Abstract

The nonlinear collective electron dynamics inside a large heated cluster irradiated by a strong linearly polarized short laser pulse are considered in the approximation of an incompressible medium. When the incident radiation frequency is near three-photon resonance with the Mie frequency, the field inside the cluster exhibits a third harmonic with an amplitude comparable with that of the fundamental. In the same parameter range, due to shielding, the field inside the cluster at the fundamental frequency is strongly reduced with respect to the incident field. The presence of the third harmonic can lead to a strong enhancement of the production of multiply charged ions. Third-harmonic generation by a cluster under the same conditions is analyzed, too.

© 2003 Optical Society of America

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  1. T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
    [Crossref] [PubMed]
  2. T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
    [Crossref]
  3. M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
    [Crossref]
  4. A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
    [Crossref]
  5. S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
    [Crossref]
  6. T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
    [Crossref] [PubMed]
  7. T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
    [Crossref]
  8. F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
    [Crossref]
  9. V.P. Krainov and M.B. Smirnov, “Cluster beams in the super-intense femtosecond laser pulse,” Phys. Rep. 370, 237–331 (2002).
    [Crossref]
  10. L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
    [Crossref]
  11. I. Last and J. Jortner, “Nuclear fusion driven by Coulomb explosion of homonuclear and heteronuclear deuteriumand tritium-containing clusters,” Phys. Rev. A 64, 063201 (1–11) (2001).
    [Crossref]
  12. T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
    [Crossref] [PubMed]
  13. E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
    [Crossref] [PubMed]
  14. K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
    [Crossref]
  15. C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
    [Crossref]
  16. I. Last and J. Jortner, “Dynamics of the Coulomb explosion of large clusters in a strong laser field,” Phys. Rev. A 62, 013201 (1–9) (2000).
    [Crossref]
  17. K. Ishikawa and T. Blenski, “Explosion dynamics of rare-gas clusters in an intense laser field,” Phys. Rev. A 62, 063204 (1–11) (2000).
    [Crossref]
  18. J.-P. Connerade and A. V. Solov’yov, “Formalism for multiphoton plasmon excitation in jellium clusters,” Phys. Rev. A 66, 013207 (1–16) (2002).
    [Crossref]
  19. P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
    [Crossref]
  20. J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
    [Crossref]
  21. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
  22. F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
    [Crossref]
  23. S. V. Fomichev and D. F. Zaretsky, “Vlasov theory of Mie resonance broadening in metal clusters,” J. Phys. B 32, 5083–5102 (1999).
    [Crossref]
  24. S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
    [Crossref]
  25. L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
    [Crossref]
  26. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, 1983).
  27. F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” J. Opt. Soc. Am. B 3, 1647–1655 (1986).
    [Crossref]
  28. J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
    [Crossref]

2003 (2)

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

2002 (3)

L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
[Crossref]

J.-P. Connerade and A. V. Solov’yov, “Formalism for multiphoton plasmon excitation in jellium clusters,” Phys. Rev. A 66, 013207 (1–16) (2002).
[Crossref]

V.P. Krainov and M.B. Smirnov, “Cluster beams in the super-intense femtosecond laser pulse,” Phys. Rep. 370, 237–331 (2002).
[Crossref]

2001 (4)

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

I. Last and J. Jortner, “Nuclear fusion driven by Coulomb explosion of homonuclear and heteronuclear deuteriumand tritium-containing clusters,” Phys. Rev. A 64, 063201 (1–11) (2001).
[Crossref]

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

2000 (3)

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

I. Last and J. Jortner, “Dynamics of the Coulomb explosion of large clusters in a strong laser field,” Phys. Rev. A 62, 013201 (1–9) (2000).
[Crossref]

K. Ishikawa and T. Blenski, “Explosion dynamics of rare-gas clusters in an intense laser field,” Phys. Rev. A 62, 063204 (1–11) (2000).
[Crossref]

1999 (3)

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

S. V. Fomichev and D. F. Zaretsky, “Vlasov theory of Mie resonance broadening in metal clusters,” J. Phys. B 32, 5083–5102 (1999).
[Crossref]

1998 (2)

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

1997 (1)

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

1996 (4)

E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
[Crossref] [PubMed]

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
[Crossref]

1995 (1)

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

1994 (2)

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

1986 (1)

Barty, C.P.J.

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

Becker, W.

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

Belkacem, M.

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

Bennemann, K.H.

J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
[Crossref]

Blenski, T.

K. Ishikawa and T. Blenski, “Explosion dynamics of rare-gas clusters in an intense laser field,” Phys. Rev. A 62, 063204 (1–11) (2000).
[Crossref]

Borisov, A.B.

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

Bouchene, M. A.

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

Boyer, K.

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

Buzza, S. A.

E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
[Crossref] [PubMed]

Calvayrac, F.

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

Campbell, E.M.

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

Castleman, Jr., A.W.

E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
[Crossref] [PubMed]

Connerade, J.-P.

J.-P. Connerade and A. V. Solov’yov, “Formalism for multiphoton plasmon excitation in jellium clusters,” Phys. Rev. A 66, 013207 (1–16) (2002).
[Crossref]

Cowan, T.E.

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Dewitz, J.P.

J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
[Crossref]

Ditmire, T.

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

Dobosz, S.

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

Donnelly, T.

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

Donnelly, T.D.

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Donnelly, Y.T.

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

Falcone, R.W.

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

Flytzanis, C.

Fomichev, S. V.

S. V. Fomichev and D. F. Zaretsky, “Vlasov theory of Mie resonance broadening in metal clusters,” J. Phys. B 32, 5083–5102 (1999).
[Crossref]

Fomichev, S.V.

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

Gerchikov, L. G.

L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
[Crossref]

Guet, C.

L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
[Crossref]

Hache, F.

Hay, N.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Hays, G.

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Hübner, W.

J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
[Crossref]

Hutchinson, M.H.R.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Ipatov, A.N.

L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
[Crossref]

Ishikawa, K.

K. Ishikawa and T. Blenski, “Explosion dynamics of rare-gas clusters in an intense laser field,” Phys. Rev. A 62, 063204 (1–11) (2000).
[Crossref]

Jortner, J.

I. Last and J. Jortner, “Nuclear fusion driven by Coulomb explosion of homonuclear and heteronuclear deuteriumand tritium-containing clusters,” Phys. Rev. A 64, 063201 (1–11) (2001).
[Crossref]

I. Last and J. Jortner, “Dynamics of the Coulomb explosion of large clusters in a strong laser field,” Phys. Rev. A 62, 013201 (1–9) (2000).
[Crossref]

Karpov, W.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Köhn, J.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Köller, L.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Krainov, V.P.

V.P. Krainov and M.B. Smirnov, “Cluster beams in the super-intense femtosecond laser pulse,” Phys. Rep. 370, 237–331 (2002).
[Crossref]

Kreibig, U.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

Last, I.

I. Last and J. Jortner, “Nuclear fusion driven by Coulomb explosion of homonuclear and heteronuclear deuteriumand tritium-containing clusters,” Phys. Rev. A 64, 063201 (1–11) (2001).
[Crossref]

I. Last and J. Jortner, “Dynamics of the Coulomb explosion of large clusters in a strong laser field,” Phys. Rev. A 62, 013201 (1–9) (2000).
[Crossref]

Lezius, M.

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

Li, R.

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, 1983).

Lieberman, M. A.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, 1983).

Liu, J.

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

Marangos, J.P.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Mason, M.B.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

McPherson, A.

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

Megi, F.

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

Meiwes-Broer, K.H.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Neuman, K.

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Nickles, P.V.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Normand, D.

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

Parks, P. B.

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

Perry, M.D.

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

Popruzhenko, S.V.

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

Radloff, W.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Reinhard, P.-G.

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

Rhodes, C.K.

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

Ricard, D.

Rose-Petruck, C.

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

Rubenchik, A.M.

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

Sandner, W.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Schafer, K.J.

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

Schmidt, M.

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

Schnürer, M.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Schumacher, M.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Shao, Y.L.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Smirnov, M.B.

V.P. Krainov and M.B. Smirnov, “Cluster beams in the super-intense femtosecond laser pulse,” Phys. Rep. 370, 237–331 (2002).
[Crossref]

Snyder, E. M.

E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
[Crossref] [PubMed]

Solov’yov, A. V.

J.-P. Connerade and A. V. Solov’yov, “Formalism for multiphoton plasmon excitation in jellium clusters,” Phys. Rev. A 66, 013207 (1–16) (2002).
[Crossref]

Springate, E.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Stephens, R.B.

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

Stiel, H.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Suraud, E.

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

Ter-Avetisyan, S.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Teuber, S.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Thompson, B.D.

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

Tiggesbäumker, J.

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

Tisch, J.W.G.

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

Ullrich, C.A.

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

Vogt, U.

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Vollmer, M.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

Wharton, K.B.

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Wilson, K.R.

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

Xu, Z.

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

Yanovsky, V.P.

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Zaretsky, D. F.

S. V. Fomichev and D. F. Zaretsky, “Vlasov theory of Mie resonance broadening in metal clusters,” J. Phys. B 32, 5083–5102 (1999).
[Crossref]

Zaretsky, D.F.

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

Zhu, P.

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

Zweiback, J.

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Zwicknagel, G.

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. B (4)

F. Megi, M. Belkacem, M. A. Bouchene, E. Suraud, and G. Zwicknagel, “On the importance of damping phenomena in clusters irradiated by intense laser fields,” J. Phys. B 36, 273–282 (2003).
[Crossref]

S. V. Fomichev and D. F. Zaretsky, “Vlasov theory of Mie resonance broadening in metal clusters,” J. Phys. B 32, 5083–5102 (1999).
[Crossref]

S.V. Fomichev, S.V. Popruzhenko, D.F. Zaretsky, and W. Becker, “Laser-induced nonlinear excitation of collective electron motion in a cluster,” J. Phys. B 36, 3817–3834 (2003).
[Crossref]

K. Boyer, B.D. Thompson, A. McPherson, and C.K. Rhodes, “Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters,” J. Phys. B 27, 4373–4389 (1994).
[Crossref]

Nature (2)

A. McPherson, B.D. Thompson, A.B. Borisov, K. Boyer, and C.K. Rhodes, “Multi-photon induced x-ray emission at 4-5 keV from Xe atoms with multiple core vacancies,” Nature (London)  370, 631–634 (1994).
[Crossref]

T. Ditmire, J. Zweiback, V.P. Yanovsky, T.E. Cowan, G. Hays, and K.B. Wharton, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature (London)  398, 489–492 (1999).
[Crossref]

Phys. Rep. (2)

F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, “Nonlinear electron dynamics in metal clusters,” Phys. Rep. 337, 493–578 (2000).
[Crossref]

V.P. Krainov and M.B. Smirnov, “Cluster beams in the super-intense femtosecond laser pulse,” Phys. Rep. 370, 237–331 (2002).
[Crossref]

Phys. Rev. A (10)

T. Ditmire, T. Donnelly, A.M. Rubenchik, R.W. Falcone, and M.D. Perry, “Interaction of intense laser pulses with atomic clusters,” Phys. Rev. A 53, 3379–3402 (1996).
[Crossref] [PubMed]

T. Ditmire, E. Springate, J.W.G. Tisch, Y.L. Shao, M.B. Mason, N. Hay, J.P. Marangos, and M.H.R. Hutchinson, “Explosion of atomic clusters heated by high-intensity femtosecond laser pulses,” Phys. Rev. A 57, 369–382 (1998).
[Crossref]

C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. Barty, “Ultrafast electron dynamics and inner-shell ionization in laser driven clusters,” Phys. Rev. A 55, 1182–1190 (1997).
[Crossref]

I. Last and J. Jortner, “Dynamics of the Coulomb explosion of large clusters in a strong laser field,” Phys. Rev. A 62, 013201 (1–9) (2000).
[Crossref]

K. Ishikawa and T. Blenski, “Explosion dynamics of rare-gas clusters in an intense laser field,” Phys. Rev. A 62, 063204 (1–11) (2000).
[Crossref]

J.-P. Connerade and A. V. Solov’yov, “Formalism for multiphoton plasmon excitation in jellium clusters,” Phys. Rev. A 66, 013207 (1–16) (2002).
[Crossref]

P. B. Parks, T.E. Cowan, R.B. Stephens, and E.M. Campbell, “Model of neutron-production rates from femtosecond-laser-cluster interactions,” Phys. Rev. A 63, 063203 (1–12) (2001).
[Crossref]

J. Liu, R. Li, P. Zhu, Z. Xu, and J. Liu, “Modified hydrodynamic model and its application in the investigation of laser-cluster interactions,” Phys. Rev. A 64, 033426 (1–7) (2001).
[Crossref]

L. G. Gerchikov, C. Guet, and A.N. Ipatov, “Multiple plasmons and anharmonic effects in small metallic clusters,” Phys. Rev. A 66, 053202 (1–8) (2002).
[Crossref]

I. Last and J. Jortner, “Nuclear fusion driven by Coulomb explosion of homonuclear and heteronuclear deuteriumand tritium-containing clusters,” Phys. Rev. A 64, 063201 (1–11) (2001).
[Crossref]

Phys. Rev. E (1)

S. Ter-Avetisyan, M. Schnürer, H. Stiel, U. Vogt, W. Radloff, W. Karpov, W. Sandner, and P.V. Nickles, “Absolute extreme ultraviolet yield from femtosecond-laser-excited Xe clusters,” Phys. Rev. E 64, 036404 (1–8) (2001).
[Crossref]

Phys. Rev. Lett. (5)

T.D. Donnelly, T. Ditmire, K. Neuman, M.D. Perry, and R.W. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998).
[Crossref]

L. Köller, M. Schumacher, J. Köhn, S. Teuber, J. Tiggesbäumker, and K.H. Meiwes-Broer, “Plasmon-enhanced multi-ionization of small metal clusters in strong femtosecond laser fields,” Phys. Rev. Lett. 82, 3783–3786 (1999).
[Crossref]

T. Ditmire, Y.T. Donnelly, R.W. Falcone, and M.D. Perry, “Strong x-ray-emission from high-temperature plasmas produced by intense irradiation of clusters,” Phys. Rev. Lett. 75, 3122–3125 (1995).
[Crossref] [PubMed]

E. M. Snyder, S. A. Buzza, and A.W. Castleman, Jr. “Intense field-matter interactions: multiple ionization of clusters,” Phys. Rev. Lett. 77, 3347–3350 (1996).
[Crossref] [PubMed]

Z. Phys. D (1)

J.P. Dewitz, W. Hübner, and K.H. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Z. Phys. D 37, 75–84 (1996).
[Crossref]

Other (2)

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, 1983).

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

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Figures (2)

Fig. 1.
Fig. 1.

Relative peak values of the electric-field envelopes at the fundamental frequency and at the third harmonic inside the cluster, Eω0 /E 0 (dashed curves) and E3ω0/E 0 (solid curves), extracted from a numerical solution of Eq. (4), versus the frequency ratio β=ωM /ω, in the third-order resonance range. The value of k corresponds to the cluster ionization degree η=0.3. The other parameters are: Ti:Sa laser, R=50 Å, Γ=0.1eV, τ=100fs, I=1016 (a) and 3×1016 W/cm2 (b).

Fig. 2.
Fig. 2.

Total production probability w 3ω and wω of Ar n+ ions inside the cluster (n is given next to the curves) by the third-harmonic (a), and by the fundamental (b), respectively, calculated according to the Coulomb-corrected Keldysh formula with the time-dependent envelopes, as a function of β=ωM /ω. The incoming laser intensity is 3×1016 W/cm2, corresponding to part (b) of Fig. 1.

Equations (7)

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

n e t + · ( v n e ) = 0 , v t + ( v · ) v = 1 m e F ,
m e x ̈ = F L ( t ) + 1 N e F ei ( x ) ,
F ei ( x ) = x { e d 3 r ϕ i ( 0 ) ( r ) n e 0 ( r x ) }
x ̈ + 2 Γ x ̇ + ω M 2 x 3 10 ω M 2 k R 2 ( 1 η ) x x 2 = e m e E L ( t ) ,
x 0 = ( e E 0 m e ω 2 ) [ ( β 2 1 ) 2 + 4 Γ 2 ω 2 ] 1 2 ,
2 ϕ 0 ( r ) = 4 π e [ z i n i 0 ( r ) n e 0 ( r ) ] , n e 0 ( r ) = n e 0 ( R ) exp { e [ ϕ 0 ( r ) ϕ 0 ( R ) ] T } ,
σ ( ω 3 ω ) = 8 π r 0 2 3 ( e 2 h ¯ ω ) 8 ( a B R ) 4 ( 27 k β 2 40 ( 1 η ) ( β 2 1 ) 3 ) 2 N e 2 ( I I at ) 2 [ ( β 2 β r 2 ) 2 + 36 Γ 2 ω 2 ] ,

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