Raman data and validity#

The response strengths, line positions, damping and collision partners must describe the chosen material and state. Built-in gases use optmat data with recorded sources and qualification ranges. This chapter explains how those data enter a run; the model chapter gives the equations and the workflow shows how to select a response.

Raman responses from optmat#

Every built-in gas’s Raman response is optmat’s (SEP 3, materials from optmat). optmat’s gas Raman responses are its per-line models of N₂, O₂, H₂, D₂, CH₄, SF₆ and N₂O, and of their mixtures, with the available gain and revival evidence recorded per response. The coverage is not uniform: N₂O retains provisional bands and requires opt-in. They are the response of every capillary and waveguide fill of built-in gases; the molecular models are a supplied species’ route.

Selecting one#

A run takes optmat’s response when its populations are built-in gases and its MaterialSelection — MaterialSelection() unless given — takes its Raman family from optmat: on prop_capillary() and prop_waveguide(), and in responses_for(). The raman keyword keeps its vocabulary (True, labels, a mapping to MolecularOptions). For one gas outside a mixture, response_for(gas, selection=..., density=...) builds it (gas_raman()). The selection’s raman field chooses the response:

MaterialSelection.raman

response

"recommended" (default)

optmat’s curated response for the run’s composition, temperature and carrier

"provisional"

the same, and admits N₂O’s provisional vibrational bands, with a one-time MaterialStatusWarning; a capillary’s raman=None then turns N₂O’s response on. Without it N₂O is refused by name, and raman=None leaves it off with a warning

an optmat response object

taken as it is, for instance raman_response("N2", 293.15, wavelength=800e-9).with_parameters(width_scale=0.8); it must be for the run’s composition, reference temperature and carrier, and is refused otherwise

optmat’s overrides (width_scale, amplitude_scale, and width_law, a law \((c, d, \beta, \Delta\nu_D)\) per band) are the one-line path for a modified width or strength; the response then has the status user and keeps the digest of the curated response it started from. A Component’s dephasing is mapped onto width_law for every band of its species. A supplied law with a Dicke term is refused, because optmat caps that term at a Doppler width the legacy law does not state; pass the four numbers through width_law instead.

The same convention, mapped onto terms#

optmat’s convention is this page’s: for a medium of total number density \(N\),

\[ P_{R,x}(t) = \varepsilon_0 N E_x(t) \int_0^\infty h(\tau; N)\, E_x^2(t-\tau)\, \mathrm{d}\tau, \qquad h(\tau; N) = \Theta(\tau) \sum_k A_k \sin(\Omega_k \tau)\, e^{-\Gamma_k(N)\tau}, \]

with \(A_k\) in m⁵ V⁻² s⁻¹ and \(\Gamma_k = \pi\Delta\nu_k\), the full width at half maximum \(\Delta\nu_k\) in Hz: the amplitude decay \(e^{-\Gamma t}\) has a Lorentzian half width \(\Gamma\) in angular frequency, so a full width \(\Gamma/\pi\) in Hz. The widths are \(\Delta\nu = c + \min(d/n, \Delta\nu_D) + \beta n\) with \(n\) in amagat: a constant, a Dicke-narrowing term capped at the forward-scattering Doppler width, and collisional broadening. One response is built for one composition, one temperature and one pump wavelength, and its amplitudes already carry each constituent’s mole fraction, so a mixture’s response is driven by its total density.

Lines that share a width law and a shift form a damping group. Each group \(g\) and tensor family \(\kappa\) becomes one existing Raman term,

\[ u_{g\kappa}(t) = w(t) \sum_{k\in g} A_{k\kappa} \sin(\Omega_k t), \qquad \gamma_g(z) = \pi\,\Delta\nu_g\big(n(z)\big), \]

with \(w\) the grid’s causal taper (sampling the kernel): an undamped row and a decay-rate path, exactly the two arrays the kernel contract already carries. The mapping is exact for every gas whose lines do not shift, at any density path: the rows do not depend on the density, and the decay rate is optmat’s own group law, evaluated by optmat. An all-zero row is dropped and groups are never merged. The counts at 293.15 K and 800 nm:

response

lines

damping groups

terms (rows)

N₂

106

43

72

O₂

64

39

54

H₂

20

3

4

D₂

30

12

17

CH₄, SF₆

1 each

1

1

dry air (N₂, O₂, Ar)

170

82

126

against two terms for solprop’s own N₂ model.

Tensor form. optmat gives each line an isotropic and an anisotropic amplitude, in Hellwarth’s decomposition \(R_{ijkl} = a\,\delta_{ij}\delta_{kl} + \tfrac12 b(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\) with \(a + b\) = iso + aniso, \(a\) = iso − aniso/2 and \(b\) = 3 aniso/2. The isotropic amplitudes become an isotropic term and the anisotropic ones a rank-two term, with the tensor weights \((1, 0)\) and \((1/4, 3/4)\). The tests verify this against the tensor directly: the two-component polarisation equals Hellwarth’s with optmat’s \(a\) and \(b\) summed by a direct causal sum, and a probe polarised across a rotational response sees \(-\tfrac12\) of what a probe along it sees. The rotational S lines are anisotropic only; a Q line’s anisotropic share follows its Placzek–Teller factor. optmat leaves out the vibrational O and S branches, which carry 3 % (H₂, D₂) to 19 % (O₂) of a band’s anisotropic share.

Pump dispersion. optmat disperses the vibrational strengths to the pump wavelength it is given. The response always takes the run’s carrier: the selection’s carrier_wavelength when it names one, otherwise the grid reference lambda0.

Temperature. A response fixes its temperature, which sets both the populations and the widths’ temperature laws. A temperature path along z takes the existing reference-temperature rule, weighted now by the composition’s summed density, or an explicit MolecularOptions(temperature=...), and warns that the populations and the width laws are held there.

Mixtures and their collision partners#

A mixture’s response is one response, for the composition of all its labelled populations, Raman-active or not: an atom is a collision partner. Two labels of one species merge into one constituent. The labels select whose lines are kept (raman=("N2",) keeps nitrogen’s lines, still broadened by oxygen and argon), and a mapping’s MolecularOptions which bands. In the translation, the response’s one leaf is driven and damped by the summed density of the labels that form it: one amplitude table and one decay-rate path per term, within the kernel contract.

The composition must not vary along z. Populations whose densities are not proportional, independent partial pressures for instance, are refused by name. The partial-pressure rule keeps the fractions of a gradient fill proportional only to the gases’ differing non-ideality, which moves argon’s share of dry air by \(1.6\times10^{-3}\) of itself between 10 bar and vacuum, and the composition’s measured spread is recorded; the tolerance is COMPOSITION_TOLERANCE, 1 %.

optmat builds a mixture’s response only from the collision partners whose broadening it has curated: N₂ and O₂ with each other and with Ar, SF₆ with any atom, and H₂, D₂ and CH₄ with no other gas. Any other mixture with Raman is refused by name, naming the pair optmat lacks (optmat has curated no Raman broadening of H2 by He). solprop does not substitute self-broadening for the missing partner: the per-component, self-broadened responses of several responding species are not available under a selection. A fill whose populations are all atoms has no Raman response and is not refused. A population optmat does not resolve (a supplied species, or the dispersion-only Gas.Air) cannot be a partner beside a Raman-active molecule, and is refused there.

H₂’s line shift#

H₂’s Q lines shift with density, \(\Omega_k(N) = \Omega_{k,0} + 2\pi s n\), so their row depends on the density.

  • At a constant density the row is built at that density, which is exact.

  • On a varying density the response is refused by name, unless the selection names the approximation raman_shift="reference-density". That holds the shift at the density-weighted reference density \(N_\text{ref} = \int N^2\,\mathrm{d}z / \int N\,\mathrm{d}z\), the analogue of the reference temperature, and records the phase bound \(2\pi|s|\max_z|n(z) - n_\text{ref}|\,T_w\) over the grid’s window \(T_w\). Each line’s phase is then wrong by at most the bound, and the response by at most \(\sum_k |A_k| e^{-\Gamma_k t}\) times the frequency error times \(t\), which the tests check against optmat’s direct evaluation along the path.

SEP 10 removes this restriction.

Validated ranges, and vacuum#

Each response holds between two densities, found for the strongest line of each band: below, the motion of the molecules sets the widths (the Dicke regime); above, the lines overlap and their mixing is not modelled. Among the stated limits:

  • N₂ and dry air from 0.054 to 2.09 amagat, that is up to about 2.3 bar at 293.15 K, and between 293 and 300 K;

  • O₂ from 0.034 to 4.2 amagat;

  • H₂ above 0.94 amagat and D₂ above 0.61 amagat, the Dicke cutoffs.

The run declares the non-zero densities its path takes (the polarisation is exactly zero in vacuum) to the response’s range check, every density down to zero included when the path falls to vacuum, and its temperature, once each, and warns once per run (MaterialRangeWarning). Nothing is suppressed. A capillary of several bar of N₂ or air therefore runs outside the independent-line model’s validated range, and so does H₂ at 1 bar and 293.15 K, which is 0.92 amagat; a gradient that falls into the Dicke regime warns as well. The material manifest records the states outside.

The decay paths are evaluated at the actual density, vacuum included. optmat’s group law has a finite limit at zero density, \(\pi(c + \Delta\nu_D)\) with a Dicke term and \(\pi c\) without one: zero for N₂’s and O₂’s lines, whose widths are pure broadening. No floor is applied; the cap at MAX_DECAY_EXPONENT over the fine sample spacing stays as a numerical guard.

Cost, and what a run records#

The kernel rebuilds \(\sum_g e^{-\gamma_g t} u_g\) and transforms it whenever a decay rate changes: once per run at constant density, and at every stage on a density gradient. The rebuild scales with the number of rows: on this page’s reference measurement, at \(2^{14}\) fine samples, 6 ms for N₂’s 72 rows and 10 ms for dry air’s 126, against 0.3 ms for solprop’s own two-term N₂ model and 0.25 ms for one convolution (benchmarks). SEP 10’s low-rank representation removes this cost.

The run’s manifest records the response’s identity, the reference temperature (raman_reference_temperature) and, for each response, its labels, damping groups, terms and shift policy (raman_terms) among the hashed policies; and beside them its composition, carrier, the densities declared to its range check, the composition’s spread, and a held shift with its phase bound over the run’s window (raman).

Sources and qualification of the molecular data#

A built-in gas’s molecular data are optmat’s: each response states its sources, its validated temperature and density ranges and its status, and optmat’s documentation records its comparisons with measured gains, revivals and linewidths. solprop keeps no molecular constants of its own.

A supplied species’ constants are the caller’s, and the models they are built into leave out what follows; read it before trusting a delayed-probe or picosecond-scale prediction from them.

  • One linewidth per ladder. A rotational ladder takes one width for every \(J\), and its temperature dependence is not modelled; the \(J\) dependence is real.

  • One oscillator per vibration. A vibrational band is one damped oscillator: a Q branch’s \(J\)-dependent frequencies, line mixing, hot bands and anisotropic vibration are absent, so the ensemble’s loss of phase coherence without collisions is not represented, and a collision time is not a coherence time of the band.

  • Spin and nuclear state. Electronic-spin fine structure (O₂) is not represented, and the rotational populations assume nuclear-spin equilibrium: cooled normal hydrogen without a conversion catalyst need not reach it.

  • Collision partners. A mixture’s widths use the total density, assuming like broadening by every partner (several responding species); a pure-gas width is not a binary or whole-air coefficient.

  • Wavelength. A polarisability derivative or anisotropy measured at one wavelength is used at the pump’s; the transfer is an approximation.

  • Coordinates. A polarisability derivative and an effective mass belong to one displacement coordinate: \((\mathrm{d}\alpha/\mathrm{d}Q)^2/\mu\) is what enters, so changing one without the other changes the strength.