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

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, Jr., Lyman; 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, Jr., Lyman; 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.CS1 maint: multiple names: authors list (link)
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.CS1 maint: multiple names: authors list (link)
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.CS1 maint: multiple names: authors list (link)
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.
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. doi:10.3847/1538-4357/aa9fec. ISSN 1538-4357.
10. Kruer, W. L.; Dawson, J. M. (1972). "Anomalous High-Frequency Resistivity of a Plasma". Physics of Fluids. 15 (3): 446. 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.
12. Papadopoulos, K. (1977). "A review of anomalous resistivity for the ionosphere". Reviews of Geophysics. 15 (1): 113–127. 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. 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. doi:10.1126/science.1080333. ISSN 0036-8075. PMID 12574625.
15. Yoon, Peter H.; Lui, Anthony T. Y. (2006). "Quasi-linear theory of anomalous resistivity". Journal of Geophysical Research: Space Physics. 111 (A2). 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. doi:10.1103/PhysRevLett.38.1283.