Spitzer resistivity

The Spitzer resistivity (or plasma resistivity), also called 'Spitzer-Harm 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 ${\displaystyle T_{\text{e}}^{-3/2}}$.

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

Formulation

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

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

where ${\displaystyle Z}$ is the ionization of nuclei, ${\displaystyle e}$ is the electron charge, ${\displaystyle m_{\text{e}}}$ 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 the Boltzmann constant, and ${\displaystyle T_{\text{e}}}$ is the electron temperature (in Kelvin).

One way to convert the ${\displaystyle \eta _{\rm {Sp}}}$ of a plasma column to its resistance is to multiply by the length of the column and divide by its area.

In CGS units, the expression is given by:

${\displaystyle \eta _{\rm {Sp}}={\frac {4{\sqrt {2\pi }}}{3}}{\frac {Ze^{2}m_{\text{e}}^{1/2}\ln \Lambda }{\left(k_{\text{B}}T_{\text{e}}\right)^{3/2}}}.}$ |^{[need to indicate how to put the result in 1/Ohm-cm or Siemens/m ]}

This formulation assumes a Maxwellian distribution, and the prediction is more accurately determined by ^[5]

${\displaystyle \eta _{\rm {Sp}}^{\prime }=\eta _{\rm {Sp}}F(Z),}$

where the factor ${\displaystyle F(1)\approx 1/1.96}$ and the classical approximation (i.e. not including neoclassical effects) of the ${\displaystyle Z}$ dependence is:

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

In the presence of a strong magnetic field (the collision rate is small compared to the gyrofrequency), there are two resistivities corresponding to the current perpendicular and parallel to the magnetic field. The transverse Spitzer resistivity is given by ${\displaystyle \eta _{\perp }=\eta _{Sp}}$, where the rotation keeps the distribution Maxwellian, effectively removing the factor of ${\displaystyle F(Z)}$.

The parallel current is equivalent to the unmagnetized case, ${\displaystyle \eta _{\parallel }=\eta _{\rm {Sp}}^{\prime }}$.

Disagreements with observation

See also: Magnetic reconnection § Anomalous resistivity and Bohm diffusion

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

1. Cohen, Robert S.; Spitzer, Lyman Jr.; McR. Routly, Paul (October 1950). "The Electrical Conductivity of an Ionized Gas" (PDF). Physical Review. 80 (2): 230–238. Bibcode:1950PhRv...80..230C. doi:10.1103/PhysRev.80.230.
2. Spitzer, Lyman Jr.; Härm, Richard (March 1953). "Transport Phenomena in a completely ionized gas" (PDF). Physical Review. 89 (5): 977–981. Bibcode:1953PhRv...89..977S. doi:10.1103/PhysRev.89.977.
3. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press, Inc., 1986
4. Trintchouk, Fedor, Yamada, M., Ji, H., Kulsrud, R. M., Carter, T. A. (2003). "Measurement of the transverse Spitzer resistivity during collisional magnetic reconnection". Physics of Plasmas. 10 (1): 319–322. Bibcode:2003PhPl...10..319T. doi:10.1063/1.1528612.
5. Kuritsyn, A., Yamada, M., Gerhardt, S., Ji, H., Kulsrud, R., Ren, Y. (2006). "Measurements of the parallel and transverse Spitzer resistivities during collisional magnetic reconnection". Physics of Plasmas. 13 (5): 055703. Bibcode:2006PhPl...13e5703K. doi:10.1063/1.2179416.
6. Davies, J. R. (2003). "Electric and magnetic field generation and target heating by laser-generated fast electrons". Physical Review E. 68 (5): 056404. Bibcode:2003PhRvE..68e6404D. doi:10.1103/physreve.68.056404. PMID 14682891.
7. Forest, C. B., Kupfer, K., Luce, T. C., Politzer, P. A., Lao, L. L., Wade, M. R., Whyte, D. G., Wroblewski, D. (1994). "Determination of the noninductive current profile in tokamak plasmas". Physical Review Letters. 73 (18): 2444–2447. Bibcode:1994PhRvL..73.2444F. doi:10.1103/physrevlett.73.2444. PMID 10057061.
8. Kaye, S. M.; Levinton, F. M.; Hatcher, R.; Kaita, R.; Kessel, C.; LeBlanc, B.; McCune, D. C.; Paul, S. (1992). "Spitzer or neoclassical resistivity: A comparison between measured and model poloidal field profiles on PBX-M". Physics of Fluids B: Plasma Physics. 4 (3): 651–658. Bibcode:1992PhFlB...4..651K. doi:10.1063/1.860263. ISSN 0899-8221. S2CID 121654553.
9. Gekelman, W.; DeHaas, T.; Pribyl, P.; Vincena, S.; Compernolle, B. Van; Sydora, R.; Tripathi, S. K. P. (2018). "Nonlocal Ohms Law, Plasma Resistivity, and Reconnection During Collisions of Magnetic Flux Ropes". The Astrophysical Journal. 853 (1): 33. Bibcode:2018ApJ...853...33G. doi:10.3847/1538-4357/aa9fec. ISSN 1538-4357. OSTI 1542014.
10. Kruer, W. L.; Dawson, J. M. (1972). "Anomalous High-Frequency Resistivity of a Plasma". Physics of Fluids. 15 (3): 446. Bibcode:1972PhFl...15..446K. doi:10.1063/1.1693927.
11. Coppi, B.; Mazzucato, E. (1971). "Anomalous Plasma Resistivity at Low Electric Fields". The Physics of Fluids. 14 (1): 134–149. Bibcode:1971PhFl...14..134C. doi:10.1063/1.1693264. ISSN 0031-9171.
12. Papadopoulos, K. (1977). "A review of anomalous resistivity for the ionosphere". Reviews of Geophysics. 15 (1): 113–127. Bibcode:1977RvGSP..15..113P. doi:10.1029/RG015i001p00113. ISSN 1944-9208.
13. Huba, J. D.; Gladd, N. T.; Papadopoulos, K. (1977). "The lower-hybrid-drift instability as a source of anomalous resistivity for magnetic field line reconnection". Geophysical Research Letters. 4 (3): 125–128. Bibcode:1977GeoRL...4..125H. doi:10.1029/GL004i003p00125. ISSN 1944-8007.
14. Drake, J. F.; Swisdak, M.; Cattell, C.; Shay, M. A.; Rogers, B. N.; Zeiler, A. (2003). "Formation of Electron Holes and Particle Energization During Magnetic Reconnection". Science. 299 (5608): 873–877. Bibcode:2003Sci...299..873D. doi:10.1126/science.1080333. ISSN 0036-8075. PMID 12574625. S2CID 15852390.
15. Yoon, Peter H.; Lui, Anthony T. Y. (2006). "Quasi-linear theory of anomalous resistivity". Journal of Geophysical Research: Space Physics. 111 (A2). Bibcode:2006JGRA..111.2203Y. doi:10.1029/2005JA011482. ISSN 2156-2202.
16. Murayama, Yoshimasa (2001-08-29). "Appendix G: Calculation of Conductivity Based on the Kubo Formula". Mesoscopic Systems: Fundamentals and Applications (1 ed.). Wiley. doi:10.1002/9783527618026. ISBN 978-3-527-29376-6.
17. DeGroot, J. S.; Barnes, C.; Walstead, A. E.; Buneman, O. (1977). "Localized Structures and Anomalous dc Resistivity". Physical Review Letters. 38 (22): 1283–1286. Bibcode:1977PhRvL..38.1283D. doi:10.1103/PhysRevLett.38.1283.