Raman lines#

Read Third-order conventions for the division between electronic and delayed response. This chapter defines the line parameters; The Raman response assembles them into a causal kernel. Examples start in Quick start, and Raman validation and limitations collects checks and unresolved questions.

The delayed, nuclear part of a molecular gas’s third-order response comes from its Raman lines:

  • the rotational S branch, J → J + 2, of a linear molecule;

  • the vibrational Q branch of each Raman-active mode.

optmat turns the curated spectroscopy into those lines, each line’s frequency, coupling amplitude and width, at the temperature, density, composition and pump wavelength asked for (TC-439, TC-442, TC-492, TC-541, TC-542). The response function built from them, with its tensor form and its density dependence, is on The Raman response (TC-685); its sampling belongs to the propagation code.

from optmat.gas.linewidths import AMAGAT
from optmat.gas.raman import raman_lines, raman_parameters

lines = raman_lines(
    "N2", 295.0, AMAGAT, {"self": 0.78, "O2": 0.21, "Ar": 0.01}, wavelength=800e-9
)
lines.branch, lines.j  # "S" or "Q", and each line's lower level
lines.frequency  # Omega, rad/s
lines.amplitude  # A, m^5 V^-2 s^-1 (the convention below)
lines.fwhm  # Hz; the dephasing rate is Gamma = pi FWHM
raman_parameters("N2").reasons  # why each constant, strength and width was chosen

The convention#

For a field linearly polarised along x, the Raman polarisation of a gas of number density N is

\[ P_x(t) = \varepsilon_0 N E_x(t) \int_0^\infty h(\tau) E_x^2(t - \tau)\, d\tau, \qquad h(\tau) = \sum_k A_k \sin(\Omega_k \tau) e^{-\Gamma_k \tau}. \]

The amplitudes are:

  • Rotational S branch: \(A_J = \frac{2}{15} \frac{\Delta\alpha^2}{\hbar\varepsilon_0} \frac{(J+1)(J+2)}{2J+3} (n_J - n_{J+2})\), with \(n_J\) the population of one sublevel.

    • The nuclear-spin weights follow from exchange symmetry: H2 1:3 (even:odd), D2 and N2 6:3, O2 odd N only, CO2 even J only.

    • The anisotropy may depend on J (H2).

  • Vibrational Q branch: \(A_J = |\langle 0|\bar\alpha|1\rangle|^2 P_J / (\hbar\varepsilon_0)\), one line per rotational level, where \(P_J\) is the fraction of molecules in level J.

    • The line positions are \(\nu_0 + (B_1 - B_0)J(J+1) - (D_1 - D_0)[J(J+1)]^2\).

    • A band whose rotational structure is not resolved is one line: the spherical tops (CH4, SF6), and the bands of CO2 and N2O. In raman.toml, lower rotational constants imply a resolved band unless resolved = false is stated. A resolved band needs both ladders, with the upper one specified by upper, fit_lines or b_difference; an incomplete request raises DataError instead of silently producing one unresolved line (TC-735).

    • The transition polarizability comes from one of three sources:

      • a curated transition element, per J where it is tabulated. H2 and D2 use Hunt et al.’s ab initio elements, whose mean is (⟨α∥⟩ + 2⟨α⊥⟩)/3.

      • \(\partial\bar\alpha/\partial Q\) along the mass-weighted normal coordinate, since \(|\langle 0|Q|1\rangle|^2 = \hbar/2\omega\). No reduced mass or bond convention enters, so polyatomic molecules need no per-bond convention.

      • Schrötter & Klöckner’s normalised cross section Σ, the band’s cross section relative to N2’s Q branch with the ν⁴ and Boltzmann factors removed. With N2’s absolute value \(C = 5.05\times10^{-48}\) cm⁶/sr and the band’s depolarisation ratio ρ, \(\bar a^2 = \Sigma C/(16\pi^4)/(1 + 7\rho/(3 - 4\rho))\).

    • Dispersion: where Bischel & Black’s single-resonance law is curated (N2, O2, H2, D2, CH4), the element is scaled from the wavelength at which it holds to the pump, by \((\nu_i^2 - \nu_{\rm ref}^2)/(\nu_i^2 - \nu_p^2)\). Pass wavelength to raman_lines. For H2 this law follows Rychlewski’s ab initio dispersion of the Q(0) element to 0.8 % from 238 to 911 nm (TC-542). The anisotropies are not dispersed.

  • Widths: \(\Delta\nu = c + A(T)/n + n\sum_p x_p\gamma_p(J, T)\), in optmat.gas.linewidths:

    • \(A\) is the Dicke term of H2 and D2;

    • \(\gamma_p\) is the broadening by collision partner p, per line where the data are per line, with its temperature law.

    The widths are per line: the overlap and mixing of lines at high density, the collapse of a Q branch, is not modelled.

The steady-state Stokes gain of a line follows as \(g = \omega_s N A / (2 c^2 \varepsilon_0 \Gamma)\) (stokes_gain()).

The parameters#

Each gas’s choices, with the reason for each, are in data/gas/raman.toml:

Gas

B (cm-1)

D (cm-1)

Delta alpha (au)

S FWHM (GHz)

nu_0 (cm-1)

alpha_01 (au)

Q FWHM (GHz)

N2

1.98957

5.760e-06

4.521

2.46-3.85

2329.917

0.3789

1.36-3.93

O2

1.43767

4.839e-06

6.883

1.05-2.95

1556.391

0.3652

1.89-3.21

H2

59.33920

4.599e-02

2.104-2.318 (per J)

0.08-0.11

4160.200

0.7392-0.7546 (per J)

0.378

D2

29.91050

1.134e-02

2.008

0.05-0.16

2993.500

0.6163-0.6229 (per J)

0.183-0.274

CH4

2916.700

1.1235

8.58

SF6

775.000

0.7293

48

CO2

0.39026

1.270e-07

14.253

5.40-6.60

1388.000

0.3995

6.6

N2O

1285.000

0.5404

5.57

(Widths at 295 K and 1 atm of the pure gas, over the lines kept. The transition polarizabilities are as curated: static for H2 and D2, near 436 nm for N2 and O2, and visible for the others.)

  • N2:

    • Bendtsen’s 1974 constants.

    • Wahlstrand et al.’s anisotropy, measured from the rotational response at 800 nm.

    • Murphy et al.’s ∂α/∂Q. Schrötter & Klöckner’s absolute cross section gives 0.375-0.384 au, which brackets it (TC-518).

    • Herring et al.’s per-J rotational widths and temperature laws.

    • Q widths (TC-541): Rosasco et al.’s measured Q(J) widths at 298 K, with a power law per J fitted to Rahn & Palmer’s tables at 500-1500 K. These replace Koszykowski et al.’s exponential-gap law, and they reproduce every table to an rms of 0.6-5.5 %.

  • O2:

    • Hajigeorgiou’s levels, which Yu et al.’s Dunham fit reproduces (TC-523).

    • Wahlstrand et al.’s anisotropy.

    • Hölzer et al.’s per-N rotational widths, self and in air, from 295 to 1900 K.

    • Millot et al.’s per-N Q widths, self and by N2.

  • H2:

    • Stoicheff’s (1957) B0, D0 and H0, from his paper (TC-586). Weber (1979) reprints them, with a misprinted B0 of 56.3392 for 59.3392 (TC-515). These reproduce Teal & MacWood’s lines to within 0.9 cm⁻¹, their scatter. Komasa et al.’s (2011) ab initio lines agree with Stoicheff’s within his stated errors.

    • Bishop & Pipin’s ab initio anisotropy per J.

    • Hunt et al.’s ab initio static transition elements per J (0.739-0.755 au), with Bischel & Black’s dispersion.

    • Bischel & Dyer’s Q-branch law.

    • Herring et al.’s rotational widths.

  • D2:

    • Stoicheff’s constants.

    • The ab initio optical anisotropy.

    • Hunt et al.’s elements, whose Q(2) agrees with Murphy et al.’s ∂α/∂Q to 0.02 %.

    • Rotational widths: Keijser et al.’s (1974) S0(J) widths (their half widths, doubled), which Knaap et al.’s figures confirm to 5 % (TC-586). Smyth et al.’s reprint of them differs by the rounding of its conversion (at most 0.4 %). Edwards et al.’s are 1.3-2.6 times lower, with no single factor (TC-530, TC-541).

    • Q widths: Smyth et al.’s per-J widths, self and by He, Ar, H2 and CH4, with Russell & Roh’s Dicke term. Above a few amagat, where the density-dependent term dominates, Q(1) is 1.5 times narrower than Q(2) and Q(0) is 1.4 times broader; at 1 amagat both ratios are 1.24.

  • CH4:

    • Bermejo et al.’s ν1.

    • The mean of Schrötter & Klöckner’s thirteen Σ (8.55, ρ = 0), 5 % above Murphy et al.’s ∂α/∂Q, with Bischel & Black’s dispersion.

    • Ottusch & Rockwell’s width law.

  • SF6:

    • Murphy et al.’s ν1.

    • Schrötter & Klöckner’s Σ (3.3-3.96), 3 % above Murphy et al.’s. The four values span 20 %, and Everall et al.’s 248 nm cross section needs 23 % more than the largest (TC-517).

    • Everall et al.’s width at 5 atm, taken as independent of density.

  • CO2:

    • Wang’s B0 and D0, as Weber reprints them.

    • Bogaard et al.’s optical anisotropy at 632.8 nm, as Gentle et al. reprint it.

    • Pinter’s per-line S widths, as Jammu et al. reprint them.

    • The two members of the ν1/2ν2 Fermi dyad (1388 and 1285 cm⁻¹) as two unresolved bands. Their positions and ρ are Danichkin’s recommended values, and their elements come from Schrötter & Klöckner’s Σ.

    • Stopgap: no Q-branch width is curated, so the S-line coefficient stands in.

  • N2O (vibrational only):

    • ν1 (1285 cm⁻¹) and ν3 (2224 cm⁻¹) at Danichkin’s positions, with elements from Schrötter & Klöckner’s Σ.

    • Stopgaps:

      • ρ = 0 is taken where no depolarisation ratio is curated, which puts an upper bound on the element.

      • The width is Chen et al.’s rotational dephasing time (23.8 ps at 2.4 atm), taken as collisional.

    • Not yet: no rotational lines. No source of N2O’s rotational constants is curated: the Weber volume has none, and Chen et al.’s B is a simulation input.

See Raman validation and limitations for comparisons with measured quantities.