shelton_1987b
=============

D\. P\. Shelton, "Rotational hyperpolarizability of a homonuclear diatomic molecule", Phys\. Rev\. A 36, 3461 \(1987\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/shelton_1987b.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/physreva\.36\.3461 <https://doi.org/10.1103/physreva.36.3461>`__

**Source kind:** theory

**Recorded technique:** Perturbation theory \(Orr & Ward\) for the off\-resonance rotational hyperpolarizability gamma^r of a homonuclear diatomic molecule, treated fully quantum mechanically\: gamma^r collects the terms with a pure rotational level of the ground electronic manifold as intermediate state; each pair of electronic sums is replaced by the static Raman transition polarizability; M sublevels averaged with the Wigner\-Eckart theorem and J with a normalised population rho\(J\), following shelton\_1987 \(pp\. 3461\-3462\)\.

**Materials in structured entries:** None recorded.

**Dataset quantities:** None recorded.

**Contents:** 0 datasets; 2 formulas.

Curation notes
--------------

A brief report with no numerical values of gamma\: only formulas \(recorded below\), Table I \(the frequency factors 6 D^2\(Omega\) for the xxxx, xxyy, xyxy and xyyx polarization combinations, in terms of D1, D2, D3 = D\(Omega; w1 \+ w2\), D\(Omega; w1 \+ w3\), D\(Omega; w2 \+ w3\), with D\(Omega; w\) = 2 Omega/\(Omega^2 \- w^2\), Eq\. 6; the same as shelton\_1987 Table 1's 6 D^2 column; image checked\) and the correction of shelton\_1987's Table 4\. Pages are the journal's \(3461\-3463; p\. = pdf page \+ 3460\)\.

gamma^r\: only Delta J = \+\-2 transitions contribute \(Delta J = 0 elements are those of JM \-\> JM, excluded, p\. 3462\), through alpha\_0^\(2\) = \(2/3\)^1/2 \(alpha\_par \- alpha\_perp\)\. With damping, D\(Omega; w\) becomes 2 Omega/\(Omega^2 \- w^2 \+ Gamma^2\) \(1 \+ i y\)/\(1 \+ y^2\), y = 2 w Gamma/\(Omega^2 \- w^2 \+ Gamma^2\) \(Eq\. 7\), Gamma the HWHM\: significant only for Gamma \>~ Omega, e\.g\. N2 above 300 atm, where the rotational Raman spectrum collapses \(p\. 3462\)\.

Eq\. \(10\), the classical limit \(hbar B/kT \<\< 1, J \>\> 1, thermal populations, the J dependence of alpha\_par \- alpha\_perp neglected\) of the dc Kerr gamma^r\: gamma^r\_dcKerr = \(\(alpha\_par \- alpha\_perp\)^2/5kT\){3/4 \+ \(1/4\)\[1 \- \(w/Omega\_m\)^2\]^\-1}, Omega\_m = 4B\(kT/hbar B\)^1/2 when w \>\> Omega\. 'The zero\-frequency limit of Eq\. \(10\) agrees exactly with the classical result \[\(alpha\_par \- alpha\_perp\)^2/5kT\]\. However, at optical frequencies where w \>\> Omega, Eq\. \(10\) gives 3/4 times the classical result\. Thus, Eqs\. \(9\) and \(10\) disagree with the classical dc Kerr result and also with the previously calculated quantum result for the dc Kerr effect \[Buckingham & Orr 1968\]'; 'it appears that Buckingham's ansatz for calculating the dc Kerr birefringence in terms of a perturbed refractive index is only valid in the static limit' \(pp\. 3462\-3463\)\. This bears on every dc Kerr gamma obtained by subtracting a Buckingham\-Orr orientation term \(tammer\_1990, tammer\_1992, buckingham\_1968b, buckingham\_1970, gentle\_1989\); Shelton & Rugar 1993 \(shelton\_1993\) later measured H2 and used the full expression\.

Ref\. 10 \(p\. 3463\), the erratum to shelton\_1987\: 'Table 4 of Ref\. 9 contains several numerical errors\. Line 2 should read D2, 2988, \+0\.007\*, \+0\.737, \+2\.464, \-0\.031, \-0\.010\. Column 6 should read ESHG, \-0\.064, \-0\.031, \-0\.008, \-0\.005' \(page image checked; applied as errata in shelton\_1987 and verified there by recomputation\)\.

Reproductions elsewhere\: shelton\_1994 Table VIII \(gamma^v of H2, D2, N2, O2 with 'the corrections given in ref 133', and gamma^R\)\. The library's validation reports references reordered on p\. 3 \(info only\)\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

No datasets are recorded.

Formulas
--------

shelton\_1987b/eq5\_rotational
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq5_rotational",
     "equation": "Eq. (5), p. 3462",
     "printed": "<gamma^r(-w_sigma; w1, w2, w3)>_abcd = (1/(30 hbar)) sum_J (alpha_par - alpha_perp)_{J,J+2}^2 [rho(J) - ((2J + 1)/(2J + 5)) rho(J + 2)] ((J + 1)(J + 2)/((2J + 1)(2J + 3))) 6 D^2(Omega_{J,J+2})",
     "variable": "rho(J): the normalised rotational population; Omega_{J,J+2}: the J -> J+2 rotational Raman frequency; 6 D^2(Omega): the frequency factor of Table I for the polarization combination abcd, from D(Omega; w) = 2 Omega/(Omega^2 - w^2) (Eq. 6)",
     "process": "other",
     "convention": "T",
     "part": "rotational",
     "range": "off resonance, damping negligible (Gamma << Omega; Eq. 7 otherwise)",
     "check": "No numerical values in the paper to reproduce."
   }

The quantum rotational hyperpolarizability of a homonuclear diatomic molecule for any off\-resonance third\-order process \(the process enters through the frequencies in Table I's D1, D2, D3\)\.

shelton\_1987b/eq9\_dc\_kerr
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq9_dc_kerr",
     "equation": "Eq. (9), p. 3462",
     "printed": "gamma^r_dcKerr = sum_J ((alpha_par - alpha_perp)_{J,J+2}^2/(5kT)) (kT/(hbar B)) [rho(J) - ((2J + 1)/(2J + 5)) rho(J + 2)] ((J + 1)(J + 2)/((2J + 1)(2J + 3)^2)) (1/2){3 + [1 - (w/Omega_{J,J+2})^2]^-1}",
     "variable": "gamma_dcKerr = (3/2)[<gamma(-w; 0, 0, w)>_xxxx - <gamma(-w; 0, 0, w)>_xyyx] (Eq. 8); rigid-rotor levels E(J) = J(J + 1) hbar B, Omega_{J,J+2} = (6 + 4J) B; damping ignored",
     "process": "dc_kerr",
     "convention": "T",
     "part": "rotational",
     "range": "off resonance, damping negligible",
     "check": "No numerical values in the paper to reproduce."
   }

Its classical limit is Eq\. \(10\) \(in the record's notes\)\: 3/4 of Buckingham's classical \(alpha\_par \- alpha\_perp\)^2/5kT at optical frequencies, equal to it in the static limit\.

Corrections
-----------

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.