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Boltzmann relation

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In aplasma, theBoltzmann relation describes thenumber density of anisothermal charged particle fluid when the thermal and the electrostatic forces acting on the fluid have reachedequilibrium.

In many situations, the electron density of a plasma is assumed to behave according to the Boltzmann relation, due to their small mass and high mobility.^[1]

Equation

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If the localelectrostatic potentials at two nearby locations areφ₁ andφ₂, the Boltzmann relation for the electrons takes the form:^[2]

n_{\text{e}}(\phi _{2})=n_{\text{e}}(\phi _{1})e^{e(\phi _{2}-\phi _{1})/k_{\text{B}}T_{\text{e}}}

wheren_e is the electronnumber density,T_e is thetemperature of the plasma, andk_B is theBoltzmann constant.

Derivation

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A simple derivation of the Boltzmann relation for the electrons can be obtained using the momentum fluid equation of the two-fluid model ofplasma physics in absence of amagnetic field. When the electrons reachdynamic equilibrium, the inertial and the collisional terms of the momentum equations are zero, and the only terms left in the equation are the pressure and electric terms. For anisothermal fluid, thepressure force takes the form

F_{\rm {fluid}}=-k_{\text{B}}T_{\text{e}}\nabla n_{\text{e}},

while the electric term is

F_{\rm {electric}}=en_{\text{e}}\nabla \phi

Integration leads to the expression given above.

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of thePoisson equation because the electron and ion densities are not knowna priori, and if they were, because ofquasineutrality the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the electron density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation.

Inaccurate situations

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Discrepancies with the Boltzmann relation can occur, for example, when oscillations occur so fast that the electrons cannot find a new equilibrium (see e.g.plasma oscillations) or when the electrons are prevented from moving by a magnetic field (see e.g.lower hybrid oscillations).

References

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Wesson, John; et al. (2004).Tokamaks.Oxford University Press.ISBN 978-0-19-850922-6.

^Chen, Francis F. (2006).Introduction to Plasma Physics and Controlled Fusion (2nd ed.). Springer. p. 75.ISBN 978-0-306-41332-2.
^Inan, Umran S. (2011).Principles of plasma physics for engineers and scientists. Marek Gołkowski. Cambridge: Cambridge University Press.ISBN 978-0-511-91683-0.OCLC 700691127.

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