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Appleton–Hartree equation
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<h2 id="equation">Equation</h2> <p>The <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a> can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the <a href="/facts/Index_of_refraction/yjkzvgeE">index of refraction</a>: </p> n 2 = ( c k ω ) 2 . {\displaystyle n^{2}=\left({\frac {ck}{\omega }}\right)^{2}.} <p>The full equation is typically given as follows:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> n 2 = 1 − X 1 − i Z − 1 2 Y 2 sin 2 θ 1 − X − i Z ± 1 1 − X − i Z ( 1 4 Y 4 sin 4 θ + Y 2 cos 2 θ ( 1 − X − i Z ) 2 ) 1 / 2 {\displaystyle n^{2}=1-{\frac {X}{1-iZ-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X-iZ}}\pm {\frac {1}{1-X-iZ}}\left({\frac {1}{4}}Y^{4}\sin ^{4}\theta +Y^{2}\cos ^{2}\theta \left(1-X-iZ\right)^{2}\right)^{1/2}}}} <p>or, alternatively, with damping term Z = 0 {\displaystyle Z=0} and rearranging terms:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> n 2 = 1 − X ( 1 − X ) 1 − X − 1 2 Y 2 sin 2 θ ± ( ( 1 2 Y 2 sin 2 θ ) 2 + ( 1 − X ) 2 Y 2 cos 2 θ ) 1 / 2 {\displaystyle n^{2}=1-{\frac {X\left(1-X\right)}{1-X-{{\frac {1}{2}}Y^{2}\sin ^{2}\theta }\pm \left(\left({\frac {1}{2}}Y^{2}\sin ^{2}\theta \right)^{2}+\left(1-X\right)^{2}Y^{2}\cos ^{2}\theta \right)^{1/2}}}} <p>Definition of terms: </p> n {\displaystyle n} : complex refractive index i = − 1 {\displaystyle i={\sqrt {-1}}} : imaginary unit X = ω 0 2 ω 2 {\displaystyle X={\frac {\omega _{0}^{2}}{\omega ^{2}}}} Y = ω H ω {\displaystyle Y={\frac {\omega _{H}}{\omega }}} Z = ν ω {\displaystyle Z={\frac {\nu }{\omega }}} ν {\displaystyle \nu } : electron collision frequency ω = 2 π f {\displaystyle \omega =2\pi f} : angular frequency f {\displaystyle f} : ordinary frequency (cycles per second, or <a href="/facts/Hertz/6Wshj5qm">Hertz</a>) ω 0 = 2 π f 0 = N e 2 ϵ 0 m {\displaystyle \omega _{0}=2\pi f_{0}={\sqrt {\frac {Ne^{2}}{\epsilon _{0}m}}}} : electron <a href="/facts/Plasma_frequency/B7lKhaMT">plasma frequency</a> ω H = 2 π f H = B 0 | e | m {\displaystyle \omega _{H}=2\pi f_{H}={\frac {B_{0}|e|}{m}}} : electron <a href="/facts/Gyro_frequency/Sy5jknQm">gyro frequency</a> ϵ 0 {\displaystyle \epsilon _{0}} : <a href="/facts/Permittivity_of_free_space/gWRUhPGY">permittivity of free space</a> B 0 {\displaystyle B_{0}} : ambient <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> strength e {\displaystyle e} : <a href="/facts/Electron_charge/lPcj6CoQ">electron charge</a> m {\displaystyle m} : <a href="/facts/Electron_mass/xlG83aQl">electron mass</a> θ {\displaystyle \theta } : angle between the ambient <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> vector and the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> <h3>Modes of propagation</h3> <p>The presence of the ± {\displaystyle \pm } sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> For propagation perpendicular to the magnetic field, i.e., k ⊥ B 0 {\displaystyle \mathbf {k} \perp \mathbf {B} _{0}} , the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., k ∥ B 0 {\displaystyle \mathbf {k} \parallel \mathbf {B} _{0}} , the '+' sign represents a left-hand circularly polarized mode, and the '−' sign represents a right-hand circularly polarized mode. See the article on <a href="/facts/Electromagnetic_electron_wave/e69qD3RK">electromagnetic electron waves</a> for more detail. </p><p> k {\displaystyle \mathbf {k} } is the vector of the propagation plane. </p> <h2 id="reduced-forms">Reduced forms</h2> <h3>Propagation in a collisionless plasma</h3> <p>If the electron collision frequency ν {\displaystyle \nu } is negligible compared to the wave frequency of interest ω {\displaystyle \omega } , the plasma can be said to be "collisionless." That is, given the condition </p> ν ≪ ω {\displaystyle \nu \ll \omega } , <p>we have </p> Z = ν ω ≪ 1 {\displaystyle Z={\frac {\nu }{\omega }}\ll 1} , <p>so we can neglect the Z {\displaystyle Z} terms in the equation. The Appleton–Hartree equation for a cold, collisionless plasma is therefore, </p> n 2 = 1 − X 1 − 1 2 Y 2 sin 2 θ 1 − X ± 1 1 − X ( 1 4 Y 4 sin 4 θ + Y 2 cos 2 θ ( 1 − X ) 2 ) 1 / 2 {\displaystyle n^{2}=1-{\frac {X}{1-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X}}\pm {\frac {1}{1-X}}\left({\frac {1}{4}}Y^{4}\sin ^{4}\theta +Y^{2}\cos ^{2}\theta \left(1-X\right)^{2}\right)^{1/2}}}} <h3>Quasi-longitudinal propagation in a collisionless plasma</h3> <p>If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., θ ≈ 0 {\displaystyle \theta \approx 0} , we can neglect the Y 4 sin 4 θ {\displaystyle Y^{4}\sin ^{4}\theta } term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton–Hartree equation becomes, </p> n 2 = 1 − X 1 − 1 2 Y 2 sin 2 θ 1 − X ± Y cos θ {\displaystyle n^{2}=1-{\frac {X}{1-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X}}\pm Y\cos \theta }}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Mary_Taylor_Slow/k5LFLlr1">Mary Taylor Slow</a></li> <li><a href="/facts/Plasma_(physics)/1d3btMVU">Plasma (physics)</a></li> <li><a href="/facts/Waves_in_plasmas/rCaD8N8e">Waves in plasmas</a></li></ul> Citations and notes <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lassen, H., I. Zeitschrift für Hochfrequenztechnik, 1926. Volume 28, pp. 109–113 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>C. Altman, K. Suchy. Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics – Developments in Electromagnetic Theory and Application. Pp 13–15. Kluwer Academic Publishers, 1991. Also available online, Google Books Scan <a href="https://books.google.com/books?id=bQmQil-dMBUC" target="_blank">https://books.google.com/books?id=bQmQil-dMBUC</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>C. Stewart Gillmor (1982), Proc. Am. Phil. S, Volume 126. pp. 395 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Helliwell, Robert (2006), Whistlers and Related Ionospheric Phenomena (2nd ed.), Mineola, NY: Dover, pp. 23–24 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hutchinson, I.H. (2005), Principles of Plasma Diagnostics (2nd ed.), New York, NY: Cambridge University Press, p. 109 <a href="/wiki/Cambridge_University_Press" target="_blank">/wiki/Cambridge_University_Press</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bittencourt, J.A. (2004), Fundamentals of Plasma Physics (3rd ed.), New York, NY: Springer-Verlag, pp. 419–429 <a href="/wiki/Springer-Verlag" target="_blank">/wiki/Springer-Verlag</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>