Abstract

A nonperturbative analytic solution to the transient effective susceptibility tensor due to the presence of a rotational wave packet in nonrigid, symmetric-top molecules excited by intense, arbitrarily polarized laser pulses is presented. Calculations using the analytic model are carried out for the case of a linear and a symmetric-top molecule, including the effect of vibration–rotation coupling on the wave packet. We also compare the analytic results to a numeric solution of the Schrödinger equation for a linearly polarized pump pulse in order to assess the validity of the impulsive approximation.

© 2008 Optical Society of America

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  1. T. Heritage, J. P. Gustafon, and C. H. Lin, “Observation of coherent transient behavior in CS2 vapor,” Phys. Rev. Lett. 34, 1299-1302 (1975).
    [CrossRef]
  2. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, 013903 (2002).
    [CrossRef] [PubMed]
  3. K. Hartinger and R. A. Bartels, “Pulse polarization splitting in a transient wave plate,” Opt. Lett. 31, 3526-3528 (2006).
    [CrossRef] [PubMed]
  4. F. Noack, O. Steinkellner, P. Tzankov, H.-H. Ritze, J. Herrmann, and Y. Kida, “Raman induced phase modulation in nitrogen,” Opt. Express 13, 2467-2474 (2005).
    [CrossRef] [PubMed]
  5. V. Renard, O. Faucher, and B. Lavorel, “Measurement of laser-induced alignment of molecules by cross defocusing,” Opt. Lett. 30, 70-72 (2005).
    [CrossRef] [PubMed]
  6. K. Hartinger and R. A. Bartels, “Single-shot measurement of ultrafast time-varying phase modulation induced by femtosecond laser pulses with arbitrary polarization,” Appl. Phys. Lett. 92, 021126 (2008).
    [CrossRef]
  7. R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346-348 (2003).
    [CrossRef] [PubMed]
  8. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
    [CrossRef] [PubMed]
  9. F. H. M. Faisal, A. Abdurrouf, K. Miyazaki, and G. Miyaji, “Origin of anomalous spectra of dynamic alignments observed in N2 and O2,” Phys. Rev. Lett. 98, 143001 (2007).
    [CrossRef] [PubMed]
  10. Y. Suzuki and T. Seideman, “Mapping rotational coherences onto time-resolved photoelectron imaging observables,” J. Chem. Phys. 122, 234302 (2005).
    [CrossRef] [PubMed]
  11. D. Zeidler, A. B. Bardon, A. Staudte, D. M. Villeneuve, R. Dorner, and P. B. Corkum, “Alignment independence of the instantaneous ionization rate for nitrogen molecules,” J. Phys. B 39, L159-L166 (2006).
    [CrossRef]
  12. P. A. Block, E. J. Bohac, and R. E. Miller, “Spectroscopy of pendular states--the use of molecular-complexes in achieving orientation,” Phys. Rev. Lett. 68, 1303-1306 (1992).
    [CrossRef] [PubMed]
  13. M. F. Gelin, C. Riehn, V. V. Matylitsky, and B. Brutschy, “Rotational recurrences in thermal ensembles of nonrigid molecules,” Chem. Phys. 290, 307-318 (2003).
    [CrossRef]
  14. V. Aquilanti, M. Bartolomei, F. Pirani, D. Cappelletti, F. Vecchiocattivi, Y. Shimizu, and T. Kasai, “Orienting and aligning molecules for stereochemistry and photodynamics,” Phys. Chem. Chem. Phys. 7, 291-300 (2005).
    [CrossRef] [PubMed]
  15. S. Ramakrishna and T. Seideman, “Coherence spectroscopy in dissipative media: A Liouville space pathway approach,” J. Chem. Phys. 122, 084502 (2005).
    [CrossRef]
  16. R. Santra, “Imaging molecular orbitals using photoionization,” Chem. Phys. 329, 357-364 (2006).
    [CrossRef]
  17. M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
    [CrossRef] [PubMed]
  18. T. Seideman and E. Hamilton, “Nonadiabatic alignment by intense pulses, concepts, theory, and directions,” Adv. At., Mol., Opt. Phys. 52, 282-329 (2005).
  19. E. Péronne, M. D. Poulsen, C. Z. Bisgaard, H. Stapelfeldt, and T. Seideman, “Nonadiabatic alignment of asymmetric top molecules: field-free alignment of iodobenzene,” Phys. Rev. Lett. 91, 043003 (2003).
    [CrossRef] [PubMed]
  20. E. Hamilton, T. Seideman, T. Ejdrup, M. D. Poulsen, C. Z. Bisgaard, S. S. Viftrup, and H. Stapelfeldt, “Alignment of symmetric top molecules by short laser pulses,” Phys. Rev. A 72, 043402 (2005).
    [CrossRef]
  21. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford U. Press, 1999).
  22. V. V. Matylitsky, W. Jarzeba, C. Riehn, and B. Brutschy, “Femtosecond degenerate four-wave mixing study of benzene in the gas phase,” J. Raman Spectrosc. 33, 877-883 (2002).
    [CrossRef]
  23. R. Zare, Angular Momentum (Wiley, 1988).
  24. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).
  25. Z. Majcherova, P. Macko, D. Romanini, V. Perevalov, S. Tashkun, J.-L. Teffo, and A. Campargue, “High-sensitivity CW-cavity ringdown spectroscopy of CO212 near 1.5μm,” J. Mol. Spectrosc. 230, 1-21 (2005).
    [CrossRef]
  26. P. M. Felker, “Rotational coherence spectroscopy: studies of the geometries of large gas-phase species by picosecond time-domain methods,” J. Phys. Chem. 96, 7844-7857 (1992).
    [CrossRef]
  27. F. Ito, “Methyl iodide clusters observed in gas phase by infrared cavity ring-down spectroscopy: the CH3 bending mode at 8μm,” J. Chem. Phys. 124, 054309 (2006).
    [CrossRef] [PubMed]
  28. G. Herzberg, Molecular Spectra and Molecular Structure Volume II (Krieger, 1991).
  29. A. Gray and R. Butcher, “High-resolution laser-radiofrequency double resonance molecular spectroscopy,” Proc. R. Soc. London, Ser. A 445, 543-560 (1994).
    [CrossRef]
  30. E. B. Wilson, Jr., “The statistical weights of the rotational levels of polyatomic molecules, including methane, ammonia, benzene, cyclopropane, and ethylene,” J. Chem. Phys. 3, 276-285 (1935).
    [CrossRef]

2008 (1)

K. Hartinger and R. A. Bartels, “Single-shot measurement of ultrafast time-varying phase modulation induced by femtosecond laser pulses with arbitrary polarization,” Appl. Phys. Lett. 92, 021126 (2008).
[CrossRef]

2007 (1)

F. H. M. Faisal, A. Abdurrouf, K. Miyazaki, and G. Miyaji, “Origin of anomalous spectra of dynamic alignments observed in N2 and O2,” Phys. Rev. Lett. 98, 143001 (2007).
[CrossRef] [PubMed]

2006 (4)

K. Hartinger and R. A. Bartels, “Pulse polarization splitting in a transient wave plate,” Opt. Lett. 31, 3526-3528 (2006).
[CrossRef] [PubMed]

D. Zeidler, A. B. Bardon, A. Staudte, D. M. Villeneuve, R. Dorner, and P. B. Corkum, “Alignment independence of the instantaneous ionization rate for nitrogen molecules,” J. Phys. B 39, L159-L166 (2006).
[CrossRef]

R. Santra, “Imaging molecular orbitals using photoionization,” Chem. Phys. 329, 357-364 (2006).
[CrossRef]

F. Ito, “Methyl iodide clusters observed in gas phase by infrared cavity ring-down spectroscopy: the CH3 bending mode at 8μm,” J. Chem. Phys. 124, 054309 (2006).
[CrossRef] [PubMed]

2005 (8)

Z. Majcherova, P. Macko, D. Romanini, V. Perevalov, S. Tashkun, J.-L. Teffo, and A. Campargue, “High-sensitivity CW-cavity ringdown spectroscopy of CO212 near 1.5μm,” J. Mol. Spectrosc. 230, 1-21 (2005).
[CrossRef]

T. Seideman and E. Hamilton, “Nonadiabatic alignment by intense pulses, concepts, theory, and directions,” Adv. At., Mol., Opt. Phys. 52, 282-329 (2005).

E. Hamilton, T. Seideman, T. Ejdrup, M. D. Poulsen, C. Z. Bisgaard, S. S. Viftrup, and H. Stapelfeldt, “Alignment of symmetric top molecules by short laser pulses,” Phys. Rev. A 72, 043402 (2005).
[CrossRef]

V. Aquilanti, M. Bartolomei, F. Pirani, D. Cappelletti, F. Vecchiocattivi, Y. Shimizu, and T. Kasai, “Orienting and aligning molecules for stereochemistry and photodynamics,” Phys. Chem. Chem. Phys. 7, 291-300 (2005).
[CrossRef] [PubMed]

S. Ramakrishna and T. Seideman, “Coherence spectroscopy in dissipative media: A Liouville space pathway approach,” J. Chem. Phys. 122, 084502 (2005).
[CrossRef]

F. Noack, O. Steinkellner, P. Tzankov, H.-H. Ritze, J. Herrmann, and Y. Kida, “Raman induced phase modulation in nitrogen,” Opt. Express 13, 2467-2474 (2005).
[CrossRef] [PubMed]

V. Renard, O. Faucher, and B. Lavorel, “Measurement of laser-induced alignment of molecules by cross defocusing,” Opt. Lett. 30, 70-72 (2005).
[CrossRef] [PubMed]

Y. Suzuki and T. Seideman, “Mapping rotational coherences onto time-resolved photoelectron imaging observables,” J. Chem. Phys. 122, 234302 (2005).
[CrossRef] [PubMed]

2004 (1)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

2003 (4)

M. F. Gelin, C. Riehn, V. V. Matylitsky, and B. Brutschy, “Rotational recurrences in thermal ensembles of nonrigid molecules,” Chem. Phys. 290, 307-318 (2003).
[CrossRef]

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef] [PubMed]

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346-348 (2003).
[CrossRef] [PubMed]

E. Péronne, M. D. Poulsen, C. Z. Bisgaard, H. Stapelfeldt, and T. Seideman, “Nonadiabatic alignment of asymmetric top molecules: field-free alignment of iodobenzene,” Phys. Rev. Lett. 91, 043003 (2003).
[CrossRef] [PubMed]

2002 (2)

V. V. Matylitsky, W. Jarzeba, C. Riehn, and B. Brutschy, “Femtosecond degenerate four-wave mixing study of benzene in the gas phase,” J. Raman Spectrosc. 33, 877-883 (2002).
[CrossRef]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, 013903 (2002).
[CrossRef] [PubMed]

1994 (1)

A. Gray and R. Butcher, “High-resolution laser-radiofrequency double resonance molecular spectroscopy,” Proc. R. Soc. London, Ser. A 445, 543-560 (1994).
[CrossRef]

1992 (2)

P. M. Felker, “Rotational coherence spectroscopy: studies of the geometries of large gas-phase species by picosecond time-domain methods,” J. Phys. Chem. 96, 7844-7857 (1992).
[CrossRef]

P. A. Block, E. J. Bohac, and R. E. Miller, “Spectroscopy of pendular states--the use of molecular-complexes in achieving orientation,” Phys. Rev. Lett. 68, 1303-1306 (1992).
[CrossRef] [PubMed]

1975 (1)

T. Heritage, J. P. Gustafon, and C. H. Lin, “Observation of coherent transient behavior in CS2 vapor,” Phys. Rev. Lett. 34, 1299-1302 (1975).
[CrossRef]

1935 (1)

E. B. Wilson, Jr., “The statistical weights of the rotational levels of polyatomic molecules, including methane, ammonia, benzene, cyclopropane, and ethylene,” J. Chem. Phys. 3, 276-285 (1935).
[CrossRef]

Adv. At., Mol., Opt. Phys. (1)

T. Seideman and E. Hamilton, “Nonadiabatic alignment by intense pulses, concepts, theory, and directions,” Adv. At., Mol., Opt. Phys. 52, 282-329 (2005).

Appl. Phys. Lett. (1)

K. Hartinger and R. A. Bartels, “Single-shot measurement of ultrafast time-varying phase modulation induced by femtosecond laser pulses with arbitrary polarization,” Appl. Phys. Lett. 92, 021126 (2008).
[CrossRef]

Chem. Phys. (2)

M. F. Gelin, C. Riehn, V. V. Matylitsky, and B. Brutschy, “Rotational recurrences in thermal ensembles of nonrigid molecules,” Chem. Phys. 290, 307-318 (2003).
[CrossRef]

R. Santra, “Imaging molecular orbitals using photoionization,” Chem. Phys. 329, 357-364 (2006).
[CrossRef]

J. Chem. Phys. (4)

S. Ramakrishna and T. Seideman, “Coherence spectroscopy in dissipative media: A Liouville space pathway approach,” J. Chem. Phys. 122, 084502 (2005).
[CrossRef]

Y. Suzuki and T. Seideman, “Mapping rotational coherences onto time-resolved photoelectron imaging observables,” J. Chem. Phys. 122, 234302 (2005).
[CrossRef] [PubMed]

F. Ito, “Methyl iodide clusters observed in gas phase by infrared cavity ring-down spectroscopy: the CH3 bending mode at 8μm,” J. Chem. Phys. 124, 054309 (2006).
[CrossRef] [PubMed]

E. B. Wilson, Jr., “The statistical weights of the rotational levels of polyatomic molecules, including methane, ammonia, benzene, cyclopropane, and ethylene,” J. Chem. Phys. 3, 276-285 (1935).
[CrossRef]

J. Mol. Spectrosc. (1)

Z. Majcherova, P. Macko, D. Romanini, V. Perevalov, S. Tashkun, J.-L. Teffo, and A. Campargue, “High-sensitivity CW-cavity ringdown spectroscopy of CO212 near 1.5μm,” J. Mol. Spectrosc. 230, 1-21 (2005).
[CrossRef]

J. Phys. B (1)

D. Zeidler, A. B. Bardon, A. Staudte, D. M. Villeneuve, R. Dorner, and P. B. Corkum, “Alignment independence of the instantaneous ionization rate for nitrogen molecules,” J. Phys. B 39, L159-L166 (2006).
[CrossRef]

J. Phys. Chem. (1)

P. M. Felker, “Rotational coherence spectroscopy: studies of the geometries of large gas-phase species by picosecond time-domain methods,” J. Phys. Chem. 96, 7844-7857 (1992).
[CrossRef]

J. Raman Spectrosc. (1)

V. V. Matylitsky, W. Jarzeba, C. Riehn, and B. Brutschy, “Femtosecond degenerate four-wave mixing study of benzene in the gas phase,” J. Raman Spectrosc. 33, 877-883 (2002).
[CrossRef]

Nature (1)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Phys. Chem. Chem. Phys. (1)

V. Aquilanti, M. Bartolomei, F. Pirani, D. Cappelletti, F. Vecchiocattivi, Y. Shimizu, and T. Kasai, “Orienting and aligning molecules for stereochemistry and photodynamics,” Phys. Chem. Chem. Phys. 7, 291-300 (2005).
[CrossRef] [PubMed]

Phys. Rev. A (1)

E. Hamilton, T. Seideman, T. Ejdrup, M. D. Poulsen, C. Z. Bisgaard, S. S. Viftrup, and H. Stapelfeldt, “Alignment of symmetric top molecules by short laser pulses,” Phys. Rev. A 72, 043402 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

E. Péronne, M. D. Poulsen, C. Z. Bisgaard, H. Stapelfeldt, and T. Seideman, “Nonadiabatic alignment of asymmetric top molecules: field-free alignment of iodobenzene,” Phys. Rev. Lett. 91, 043003 (2003).
[CrossRef] [PubMed]

P. A. Block, E. J. Bohac, and R. E. Miller, “Spectroscopy of pendular states--the use of molecular-complexes in achieving orientation,” Phys. Rev. Lett. 68, 1303-1306 (1992).
[CrossRef] [PubMed]

M. Leibscher, I. Sh. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90, 213001 (2003).
[CrossRef] [PubMed]

F. H. M. Faisal, A. Abdurrouf, K. Miyazaki, and G. Miyaji, “Origin of anomalous spectra of dynamic alignments observed in N2 and O2,” Phys. Rev. Lett. 98, 143001 (2007).
[CrossRef] [PubMed]

T. Heritage, J. P. Gustafon, and C. H. Lin, “Observation of coherent transient behavior in CS2 vapor,” Phys. Rev. Lett. 34, 1299-1302 (1975).
[CrossRef]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88, 013903 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

A. Gray and R. Butcher, “High-resolution laser-radiofrequency double resonance molecular spectroscopy,” Proc. R. Soc. London, Ser. A 445, 543-560 (1994).
[CrossRef]

Other (4)

G. Herzberg, Molecular Spectra and Molecular Structure Volume II (Krieger, 1991).

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford U. Press, 1999).

R. Zare, Angular Momentum (Wiley, 1988).

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, 2000).

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Figures (3)

Fig. 1
Fig. 1

b L 0 coefficients for L = 2 (dotted), L = 4 (dashed), and L = 6 (solid) for circularly (a), elliptically (b), and linearly (c) polarized pump pulses with a peak intensity of 3 × 10 14 W cm 2 , a pulse duration of 20 fs , and a gas temperature of 50 K . The pump pulse propagates along z ̂ . The orientational probability distribution corresponding to the respective pump pulse polarization is shown to the right of each panel at the peak of the quarter-revival at τ = 10.6 ps .

Fig. 2
Fig. 2

(a) Alignment of CH 3 I at T = 10 K for a linearly polarized pulse resulting in P lin = 10.26 . (b) Effect due to centrifugal distortion on the full revival of the rotational wave packet comparing the first (dashed) to the 30th (solid) full revival.

Fig. 3
Fig. 3

(a) Comparison of relative error in cos 2 θ ( τ ) for three different combinations of pulse intensity and duration, each resulting in a kick strength of P lin = 5.13 : I o = 2.5 × 10 13 W cm 2 and τ = 80 fs (dotted), I o = 5 × 10 13 W cm 2 and τ = 40 fs (dashed), I o = 2 × 10 14 W cm 2 and τ = 10 fs (solid). Clearly, the relative error increases with pulse duration for a fixed kick strength. (b) Relative error for P lin = 2.56 ( I o = 2 × 10 14 W cm 2 , τ = 5 fs , solid) compared to P lin = 5.13 . Case I: I o = 4 × 10 14 W cm 2 , τ = 5 fs (dashed), Case II: I o = 2 × 10 14 W cm 2 , τ = 10 fs (dotted). The loss in accuracy is higher for longer pulse durations, rather than higher pulse intensities.

Equations (17)

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E ( t ) = 1 2 E ( t ) e i ( ω o t k o z ) ( a x ̂ + i b y ̂ ) + c.c.
H int = Δ α 8 E ( t ) 2 sin 2 θ [ 1 + A cos 2 ϕ ] ,
P = Δ α 8 t i t f E ( t ) 2 d t .
c J , M J 0 , K 0 , M 0 = J K 0 M exp ( i P sin 2 θ [ 1 + A cos ( 2 ϕ ) ] ) J 0 K 0 M 0 .
c J , M J 0 , K 0 , M 0 = ν = 0 ( i P ) ν ν ! k = 0 ν ( ν k ) A k n = 0 k b n k J K 0 M sin 2 ν ( θ ) cos ( 2 n ϕ ) J 0 K 0 M 0 ,
b n k = 1 2 k 1 + δ n , 0 ( k k 2 n 2 ) ,
c J , M J 0 , K 0 , M 0 = ( 1 ) K 0 + M 0 2 ( 1 + δ M , M 0 ) ( 2 J + 1 ) ( 2 J o + 1 ) ν = Δ M 2 ( i P ) ν ν ! k = Δ M 2 ν ( ν k ) A k b Δ M 2 k l = Max { Δ M , J J o } J + J o g l , Δ M 2 ν [ J l J 0 K 0 0 K 0 ] [ J l J 0 M Δ M M 0 ]
g l , n ν = ( 2 l + 1 ) ( l 2 n ) ! ( l + 2 n ) ! π 2 2 n 1 Γ ( ν + 1 + n ) Γ ( ν + 1 n ) Γ ( ν + 1 + 1 2 l + 1 2 ) Γ ( ν + 1 1 2 l ) Γ ( n + 1 2 l + 1 ) Γ ( n 1 2 l + 1 2 ) .
c J M 0 J 0 K 0 M 0 = ( 1 ) K 0 + M 0 ( 2 J + 1 ) ( 2 J 0 + 1 ) ν = 0 ( i P lin ) ν ν ! l = l min Min ( J + J 0 , 2 ν ) ( 2 l + 1 ) π 2 2 ν 1 Γ ( 1 + 2 ν ) Γ ( 1 + ν 1 2 l ) Γ ( ν + 1 2 l + 3 2 ) [ J l J 0 K 0 0 K 0 ] [ J l J 0 M 0 0 M 0 ] ,
ψ ( t ) = J , M c J , M J 0 , K 0 , M 0 e i E J K 0 t J K 0 M .
G ( θ , ϕ , t ) = J 0 = 0 K 0 = J 0 J 0 W ( J 0 , K 0 ) M 0 = J 0 J 0 ϕ , θ , χ ψ ( t ) 2 ,
G ( θ , ϕ , t ) = L = 0 m = L L b L m ( t ) Y L m ( θ , ϕ ) .
χ ( t ) = N ϵ o α ( ϕ , θ ) J 0 K 0 W ( J 0 , K 0 ) M 0 ψ K 0 M 0 J 0 * ( ϕ , θ , χ , t ) ψ K 0 M 0 J 0 ( ϕ , θ , χ , t ) d Ω ,
χ ( t ) = N ϵ o L = 0 m = L L b L m ( t ) α ( ϕ , θ ) Y L m ( θ , ϕ ) d Ω .
χ x x ( t ) = N ϵ o { α ¯ 2 3 π 5 Δ α [ b 2 0 ( t ) 6 b 2 2 ( t ) ] } ,
χ y y ( t ) = N ϵ o { α ¯ 2 3 π 5 Δ α [ b 2 0 ( t ) + 6 b 2 2 ( t ) ] } ,
χ z z ( t ) = N ϵ o { α ¯ + 4 3 π 5 Δ α b 2 0 ( t ) } ,

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