# Plasma oscillation

In physics , a plasma oscillation is a periodic oscillation of the charge density in a medium , for example in a plasma or a metal . The quasiparticle that results from the quantization of these oscillations is the plasmon .

## Plasma frequency

If the free electrons are locally compressed in an electron gas , the Coulomb force acts on them , which tries to restore the homogeneous charge distribution. Due to their inertia , the electrons will shoot past the neutral position and build up a new excess charge, which leads to a periodic oscillation. The angular frequency with which the electron density oscillates around the mean density is called the plasma frequency :

${\ displaystyle \ omega _ {\ mathrm {p}} = {\ sqrt {\ frac {4 \ pi n _ {\ mathrm {e}} e ^ {2}} {m _ {\ mathrm {e}}}}} }$ ( CGS units ),
${\ displaystyle \ omega _ {\ mathrm {p}} = {\ sqrt {\ frac {n _ {\ mathrm {e}} e ^ {2}} {\ varepsilon _ {0} m _ {\ mathrm {e}} }}}}$ ( SI units ),

wherein

• ${\ displaystyle n_ {e}}$ the electron density is
• ${\ displaystyle e}$ the elementary charge ,
• ${\ displaystyle \ varepsilon _ {0} \!}$ the electric field constant and
• ${\ displaystyle m _ {\ mathrm {e}}}$ the electron mass .

If one considers the charge carrier in a dielectric with a permittivity , the plasma frequency decreases: ${\ displaystyle \ varepsilon _ {\ mathrm {r}}> 1}$ ${\ displaystyle \ omega _ {\ mathrm {p}} = {\ sqrt {\ frac {n _ {\ mathrm {e}} e ^ {2}} {\ varepsilon _ {\ mathrm {r}} \ varepsilon _ { 0} m _ {\ mathrm {e}}}}}}$ (SI units).

The plasma resonance is an excitation without dispersion, i.e. independent of the expansion. An electromagnetic wave penetrating the material can stimulate the oscillation and experiences both absorption and refraction .

### Derivation

The three equations necessary to derive the plasma frequency are:

1.) The Poisson equation for electrostatics , which describes the potential as a function of the charge density:

${\ displaystyle \ Delta \ Phi (\ mathbf {r}, t) = - {\ frac {qn (\ mathbf {r}, t)} {\ varepsilon}}}$ in which

• ${\ displaystyle \ Delta}$ Laplace operator
• ${\ displaystyle \ Phi}$ Electrostatic potential
• ${\ displaystyle q}$ Electric charge of the particles
• ${\ displaystyle n}$ Particle density
• ${\ displaystyle \ varepsilon = \ varepsilon _ {0} \ varepsilon _ {\ mathrm {r}}}$ Permittivity

2.) The continuity equation , which describes the conservation of the particles:

${\ displaystyle q {\ frac {\ partial n (\ mathbf {r}, t)} {\ partial t}} + \ operatorname {div} \, \ mathbf {j} (\ mathbf {r}, t) = 0 \,}$ With

• ${\ displaystyle \ mathbf {j} = qn \ mathbf {v}}$ Electric current density with particle velocity (The equation can be formulated both for charge conservation - as here - or for particle conservation.)${\ displaystyle \ mathbf {v}}$ 3.) The Newton's second law , that the kinetic response of the particles with respect to the strength of the electric field describes: ${\ displaystyle \ mathbf {F}}$ ${\ displaystyle \ mathbf {E}}$ ${\ displaystyle \ mathbf {F} (\ mathbf {r}, t) = m {\ frac {\ partial \ mathbf {v} (\ mathbf {r}, t)} {\ partial t}}}$ With

• ${\ displaystyle \ mathbf {F} (\ mathbf {r}, t) = q \ mathbf {E} (\ mathbf {r}, t) = - q \ operatorname {grad} \, \ Phi (\ mathbf {r }, t)}$ For small density fluctuations, using the relationship for the current density shown under 2.), the time derivative of the particle speed can be expressed solely by the time derivative of the current density:

${\ displaystyle {\ frac {\ partial \ mathbf {v} (\ mathbf {r}, t)} {\ partial t}} = {\ frac {\ partial {\ frac {\ mathbf {j} (\ mathbf { r}, t)} {qn (\ mathbf {r}, t)}}} {\ partial t}} \ approx {\ frac {1} {\, qn_ {0}}} {\ frac {\ partial \ mathbf {j} (\ mathbf {r}, t)} {\ partial t}}}$ This implies the assumption that the relative density fluctuations are small compared to the relative changes in particle velocities. By inserting it back into the 3rd) equation one obtains

${\ displaystyle -q \ operatorname {grad} \, \ Phi (\ mathbf {r}, t) = {\ frac {m} {qn_ {0}}} {\ frac {\ partial \ mathbf {j} (\ mathbf {r}, t)} {\ partial t}}}$ which by applying the divergence operation to the entire equation

${\ displaystyle -q \ Delta \, \ Phi (\ mathbf {r}, t) = {\ frac {m} {qn_ {0}}} {\ frac {\ partial \ operatorname {div} \, \ mathbf { j} (\ mathbf {r}, t)} {\ partial t}}}$ inserting the Poisson equation of electrostatics on the left and the continuity equation on the right allows:

${\ displaystyle {\ frac {q ^ {2}} {\ varepsilon}} n (\ mathbf {r}, t) = - {\ frac {m} {n_ {0}}} {\ frac {\ partial ^ {2} n (\ mathbf {r}, t)} {\ partial t ^ {2}}}}$ This results in the equation for a harmonic oscillation with the natural plasma frequency

${\ displaystyle \ omega _ {\ mathrm {p}} ^ {2} = {\ frac {q ^ {2} \, n_ {0}} {m \, \ varepsilon}}}$ ## Dispersion relation

Because the plasma frequency is independent of the wavelength (!), Plasma oscillations have a phase velocity that is proportional to the wavelength and a vanishing group velocity . The electromagnetic wave incident in the example above excites the charge carriers of the plasma to oscillate (perpendicular to the direction of propagation , because the wave is transversely polarized), but does not cause any charge transport in the direction of incidence of the wave.

When the electrons have a finite thermal velocity with ${\ displaystyle v _ {\ mathrm {e, th}} = {\ sqrt {\ frac {k _ {\ mathrm {B}} T _ {\ mathrm {e}}} {m _ {\ mathrm {e}}}}} }$ • ${\ displaystyle k _ {\ mathrm {B}}}$ : Boltzmann constant
• ${\ displaystyle m _ {\ mathrm {e}}}$ : Mass of electrons
• ${\ displaystyle T _ {\ mathrm {e}}}$ : the normalized electron temperature ,${\ displaystyle m _ {\ mathrm {e}}}$ ${\ displaystyle T}$ the electron pressure acts as a restoring force in addition to the electric field . Then the oscillations propagate with the Bohm - Gross - dispersion relation

${\ displaystyle \ omega ^ {2} = \ omega _ {\ mathrm {pe}} ^ {2} +3 (k \ cdot v _ {\ mathrm {e, th}}) ^ {2}}$ ( k : wave number ).

If the spatial scale is large compared to the Debye longitude , then the pressure plays a subordinate role:

${\ displaystyle \ omega \ approx \ omega _ {\ mathrm {pe}}.}$ On the other hand, pressure dominates on small scales:

{\ displaystyle {\ begin {aligned} \ omega ^ {2} & \ approx 3 (k \ cdot v _ {\ mathrm {e, th}}) ^ {2} \\\ Leftrightarrow \ omega & \ approx k {\ sqrt {3}} \ cdot v _ {\ mathrm {e, th}} \\\ Leftrightarrow v _ {\ mathrm {ph}} \ equiv {\ frac {\ omega} {k}} & \ approx {\ sqrt {3 }} \ cdot v _ {\ mathrm {e, th}}, \ end {aligned}}} d. H. the waves become dispersion- less with the phase velocity , so that the plasma wave can accelerate individual electrons. This process is a type of collisionless damping called Landau damping . For this reason, the dispersion relationship at large is  difficult to observe and seldom important. ${\ displaystyle {\ sqrt {3}} \ cdot v _ {\ mathrm {e, th}},}$ ${\ displaystyle k}$ ## application

Electrons with a certain plasma frequency can therefore carry out almost instantaneous movements that run “slower” than the plasma frequency. This means in particular that plasmas almost completely reflect electromagnetic waves with frequencies below the plasma frequency, whereas they are transparent for waves with frequencies above the plasma frequency .

### Reflection of light on metals

The plasma frequency in metallic solids with typical electron densities is in the range of , which can be converted into a wavelength of which is in the UV range via the phase velocity for electromagnetic waves . Metals therefore reflect light in the optical range and especially radio and radar waves. In contrast, electromagnetic waves with a higher frequency, such as UV or X-ray radiation, are transmitted as long as no other resonances above the plasma frequency (e.g. electronic transitions from low-energy shells) absorb them. ${\ displaystyle n _ {\ mathrm {e}} = 10 ^ {28} \, \ mathrm {m} ^ {- 3}}$ ${\ displaystyle \ omega _ {\ mathrm {p}} = 5 \ cdot 10 ^ {15} \, \ mathrm {s} ^ {- 1}}$ ${\ displaystyle \ lambda _ {\ mathrm {p}} \ approx 300 \, \ mathrm {nm}}$ ### Reflection of radio waves on the atmosphere

Plasma oscillations in the earth's ionosphere are the reason why shortwave radio programs have a very long range . The radio waves hit the ionosphere and stimulate the electrons to vibrate. From the relatively low electron density of the F-layer of only 10 12  m −3 , a plasma frequency of about 9 MHz can be calculated. This leads to a reflection of all vertically incident waves with lower frequencies in the ionosphere. With a shallower angle of incidence, the usable cut -off frequency can increase to values ​​up to over 50 MHz. Programs transmitted via shortwave can therefore also be received at locations that are actually in the shadow of the transmitter. Communication with higher flying satellites or GPS is only possible via even higher frequencies in the VHF band.