optmat.gas.mixture#

The refractive index of a gas mixture at finite density (OEP 4, sections 4.1, 4.3).

In the low-density cluster expansion the Lorentz-Lorenz function of a mixture is

\[y = \frac{n^2 - 1}{n^2 + 2} = \sum_s A_s \rho_s + \sum_{s,t} B_{st}\,\rho_s \rho_t, \qquad A_s = \frac{N_A \alpha_s}{3\varepsilon_0},\]

with \(\rho_s = N_s/N_A\) the molar densities and \(B_{ss}\) each gas’s own refractivity virial coefficient, so that one species is optmat’s pure law exactly. The whole mixture’s \(y\) is inverted once, \(\chi = 3y/(1-y)\).

Like terms disperse with their gas, \(B_{ss} = b_s A_s\), as in the pure models. O2’s is a published estimate, provisional with its 50 % uncertainty. A gas whose \(B_R\) optmat has not curated (D2, N2O, CO, O3) is unknown: it enters with \(b = 0\) and the declared scale \(|b| \le 2.5\) cm^3/mol.
Unlike terms are not curated for any pair. Each is estimated as the mean of its like terms, \(B_{st} = \tfrac12(B_{ss} + B_{tt})\), each like term with its own \(A\), with standard uncertainty \(\max(|B_{ss}|, |B_{tt}|)\), taking an unknown like term at its scale. The status is estimate, or unknown where a like term is unknown (section 7.2).
The uncertainty of the estimated and unknown terms is combined linearly, as fully correlated (MixtureOptics.refractivity_uncertainty()).

The optical domain (MixtureOptics.density_bound()) is the lower of two densities. One is where the neglected third term reaches 1e-3 of \(y\), from the largest constituent \(|C_R/A_R|\): measured, a measured bound, or the declared scale \(600 B_R^2/A_R\). The other is where the estimated and unknown terms’ uncertainty reaches 1e-3 of \(y\). Beyond the bound the optics warn. In wavelength, a mixture holds on the intersection of its constituents’ qualified intervals.

Functions

mixture_optics(species)

Return the optics of a mixture of gases (OEP 4).

Classes

`MixtureOptics`(species, models, pairs, ...[, ...])	The optics of a gas mixture at finite density (OEP 4).
`PairTerm`(species, partner, b, sigma, status, ...)	One pair's refractivity virial term, like or unlike (OEP 4, section 6).

class optmat.gas.mixture.MixtureOptics(species, models, pairs, intervals, *, status='user', status_reason='', modified=(), base='')[source]#

Bases: Identified

The optics of a gas mixture at finite density (OEP 4).

Attributes:

speciestuple of str: The species, sorted.
modelstuple of Model: Each species’ recommended dispersion model, in the same order.
pairstuple of PairTerm: Every like and unlike pair, like terms first.
intervalstuple of tuple of float: The intersection of the models’ qualified intervals, m.
statusstr: provisional where any like term is unknown or provisional, else validated.

EVALUATES_THROUGH: ClassVar[tuple[str, ...]] = ()#

EVALUATION_VERSION: ClassVar[int] = 1#

IDENTITY_KIND: ClassVar[str] = 'gas_refractive_mixture'#

density_bound(composition)[source]#

Return the optical domain’s bound on the total number density.

The lower of the density at which the third term reaches 1e-3 of y (the largest |C_R/A_R| of the species present) and that at which the estimated and unknown terms’ uncertainty does, the smallest over the qualified intervals (section 4.3).

Parameters:

compositionMapping: Species to mole fraction, summing to one; a species left out is absent.

Returns:

float: The bound, m^-3; inf where nothing bounds it.

Raises:

OptmatError: For a species not in the mixture, a negative fraction, or fractions that do not sum to one.

Examples

>>> f"{mixture_optics(['Ar']).density_bound({'Ar': 1.0}) / 6.02214076e23:.0f}"
'6961'

dsusceptibility_ddensity(wavelength, densities)[source]#

Return d chi / d N_s for every species.

Parameters:

wavelengtharray_like or float: Vacuum wavelength, m.
densitiesMapping: Each species’ number density, m^-3.

Returns:

dict: Species to the derivative with respect to its number density, m^3.

Warns:

ModelRangeWarning: Outside the constituents’ common intervals or the optical domain.

dsusceptibility_domega(wavelength, densities)[source]#

Return d chi / d omega of the mixture.

Parameters:

wavelengtharray_like or float: Vacuum wavelength, m.
densitiesMapping: Each species’ number density, m^-3.

Returns:

numpy.ndarray: The derivative, s.

Warns:

ModelRangeWarning: Outside the constituents’ common intervals or the optical domain.

intervals: tuple[tuple[float, float], ...]#

models: tuple[Model, ...]#

pairs: tuple[PairTerm, ...]#

refractivity(wavelength, densities)[source]#

Return n - 1 of the mixture, in the real-polarizability convention.

For one species it is the pure model’s refractivity at that density.

Parameters:

wavelengtharray_like or float: Vacuum wavelength, m; broadcast with the densities.
densitiesMapping: Each species’ number density, m^-3.

Returns:

numpy.ndarray: n - 1 = chi / (1 + sqrt(1 + chi)).

Warns:

ModelRangeWarning: Outside the constituents’ common intervals or the optical domain.
ModelAbsorptionWarning: Where a constituent absorbs, as its model’s polarizability does.

Examples

>>> he_ar = mixture_optics(["He", "Ar"])
>>> n = 1.25e25  # m^-3 each: about 1 bar together at 294 K
>>> f"{float(he_ar.refractivity(800e-9, {'He': n, 'Ar': n})):.5e}"
'1.46154e-04'

refractivity_uncertainty(wavelength, densities)[source]#

Return the estimated and unknown terms’ effect on n - 1.

Their uncertainties are combined linearly, as fully correlated (section 7.2): a sensitivity, not a statistical uncertainty.

Parameters:

wavelengtharray_like or float: Vacuum wavelength, m.
densitiesMapping: Each species’ number density, m^-3.

Returns:

numpy.ndarray: The change in n - 1 they span.

Warns:

ModelRangeWarning: Outside the constituents’ common intervals or the optical domain.

species: tuple[str, ...]#

susceptibility(wavelength, densities)[source]#

Return the complex susceptibility chi = n^2 - 1 of the mixture.

For one species, sqrt(1 + chi) is the pure model’s refractive_index.

Parameters:

wavelengtharray_like or float: Vacuum wavelength, m; broadcast with the densities.
densitiesMapping: Each species’ number density, m^-3.

Returns:

numpy.ndarray: chi, from each constituent’s complex polarizability.

Warns:

ModelRangeWarning: Outside the constituents’ common intervals or the optical domain.

class optmat.gas.mixture.PairTerm(species, partner, b, sigma, status, source, reason, frequency, temperature, contribution)[source]#

Bases: object

One pair’s refractivity virial term, like or unlike (OEP 4, section 6).

Attributes:

species, partnerstr: The pair, in sorted order; the same species twice for a like term.
bfloat: B_st / mean(A_s, A_t), m^3/mol: a like term’s curated ratio, or 0 for an unknown one. NaN for an unlike pair, whose rule is named in source.
sigmafloat: The standard uncertainty of b, m^3/mol: the curated one, or the declared scale for an unknown like term. NaN for an unlike pair (see reason).
statusstr: curated (a like term optmat curates), estimate (the mean rule) or unknown (a like term optmat has not curated, and its pairs); a like term that is a published estimate is provisional.
sourcestr: The curated dataset, or rule: mean.
reasonstr: Why the term is what it is.
frequencystr: static or the source’s vacuum wavelength, nm; empty where there is none.
temperaturefloat: The source’s temperature, K; NaN where there is none.
contributionstr: induced, or total for a static value that includes orientation.

b: float#

contribution: str#

frequency: str#

partner: str#

reason: str#

sigma: float#

source: str#

species: str#

status: str#

temperature: float#

optmat.gas.mixture.mixture_optics(species)[source]#

Return the optics of a mixture of gases (OEP 4).

Parameters:

speciessequence of Gas or str: The mixture’s species, each once.

Returns:

MixtureOptics: With each species’ recommended dispersion model; one object per set of species, whatever their order.

Raises:

OptmatError: If the list is empty or a species repeats.

Examples

>>> optics = mixture_optics(["Ar", "He"])
>>> optics.species, [p.status for p in optics.pairs]
(('Ar', 'He'), ['curated', 'curated', 'estimate'])