Abstract

A theoretical study of the optical properties of a relativistic Fermi gas at a finite temperature is performed. The results are obtained from Maxwell’s equations, assuming bianisotropic constitutive relations for the relativistic gas. The longitudinal and transverse spatial directions are defined according to the properties of the electric permittivity tensor. The electromagnetic modes and the refractive index of the relativistic gas are calculated along such directions. The longitudinal and transverse group velocities corresponding to the respective propagation modes are also obtained. No typical characteristics of metamaterials are observed, neither in the refractive index nor in the group velocities corresponding to the transverse modes. However, in the case of longitudinal modes, negative values of the group velocity are observed in a region of the reciprocal space where particle–antiparticle excitations can occur.

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [Crossref]
  2. J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
    [Crossref]
  3. Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
    [Crossref]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [Crossref]
  5. P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
    [Crossref]
  6. C. A. A. de Carvalho, “Relativistic electron gas: a candidate for nature’s left-handed materials,” Phys. Rev. D 93, 105005 (2016).
    [Crossref]
  7. J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 28, 8 (1954).
  8. E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
    [Crossref]
  9. D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
    [Crossref]
  10. D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
    [Crossref]
  11. D. B. Nguyen, “Relativistic constitutive relations, differential forms, and the p-compound,” Am. J. Phys. 60, 1137–1144 (1992).
    [Crossref]
  12. T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy. A Field Guide (World Scientific, 2010).
  13. T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
    [Crossref]
  14. P.-H. Chang, C.-Y. Kuo, and R.-L. Chern, “Wave propagation in bianisotropic metamaterials: angular selective transmission,” Opt. Express 22, 25710–25721 (2014).
    [Crossref]
  15. F. Wooten, Optical Properties of Solids (Academic, 1972).

2018 (1)

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

2017 (1)

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

2016 (2)

C. A. A. de Carvalho, “Relativistic electron gas: a candidate for nature’s left-handed materials,” Phys. Rev. D 93, 105005 (2016).
[Crossref]

E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
[Crossref]

2014 (1)

2013 (1)

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

2011 (1)

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

2009 (1)

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

1992 (1)

D. B. Nguyen, “Relativistic constitutive relations, differential forms, and the p-compound,” Am. J. Phys. 60, 1137–1144 (1992).
[Crossref]

1968 (2)

D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1954 (1)

J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 28, 8 (1954).

Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Briggs, D. P.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Chang, P.-H.

Chen, J.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Chen, M.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Chen, X.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Cheng, D. K.

D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

Chern, R.-L.

de Carvalho, C. A. A.

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
[Crossref]

C. A. A. de Carvalho, “Relativistic electron gas: a candidate for nature’s left-handed materials,” Phys. Rev. D 93, 105005 (2016).
[Crossref]

Duan, Z.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Feng, L.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Geng, T.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Gong, Y.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Gu, M.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Kong, J.-A.

D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

Kravchenko, I. I.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Kuo, C.-Y.

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy. A Field Guide (World Scientific, 2010).

Li, X.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Liang, B.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Lindhard, J.

J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 28, 8 (1954).

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy. A Field Guide (World Scientific, 2010).

Moitra, P.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Nguyen, D. B.

D. B. Nguyen, “Relativistic constitutive relations, differential forms, and the p-compound,” Am. J. Phys. 60, 1137–1144 (1992).
[Crossref]

Oliveira, L. E.

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Qian, W.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Reis, D. M.

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

Reyes-Gómez, E.

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
[Crossref]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Tang, X.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Valentine, J.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Wang, Y.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Wang, Z.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Wooten, F.

F. Wooten, Optical Properties of Solids (Academic, 1972).

Yang, Y.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Zhang, X.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Zhang, Y.

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Zhuang, S.

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

Am. J. Phys. (1)

D. B. Nguyen, “Relativistic constitutive relations, differential forms, and the p-compound,” Am. J. Phys. 60, 1137–1144 (1992).
[Crossref]

Ann. Phys. (1)

D. M. Reis, E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “Electromagnetic propagation in a relativistic electron gas at finite temperatures,” Ann. Phys. 530, 1700443 (2018).
[Crossref]

Europhys. Lett. (1)

E. Reyes-Gómez, L. E. Oliveira, and C. A. A. de Carvalho, “The electromagnetic response of a relativistic fermi gas at finite temperatures: Applications to condensed-matter systems,” Europhys. Lett. 114, 17009 (2016).
[Crossref]

Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. (1)

J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Danske Videnskab. Selskab Mat.-fys. Medd. 28, 8 (1954).

Nat. Commun. (1)

Z. Duan, X. Tang, Z. Wang, Y. Zhang, X. Chen, M. Chen, and Y. Gong, “Observation of the reversed Cherenkov radiation,” Nat. Commun. 8, 14901 (2017).
[Crossref]

Nat. Photonics (2)

J. Chen, Y. Wang, Y. Wang, Y. Wang, T. Geng, X. Li, L. Feng, W. Qian, B. Liang, X. Zhang, M. Gu, and S. Zhuang, “Observation of the inverse doppler effect in negative-index materials at optical frequencies,” Nat. Photonics 5, 239–245 (2011).
[Crossref]

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–793 (2013).
[Crossref]

Opt. Express (1)

Phys. Rev. B (1)

T. G. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

Phys. Rev. D (1)

C. A. A. de Carvalho, “Relativistic electron gas: a candidate for nature’s left-handed materials,” Phys. Rev. D 93, 105005 (2016).
[Crossref]

Phys. Rev. Lett. (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[Crossref]

Proc. IEEE (1)

D. K. Cheng and J.-A. Kong, “Covariant descriptions of bianisotropic media,” Proc. IEEE 56, 248–251 (1968).
[Crossref]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (2)

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy. A Field Guide (World Scientific, 2010).

F. Wooten, Optical Properties of Solids (Academic, 1972).

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Figures (6)

Fig. 1.
Fig. 1. Real and imaginary parts of $ \varepsilon $, $ {\mu ^{ - 1}} $, $ {\varepsilon ^\prime } $, and $ \tau $ as functions of $ \tilde \beta $. Calculations were performed for $ \tilde \eta = 0.01 $, $ \tilde q = 0.07 $, and $ \tilde \omega = 0.04 $.
Fig. 2.
Fig. 2. (a) Upper (longitudinal plasmon frequency) and lower frequency zeros (solid and dashed lines, respectively) of the real part of $ {\varepsilon _\parallel } $. Dotted lines for each value of $ \tilde \beta $ correspond to the boundary between the regions where $ {\rm Im}({\varepsilon _\parallel }) \ne 0 $ (shaded areas) and $ {\rm Im}({\varepsilon _\parallel }) = 0 $. (b) Transverse electromagnetic modes (solid lines). The dashed line corresponds to a photonic mode that propagates with the speed of light in vacuum. Results were obtained as functions of the $ q $ wave vector in units of the $ {q_c} $ Compton wave vector, for different values of $ \tilde \beta $, and for $ \tilde \eta = 0.01 $. The wave frequency is given in units of the $ {\omega _c} $ Compton frequency.
Fig. 3.
Fig. 3. Real [panels (a) and (c)] and imaginary [panels (b) and (d)] parts of $ n_\parallel ^2 $ [cf. Eq. (29)] as functions of the $ \omega $ frequency in units of the $ {\omega _c} $ Compton frequency. Results of panels (a) and (b) [(c) and (d)] were obtained for $ \tilde \beta = 1 $ ($ \tilde \beta = 1000 $) and $ \tilde \eta = 0.01 $. Solid, dashed, dotted, and dotted-dashed lines in panels (a) and (b) [(c) and (d)] correspond to $ q = 0.001\; {q_c} $, $ q = 0.002\; {q_c} $, $ q = 0.003\; {q_c} $, and $ q = 0.004\; {q_c} $ ($ q = 0.002\; {q_c} $, $ q = 0.004\; {q_c} $, $ q = 0.006\; {q_c} $, and $ q = 0.008\; {q_c} $), respectively, where $ {q_c} $ is the Compton wave vector.
Fig. 4.
Fig. 4. As in Fig. 3, but for $ n_ \bot ^2 $ [cf. Eq. (31)].
Fig. 5.
Fig. 5. (a) Real and (b) imaginary parts of $ n_\parallel ^2 = n_\parallel ^2[\tilde q,\tilde \omega (\tilde q)] $, and (c) real part of $ n_ \bot ^2 = n_ \bot ^2[\tilde q,\tilde \omega (\tilde q)] $ as functions of the $ q $ wave vector in units of the $ {q_c} $ Compton wave vector. The $ \tilde \omega = \tilde \omega (\tilde q) $ frequency values used to evaluate the functions $ n_\parallel ^2 = n_\parallel ^2(\tilde q,\tilde \omega ) $ and $ n_ \bot ^2 = n_ \bot ^2(\tilde q,\tilde \omega ) $ correspond to the dispersion relation of the longitudinal and transverse plasmon modes, respectively [see solid lines in Figs. 2(a) and 2(b), respectively]. The imaginary part of $ n_ \bot ^2 = n_ \bot ^2[\tilde q,\tilde \omega (\tilde q)] $ vanishes.
Fig. 6.
Fig. 6. (a) Longitudinal and (b) transverse group velocity, in units of the speed of light in vacuum, obtained for different values of the gas temperature. Shadow areas correspond to the regions in which the group velocity is negative.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

q i D i = 0 ,
q i B i = 0 ,
ϵ ijk q j E k = ω c B i ,
ϵ ijk q j H k = ω c D i ,
D i = ε ij E j + τ ij B j a n d H i = ( μ 1 ) ij B j + τ ij E i ,
ε ij = ε δ ij + ε q ^ i q ^ j , ( μ 1 ) ij = μ 1 δ ij + μ 1 q ^ i q ^ j , a n d τ ij = τ ϵ ijk q ^ k ,
ε = 1 + A + ( 1 ω ~ 2 q ~ 2 ) B + ( 2 + ω ~ 2 q ~ 2 ω ~ 2 ) C ,
μ 1 = 1 + A 2 ω ~ 2 q ~ 2 B + ( 2 q ~ 2 q ~ 2 ω ~ 2 ) C ,
ε = μ 1 = [ A + q ~ 2 q ~ 2 ω ~ 2 C ] ,
τ = ω ~ q ~ [ B + q ~ 2 q ~ 2 ω ~ 2 C ] ,
A = A α q ~ 2 ω ~ 2 I + [ 1 3 2 q ~ 2 ω ~ 2 q ~ 2 ] B ,
B = A α q ~ 2 ω ~ 2 J ,
C = A α 12 { 1 3 + ( 3 + γ 2 ) [ γ a r c c o t ( γ ) 1 ] } ,
I = 0 d y y 2 y 2 + 1 F ( y , β ~ , ξ ~ ) × [ 1 + 2 q ~ 2 + ω ~ 2 8 y q ~ F 1 ( y , q ~ , ω ~ ) ]
J = 0 d y y 2 y 2 + 1 F ( y , β ~ , ξ ~ ) × [ ω ~ y 2 + 1 2 y q ~ F 2 ( y , q ~ , ω ~ ) ] 1 + 4 ( y 2 + 1 ) q ~ 2 + ω ~ 2 8 y q ~ F 1 ( y , q ~ , ω ~ ) ω ~ y 2 + 1 2 y q ~ F 2 ( y , q ~ , ω ~ ) ] ,
F ( y , β ~ , ξ ~ ) = 1 e β ~ ( y 2 + 1 ξ ~ ) + 1 1 e β ~ ( y 2 + 1 + ξ ~ ) + 1 ,
F 1 ( y , q ~ , ω ~ ) = ln [ ( q ~ 2 ω ~ 2 + 2 y q ~ ) 2 4 ( y 2 + 1 ) ω ~ 2 ( q ~ 2 ω ~ 2 2 y q ~ ) 2 4 ( y 2 + 1 ) ω ~ 2 ] ,
F 2 ( y , q ~ , ω ~ ) = ln [ ω ~ 4 4 ( ω ~ y 2 + 1 + y q ~ ) 2 ω ~ 4 4 ( ω ~ y 2 + 1 y q ~ ) 2 ] ,
1 = 1 η ~ 0 + d y y 2 F ( y , β ~ , ξ ~ ) ,
T 1 [ ε ij ] T = d i a g [ ε , ε , ε + ε ] ,
T = [ sec ( φ ) cot ( θ ) tan ( φ ) cos ( φ ) tan ( θ ) 0 1 sin ( φ ) tan ( θ ) 1 0 1 ] .
T 1 [ ( μ 1 ) ij ] T = d i a g [ μ 1 , μ 1 , μ 1 + μ 1 ] .
ε = ε + ε = 0.
M [ E B ] = [ P Q R S ] [ E B ] = 0 ,
P ij = q ϵ ijk q ^ k , Q ij = ω c δ ij , R ij = ( q τ + ω c ε ) δ ij , a n d S ij = ( ω c τ q μ ) ϵ ijk q ^ k ,
μ 2 ω c ( ω c ε + q τ ) ( q 2 ε μ ω 2 c 2 2 q τ μ ω c ) 2 = 0.
ω c ε + q τ = 0
q 2 ε μ ω 2 c 2 2 q τ μ ω c = 0.
ν = 1 μ + 1 μ = 1.
( 1 + A + C ) q ~ 2 B ω ~ 2 = 0
1 D ( q ~ 2 ω ~ 2 ) 2 [ ( 1 + A + C ) q ~ 2 B ω ~ 2 ] = 0 ,
q 2 ε μ ω 2 c 2 = 0 ,
ε = 0 = 1 + B + C B ω ~ 2 q ~ 2 ,
q ~ 2 n 2 ω ~ 2 = 0 ,
n 2 = B 1 + B + C .
q ~ 2 n 2 ω ~ 2 = 0 ,
n 2 = B 1 + A + C
β g = 1 c ω ( q ) q = ω ~ ( q ~ ) q ~ .

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