time for a beam to become neutralized by ionization
traversing a gas-filled region is given by

$\tau_{\text{neut}} = \frac{1}{n_{g}\sigma_{\text{ionization}}v_{o}}$

with a characteristic length

$L_{\text{neut}} = \frac{1}{n_{g}\sigma_{\text{ionization}}}$

![](media/image10.jpeg){width="5.997222222222222in"
height="3.6534722222222222in"}

Ref: Reiser

> ![](media/image11.jpeg){width="4.197222222222222in"
> height="3.497916666666667in"}

As the beam enters the tube, the potential decreases along

the beam owing to the presence of unneutralized electrons.

If the beam current is greater than a limiting current I, the electrons
accumulate near the entrance and form a virtual cathode that reflects
the electrons backward.

However, if the tube is gassy, the beam neutralization increases until
the electron beam can propagate. So why not always use gassy tubes? 
Because the presence of plasma causes a whole new set of possible
collective modes resulting in over-focusing and instability, not
surprisingly at the plasma frequency.

Ion acceleration can also occur due to the large virtual cathode
potential. This effect in dense plasmas is the basis for plasma
accelerators, to be covered later in these lectures.

DC Glow Discharges

Discharge formation by electrical breakdown

![](media/image12.jpeg){width="3.995833333333333in" height="2.05in"}

Initially, the resistance of the gas is much greater than that of the
resistor. At first, only a small current will flow due to any free
electrons and ions from cosmic radiation.

![](media/image13.jpeg){width="3.5493055555555557in"
height="2.6055555555555556in"}

As V increases, the number of charge carriers increases and the
discharge resistance decreases until the voltage reaches a constant
value at V~B~. The is the region of initial breakdown called the
Townsend discharge.

![](media/image14.jpeg){width="3.7493055555555554in"
height="3.0993055555555555in"}

The value of V~B~ depends on the gas species and follows [Paschen\'s
Law]{.underline} which relates V~B~ to the product of gas pressure and
electrode spacing (i.e. the electric field).

As V is further increased, the rate of production of Ar^+^ by electron
impact is sufficient to produce cathode secondary electrons to cause the
ionization. Since a typical ion impact induced secondary emission
coefficient is 0.1, each secondary electron thus produced must be
responsible for 10 - 20 ionizations.

In the initial stage of glow formation, the discharge covers only part
of the cathode surface and the voltage V and current density J are
roughly constant. This is the so-called [normal glow]{.underline} used
in voltage regulator tubes. As the current is increased, the cathode
coverage increases until it is fully covered. Then the Child-Langmuir
law takes over and this regime is called the [abnormal
glow]{.underline}.

![](media/image15.jpeg){width="4.05in" height="2.6979166666666665in"}

Structure of the discharge:

The DC glow arranges itself into several characteristic regions as shown
below

> ![](media/image16.png)

Cathode Region:

A narrow luminous layer at the cathode surface resulting from excitation
and surface bombardment. A large voltage drop occurs across the cathode
sheath. The electrons produced at the cathode are accelerated away from
the sheath and the ion toward the cathode according to the C-L law.

Negative Glow Region:

A plasma with $n_{e} \simeq n_{i}$. The glow results from Ar excitation
by electron impact. Electrons from the cathode are accelerated by V and
provide the main energy input to the discharge. They are in local
thermodynamic equilibrium with typical values

$n_{e} \simeq \text{10}^{\text{10}}\text{cm}^{- 3}$

$T_{e} \simeq 2 - \text{10}\text{eV}$

$T_{i} \simeq 0\text{.}1\text{eV}$

The energy transfer to the Ar^+^ ions is aided by collective effects.
The electrons are rapidly thermalized (at typically 2-5 eV) and carry
out the bulk of the ionization in the positive column.

Positive Column

A luminous region with $n_{e} \simeq n_{i}$. Energy and momentum
imparted by the electric field to the electrons are lost through
collisions with the Ar. Charged particle creation is balanced by losses
to the wall. Manifest collective behavior: instabilities and anomalous
transport.

Anode Region:

A sheath exists such that the plasma potential is about 10 eV above the
anode potential. Secondary electrons are created by electron
bombardment. These are accelerated by the anode sheath and become a
source of particles and energy for the positive column.

The DC discharges just described occur in many applications, but are
limited by rather poor confinement of particles and energy (actually no
confinement at all!). Also these discharges produce hot electrodes with
the associated metallic erosion. Other types of sources have been
developed which improve confinement by the use of magnetic fields, and
eliminate electrode erosion issues by using RF electric fields. These
will be covered later in this course.

Summary of Lecture #1

Plasma are composed of ensembles of charged (and neutral) particles
which may, or may not, be in charge and thermal equilibrium. If it is,
the result is a [Maxwellian distribution]{.underline}.

As a result of the free motion of charges, Debye shielding will occur
over a characteristic length known as the [Debye length]{.underline}
(what else?).

The fields produced by perturbations of the charge neutrality cause a
restoring force which acts over a characteristic time, causing plasma
oscillations at the [plasma frequency]{.underline}.

The state of a plasma depends on an anlysis of the mechanisms by which
particles and energy are fed into and lost from the system. Many types
of collisions are possible, often simultaneously at work.

References:

1\. F. Chen, Introduction to Plasma Physics and Controlled Fusion,
Plenum Press, 1990

2\. S. Tanenbaum, Plasma Physics, McGraw-Hill, 1967

3\. M. Reiser, Theory and Design of Charged Particle Beams, Wiley, 1994

4\. J. Cobine, Gaseous Conductors, Dover, 1958

5\. I. H. Hutchinson, Principles of Plasma Diagnostics, Cambridge
University Press, 1987