Spitzer resistivity

The Spitzer resistivity (or plasma resistivity) is an expression describing the electrical resistance in a plasma, which was first formulated by Lyman Spitzer in 1950.^[1][2] The Spitzer resistivity of a plasma decreases in proportion to the electron temperature as ${\displaystyle T_{e}^{-3/2}}$.

The inverse of the Spitzer resistivity ${\displaystyle \eta _{Sp}}$ is known as the Spitzer conductivity ${\displaystyle \sigma _{Sp}=1/\eta _{Sp}}$.

Formulation[]

The Spitzer restitivity is classical model of electrical resistivity based upon electron-ion collisions and it is commonly used in plasma physics.^{[3][4][5][6][7]} The transverse Spitzer resistivity is given by:

${\displaystyle \eta _{\perp }={\frac {4{\sqrt {2\pi }}}{3}}{\frac {Ze^{2}m_{e}^{1/2}\ln \Lambda }{\left(4\pi \varepsilon _{0}\right)^{2}\left(k_{\text{B}}T_{e}\right)^{3/2}}},}$

and the parallel Spitzer resistivity by:

${\displaystyle \eta _{\parallel }=\eta _{\perp }/1.96\qquad ({\text{for }}Z=1),}$

where ${\displaystyle Z}$ is the ionization of nuclei, ${\displaystyle e}$ is the electron charge, ${\displaystyle m_{e}}$ is the electron mass, ${\displaystyle \ln \Lambda }$ is the Coulomb logarithm, ${\displaystyle \varepsilon _{0}}$ is the electric permittivity of free space, ${\displaystyle k_{\text{B}}}$ is Boltzmann's constant, and ${\displaystyle T_{e}}$ is the electron temperature in kelvins. The two resistivities correspond to current perpendicular and parallel to a strong magnetic field (the collision rate is small compared to the gyrofrequency). In an unmagnetized case the resistivity is ${\displaystyle \eta _{\parallel }}$.

In CGS units, the expression is given by:

${\displaystyle \eta _{\perp }={\frac {4{\sqrt {2\pi }}}{3}}{\frac {Ze^{2}m_{e}^{1/2}\ln \Lambda }{\left(k_{\text{B}}T_{e}\right)^{3/2}}}.}$

For arbitrary ${\displaystyle Z}$,

${\displaystyle \eta _{\parallel }=\eta _{\perp }F(Z),}$

where

${\displaystyle F(Z)={\frac {1+1.198Z+0.222Z^{2}}{1+2.966Z+0.753Z^{2}}}}$.

Disagreements with observation[]

Measurements in laboratory experiments and computer simulations have shown that under certain conditions, the resistivity of a plasma tends to be much higher than the Spitzer resistivity.^[8][9][10] This effect is sometimes known as anomalous resistivity or neoclassical resistivity.^[11] It has been observed in space and effects of anomalous resistivity have been postulated to be associated with particle acceleration during magnetic reconnection.^[12][13][14] There are various theories and models that attempt to describe anomalous resistivity and they are frequently compared to the Spitzer resistivity.^{[9][15][16][17]}

References[]

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