Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field

Jisoo Hwang; Byoungchoo Park; H. J. Chang; J. W. Wu

doi:10.1364/JOSAB.19.001620

Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field

Jisoo Hwang, Byoungchoo Park, H. J. Chang, and J. W. Wu

Journal of the Optical Society of America B
Vol. 19,
Issue 7,
pp. 1620-1626
(2002)
•https://doi.org/10.1364/JOSAB.19.001620

Get Citation
Copy Citation Text
Jisoo Hwang, Byoungchoo Park, H. J. Chang, and J. W. Wu, "Unique determination of the hyperpolarizability tensor ratio through depolarized hyper-Rayleigh scattering under an external electric field," J. Opt. Soc. Am. B 19, 1620-1626 (2002)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (169 KB)
Set citation alerts
Save article

Related Topics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Harmonic generation and mixing (190.2620)
- Multiharmonic generation (190.4160)
- Optical nonlinearities in organic materials (190.4710)
- Scattering, Rayleigh (290.5870)

About this Article
History
- Original Manuscript: July 30, 2001
- Revised Manuscript: November 26, 2001
- Published: July 1, 2002

Not Accessible

Your account may give you access

Abstract

We have analyzed the depolarized hyper-Rayleigh scattering from a molecular system with partial macroscopic polar ordering. It is shown that the hyperpolarizability tensor ratio and the molecular dipole moment can be determined simultaneously by the measurement of depolarization as a function of external field strength. As an experimental example, we performed a quantitative analysis of the electric-field-dependent depolarized hyper-Rayleigh signal from a poly-γ-benzyl-l-glutamate solution, obtaining the dipole moment and the ratio of hyperpolarizability components as 4.0 D and $β_{311} / β_{333} = - 0.81,$ respectively.

Full Article | PDF Article

More Like This

Measurements of the hyperpolarizability tensor by means of hyper-Rayleigh scattering

V. Ostroverkhov, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien
J. Opt. Soc. Am. B 17(9) 1531-1542 (2000)

Polarization and angle dependence for hyper-Rayleigh scattering from local and nonlocal modes of isotropic fluids

David P. Shelton
J. Opt. Soc. Am. B 17(12) 2032-2036 (2000)

Optimization of a dual-rotating-retarder polarimeter designed for hyper-Rayleigh scattering

Paul Lemaillet, Fabrice Pellen, Sylvain Rivet, Bernard Le Jeune, and Jack Cariou
J. Opt. Soc. Am. B 24(3) 609-614 (2007)

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Figures (2)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Tables (2)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Equations (7)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Metrics

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

$〈 ϕ_{I_{p}}^{6} 〉_{f (θ)}$	$\frac{1}{7} i_{0} (x) + \frac{10}{21} i_{2} (x) + \frac{24}{77} i_{4} (x) + \frac{16}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{4} ϕ_{I_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{21} i_{2} (x) - \frac{16}{385} i_{4} (x) - \frac{8}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{I_{q}}^{2} ϕ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{55} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{4} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) - \frac{3}{55} i_{4} (x) + \frac{2}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{4} 〉_{f (θ)}$	$\frac{3}{35} i_{0} (x) + \frac{3}{14} i_{2} (x) + \frac{201}{3080} i_{4} (x) + \frac{3}{308} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{J_{q}}^{2} 〉_{f (θ)}$	$\frac{2}{105} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{23}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{14} i_{2} (x) + \frac{67}{3080} i_{4} (x) + \frac{1}{308} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}}^{3} ϕ_{J_{q}} 〉_{f (θ)}$	$- \frac{1}{70} i_{0} (x) + \frac{3}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) - \frac{1}{168} i_{2} (x) + \frac{23}{3080} i_{4} (x) + \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{p}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{385} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{37}{770} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{K_{r}}^{2} 〉_{f (θ)}$	$\frac{8}{105} i_{0} (x) + \frac{9}{42} i_{2} (x) + \frac{529}{6160} i_{4} (x) + \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{K_{p}} ϕ_{K_{q}} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) + \frac{1}{55} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$- \frac{1}{42} i_{0} (x) - \frac{1}{14} i_{2} (x) - \frac{19}{616} i_{4} (x) + \frac{4}{231} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) + \frac{1}{168} i_{2} (x) - \frac{51}{3080} i_{4} (x) + \frac{1}{231} i_{6} (x)$

$〈 ϕ_{I_{p}}^{6} 〉_{f (θ)}$	$\frac{1}{7} i_{0} (x) + \frac{10}{21} i_{2} (x) + \frac{24}{77} i_{4} (x) + \frac{16}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{4} ϕ_{I_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{21} i_{2} (x) - \frac{16}{385} i_{4} (x) - \frac{8}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{I_{q}}^{2} ϕ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{55} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{4} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) - \frac{3}{55} i_{4} (x) + \frac{2}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{4} 〉_{f (θ)}$	$\frac{3}{35} i_{0} (x) + \frac{3}{14} i_{2} (x) + \frac{201}{3080} i_{4} (x) + \frac{3}{308} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{J_{q}}^{2} 〉_{f (θ)}$	$\frac{2}{105} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{23}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{14} i_{2} (x) + \frac{67}{3080} i_{4} (x) + \frac{1}{308} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}}^{3} ϕ_{J_{q}} 〉_{f (θ)}$	$- \frac{1}{70} i_{0} (x) + \frac{3}{770} i_{4} (x) - \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{J_{r}}^{2} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) - \frac{1}{168} i_{2} (x) + \frac{23}{3080} i_{4} (x) + \frac{1}{77} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{p}}^{2} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) - \frac{1}{385} i_{4} (x) + \frac{2}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}}^{2} ϕ_{K_{q}}^{2} 〉_{f (θ)}$	$\frac{1}{35} i_{0} (x) + \frac{1}{42} i_{2} (x) - \frac{37}{770} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}}^{2} ϕ_{K_{r}}^{2} 〉_{f (θ)}$	$\frac{8}{105} i_{0} (x) + \frac{9}{42} i_{2} (x) + \frac{529}{6160} i_{4} (x) + \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{p}} ϕ_{J_{q}} ϕ_{K_{p}} ϕ_{K_{q}} 〉_{f (θ)}$	$- \frac{1}{210} i_{0} (x) + \frac{1}{55} i_{4} (x) - \frac{1}{231} i_{6} (x)$
$〈 ϕ_{I_{p}}^{2} ϕ_{J_{q}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$- \frac{1}{42} i_{0} (x) - \frac{1}{14} i_{2} (x) - \frac{19}{616} i_{4} (x) + \frac{4}{231} i_{6} (x)$
$〈 ϕ_{I_{p}} ϕ_{I_{q}} ϕ_{J_{p}} ϕ_{J_{r}} ϕ_{K_{q}} ϕ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{105} i_{0} (x) + \frac{1}{168} i_{2} (x) - \frac{51}{3080} i_{4} (x) + \frac{1}{231} i_{6} (x)$

	$α = \sum_{p} β_{ppp}^{2}$	$β_{1} = \sum_{p, q} β_{ppp} β_{pqq}$	$β_{2} = \sum_{p, q} β_{ppp} (β_{qpq} + β_{qqp})$	$β_{3} = \sum_{p, q} β_{pqq}^{2}$
$A 1$	$〈 Φ_{I_{p}}^{6} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$
$A 2$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)}$
$A 3$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{p}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 4$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$
$A 5$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{4} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$
$A 6$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{p}}^{2} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)}$
	$β_{4} = \sum_{p, q} β_{pqq} (β_{qpq} + β_{qqp})$	$β_{5} = \sum_{p, q} (β_{qpq} + β_{qqp})^{2}$	$γ_{1} = \sum_{p, q, r} β_{pqq} β_{prr}$	$γ_{2} = \sum_{p, q, r} (β_{qpq} + β_{qqp}) (β_{rpr} + β_{rrp})$

$A 1$	$2 〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{4} Φ_{I_{q}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$
$A 2$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 3$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{K_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
$A 4$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 5$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 4 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}}^{3} Φ_{J_{q}} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{4} 〉_{f (θ)} + 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{J_{q}}^{2} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 3 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 6$	$8 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}}^{2} Φ_{K_{q}}^{2} 〉_{f (θ)} + 2 〈 Φ_{I_{p}}^{2} Φ_{J_{p}} Φ_{J_{q}} Φ_{K_{p}} Φ_{K_{q}} 〉_{f (θ)}$	$4 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)} + 2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
	$γ_{3} = \sum_{p, q, r} β_{pqq} (β_{rpr} + β_{rrp})$	$δ_{1} = \sum_{p, q, r} (β_{pqr} + β_{prq})^{2}$		$δ_{2} = \sum_{p, q, r} (β_{qpq} + β_{prp}) (β_{qpr} + β_{qrp})$
$A 1$	$2 〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{I_{q}}^{2} Φ_{I_{r}}^{2} 〉_{f (θ)}$
$A 2$	$2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 3$	$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$
$A 4$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$\frac{1}{2} 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 5$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$		$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{J_{r}}^{2} 〉_{f (θ)} + 3 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{q}} Φ_{J_{r}}^{2} 〉_{f (θ)}$
$A 6$	$8 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$	$〈 Φ_{I_{p}}^{2} Φ_{J_{q}}^{2} Φ_{K_{r}}^{2} 〉_{f (θ)}$		$2 〈 Φ_{I_{p}}^{2} Φ_{J_{q}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)} + 2 〈 Φ_{I_{p}} Φ_{I_{q}} Φ_{J_{p}} Φ_{J_{r}} Φ_{K_{q}} Φ_{K_{r}} 〉_{f (θ)}$

Abstract

References

Cited By

Figures (2)

Tables (2)

Equations (7)

Metrics

Login or Create Account