In
, an ion acoustic wave is one type of
longitudinal
Longitudinal is a geometric term of location which may refer to:
* Longitude
** Line of longitude, also called a meridian
* Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...
oscillation of the
ions and
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s in a
plasma, much like
acoustic wave
Acoustic waves are a type of energy propagation through a medium by means of adiabatic loading and unloading. Important quantities for describing acoustic waves are acoustic pressure, particle velocity, particle displacement and acoustic intensi ...
s traveling in neutral gas. However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
s, as well as simple collisions. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. They commonly govern the evolution of mass density, for instance due to
pressure gradients, on time scales longer than the frequency corresponding to the relevant length scale. Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. For a single ion species plasma and in the long
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
limit, the waves are
dispersionless (
) with a speed given by (see derivation below)
:
where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
,
is the mass of the ion,
is its charge,
is the temperature of the electrons and
is the temperature of the ions. Normally γ
e is taken to be unity, on the grounds that the
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
of electrons is large enough to keep them
isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
on the time scale of ion acoustic waves, and γ
i is taken to be 3, corresponding to one-dimensional motion. In
collisionless plasmas, the electrons are often much hotter than the ions, in which case the second term in the numerator can be ignored.
Derivation
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and
ion species. We write each quantity as
where subscript 0 denotes the "zero-order" constant equilibrium value, and 1 denotes the first-order perturbation.
is an ordering parameter for linearization, and has the physical value 1. To linearize, we balance all terms in each equation of the same order in
. The terms involving only subscript-0 quantities are all order
and must balance, and terms with one subscript-1 quantity are all order
and balance. We treat the electric field as order-1 (
) and neglect magnetic fields,
Each species
is described by mass
, charge
, number density
, flow velocity
, and pressure
. We assume the pressure perturbations for each species are a
Polytropic process
A polytropic process is a thermodynamic process that obeys the relation:
p V^ = C
where ''p'' is the pressure, ''V'' is volume, ''n'' is the polytropic index, and ''C'' is a constant. The polytropic process equation describes expansion and comp ...
, namely
for species
. To justify this assumption and determine the value of
, one must use a kinetic treatment that solves for the species distribution functions in velocity space. The polytropic assumption essentially replaces the energy equation.
Each species satisfies the continuity equation
and the momentum equation
.
We now linearize, and work with order-1 equations. Since we do not work with
due to the polytropic assumption (but we do ''not'' assume it is zero), to alleviate notation we use
for
. Using the ion continuity equation, the ion momentum equation becomes
:
We relate the electric field
to the electron density by the electron momentum equation:
:
We now neglect the left-hand side, which is due to electron inertia. This is valid for waves with frequencies much less than the electron plasma frequency
. This is a good approximation for
, such as ionized matter, but not for situations like electron-hole plasmas in semiconductors, or electron-positron plasmas. The resulting electric field is
:
Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates
for each species to
:
:
We arrive at a dispersion relation via Poisson's equation:
:
The first bracketed term on the right is zero by assumption (charge-neutral equilibrium). We substitute for the electric field and rearrange to find
:
.
defines the electron Debye length. The second term on the left arises from the
term, and reflects the degree to which the perturbation is not charge-neutral. If
is small we may drop this term. This approximation is sometimes called the plasma approximation.
We now work in Fourier space, and write each order-1 field as
We drop the tilde since all equations now apply to the Fourier amplitudes, and find
:
is the wave phase velocity. Substituting this into Poisson's equation gives us an expression where each term is proportional to
. To find the dispersion relation for natural modes, we look for solutions for
nonzero and find:
where
, so the ion fractions satisfy
, and
is the average over ion species. A unitless version of this equation is
:
with
,
is the atomic mass unit,
, and
:
If
is small (the plasma approximation), we can neglect the second term on the right-hand side, and the wave is dispersionless
with
independent of k.
Dispersion relation
The general dispersion relation given above for ion acoustic waves can be put in the form of an order-N polynomial (for N ion species) in
. All of the roots should be real-positive, since we have neglected damping. The two signs of
correspond to right- and left-moving waves. For a single ion species,
:
We now consider multiple ion species, for the common case
. For
, the dispersion relation has N-1 degenerate roots
, and one non-zero root
:
This non-zero root is called the "fast mode", since
is typically greater than all the ion thermal speeds. The approximate fast-mode solution for
is
:
The N-1 roots that are zero for
are called "slow modes", since
can be comparable to or less than the thermal speed of one or more of the ion species.
A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions (
). Let us specialize to full ionization (
), equal temperatures (
), polytrope exponents
, and neglect the
contribution. The dispersion relation becomes a quadratic in
, namely:
:
Using
we find the two roots are
.
Another case of interest is one with two ion species of very different masses. An example is a mixture of gold (A=197) and boron (A=10.8), which is currently of interest in hohlraums for laser-driven inertial fusion research. For a concrete example, consider
and
for both ion species, and charge states Z=5 for boron and Z=50 for gold. We leave the boron atomic fraction
unspecified (note
). Thus,
and
.
Damping
Ion acoustic waves are damped both by
Coulomb collision A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit. Th ...
s and collisionless
Landau damping In physics, Landau damping, named after its discoverer,Landau, L. "On the vibration of the electronic plasma". ''JETP'' 16 (1946), 574. English translation in ''J. Phys. (USSR)'' 10 (1946), 25. Reproduced in Collected papers of L.D. Landau, edited a ...
. The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters.
See also
*
Waves in plasmas In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A Plasma (physics), plasma is a Plasma (physics)#Plasma_potential, quasineutral, electrical conductivity, elec ...
*
Sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
*
Alfvén wave
In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines.
Definition
An Alfvén wave is a ...
*
Magnetosonic wave
A magnetosonic wave, also called a magnetoacoustic wave, is a linear magnetohydrodynamic (MHD) wave that is driven by thermal pressure, magnetic pressure, and magnetic tension. There are two types of magnetosonic waves, the ''fast'' magnetosonic w ...
*
List of plasma (physics) articles
This is a list of plasma physics topics.
A
* Ablation
* Abradable coating
* Abraham–Lorentz force
* Absorption band
* Accretion disk
* Active galactic nucleus
* Adiabatic invariant
* ADITYA (tokamak)
* Aeronomy
* Afterglow plasma
* Airg ...
External links
Various patents and articles related to fusion, IEC, ICC and plasma physics
{{DEFAULTSORT:Ion Acoustic Wave
Waves in plasmas