Physics:Spitzer resistivity

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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 [math]\displaystyle{ T_e^{{-3/2}} }[/math]. The inverse of the Spitzer resistivity [math]\displaystyle{ \eta_{\rm Sp} }[/math] is known as the Spitzer conductivity [math]\displaystyle{ \sigma_{\rm Sp}=1/\eta_{\rm Sp} }[/math].

Formulation

The Spitzer resistivity 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 Spitzer resistivity is given by:

[math]\displaystyle{ \eta_{\rm Sp} = \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}} , }[/math]

where [math]\displaystyle{ Z }[/math] is the ionization of nuclei, [math]\displaystyle{ e }[/math] is the electron charge, [math]\displaystyle{ m_e }[/math] is the electron mass, [math]\displaystyle{ \ln\Lambda }[/math] is the Coulomb logarithm, [math]\displaystyle{ \varepsilon_0 }[/math] is the electric permittivity of free space, [math]\displaystyle{ k_\text{B} }[/math] is Boltzmann's constant, and [math]\displaystyle{ T_e }[/math] is the electron temperature in kelvins.

In CGS units, the expression is given by:

[math]\displaystyle{ \eta_{\rm Sp} = \frac{4\sqrt{2\pi}}{3}\frac{Ze^{2}m_e^{1/2}\ln \Lambda}{\left(k_\text{B}T_e\right)^{3/2}}. }[/math]

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

[math]\displaystyle{ \eta_{\rm Sp}^\prime = \eta_{\rm Sp} F(Z), }[/math]

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

[math]\displaystyle{ F(Z) \approx \frac{1+1.198Z+0.222Z^2}{1+2.966Z+0.753Z^2} }[/math].


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 [math]\displaystyle{ \eta_\perp = \eta_{Sp} }[/math], where the rotation keeps the distribution Maxellian, effectively removing the factor of [math]\displaystyle{ F(Z) }[/math].

The parallel current is equivalent to the unmagnetized case, [math]\displaystyle{ \eta_\parallel = \eta_{\rm Sp}^\prime }[/math].



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

  1. ↑ Cohen, Robert S.; Spitzer, Jr., Lyman; McR. Routly, Paul (October 1950). "The Electrical Conductivity of an Ionized Gas". Physical Review 80 (2): 230–238. doi:10.1103/PhysRev.80.230. Bibcode: 1950PhRv...80..230C. http://ayuba.fr/pdf/spitzer1950.pdf. 
  2. ↑ Spitzer, Jr., Lyman; Härm, Richard (March 1953). "Transport Phenomena in a completely ionized gas". Physical Review 89 (5): 977–981. doi:10.1103/PhysRev.89.977. Bibcode: 1953PhRv...89..977S. http://ayuba.fr/pdf/spitzer1953.pdf. 
  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. doi:10.1063/1.1528612. Bibcode: 2003PhPl...10..319T. https://digital.library.unt.edu/ark:/67531/metadc740671/. 
  5. ↑ 5.0 5.1 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. doi:10.1063/1.2179416. Bibcode: 2006PhPl...13e5703K. 
  6. ↑ Davies, J. R. (2003). "Electric and magnetic field generation and target heating by laser-generated fast electrons". Physical Review E 68 (5): 056404. doi:10.1103/physreve.68.056404. PMID 14682891. Bibcode: 2003PhRvE..68e6404D. 
  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. doi:10.1103/physrevlett.73.2444. PMID 10057061. Bibcode: 1994PhRvL..73.2444F. https://zenodo.org/record/1233899. 
  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. doi:10.1063/1.860263. ISSN 0899-8221. Bibcode: 1992PhFlB...4..651K. https://aip.scitation.org/doi/abs/10.1063/1.860263. 
  9. ↑ 9.0 9.1 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" (in en). The Astrophysical Journal 853 (1): 33. doi:10.3847/1538-4357/aa9fec. ISSN 1538-4357. Bibcode: 2018ApJ...853...33G. 
  10. ↑ Kruer, W. L.; Dawson, J. M. (1972). "Anomalous High-Frequency Resistivity of a Plasma" (in en). Physics of Fluids 15 (3): 446. doi:10.1063/1.1693927. Bibcode: 1972PhFl...15..446K. https://aip.scitation.org/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. doi:10.1063/1.1693264. ISSN 0031-9171. Bibcode: 1971PhFl...14..134C. https://aip.scitation.org/doi/abs/10.1063/1.1693264. 
  12. ↑ Papadopoulos, K. (1977). "A review of anomalous resistivity for the ionosphere" (in en). Reviews of Geophysics 15 (1): 113–127. doi:10.1029/RG015i001p00113. ISSN 1944-9208. Bibcode: 1977RvGSP..15..113P. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RG015i001p00113. 
  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" (in en). Geophysical Research Letters 4 (3): 125–128. doi:10.1029/GL004i003p00125. ISSN 1944-8007. Bibcode: 1977GeoRL...4..125H. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/GL004i003p00125. 
  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" (in en). Science 299 (5608): 873–877. doi:10.1126/science.1080333. ISSN 0036-8075. PMID 12574625. Bibcode: 2003Sci...299..873D. https://www.science.org/doi/10.1126/science.1080333. 
  15. ↑ Yoon, Peter H.; Lui, Anthony T. Y. (2006). "Quasi-linear theory of anomalous resistivity" (in en). Journal of Geophysical Research: Space Physics 111 (A2). doi:10.1029/2005JA011482. ISSN 2156-2202. Bibcode: 2006JGRA..111.2203Y. 
  16. ↑ Murayama, Yoshimasa (2001-08-29). "Appendix G: Calculation of Conductivity Based on the Kubo Formula" (in en). Mesoscopic Systems: Fundamentals and Applications (1 ed.). Wiley. doi:10.1002/9783527618026. ISBN 978-3-527-29376-6. https://onlinelibrary.wiley.com/doi/book/10.1002/9783527618026. 
  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. doi:10.1103/PhysRevLett.38.1283. Bibcode: 1977PhRvL..38.1283D. https://link.aps.org/doi/10.1103/PhysRevLett.38.1283. 



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