optmat.gas.linewidths#

The widths of Raman lines against density, temperature and collision partner.

A Raman line’s full width at half maximum (FWHM) in a gas of number density \(n\) (in amagat) is written

\[\Delta\nu = c + \frac{A(T)}{n} + n \sum_p x_p\, \gamma_p(J, T),\]

with \(c\) a density-independent width, \(A\) the Dicke (motional-narrowing) coefficient of the light molecules, \(x_p\) the mole fraction of the collision partner \(p\) (“self” for the gas itself) and \(\gamma_p\) its broadening coefficient. Three forms of \(\gamma_p\) are used, as the curated data allow:

LinewidthLaw: one coefficient for every line of a band, with a temperature law \(\gamma(T) = [\gamma_0 + c_1 (T - T_0) + c_2 (T - T_0)^2] (T_0 / T)^N\) (Broadening), and \(A(T) = A_0 (T / T_A)^{n_A}\);
LineTable: a coefficient per rotational level, tabulated at one temperature with a per-level exponent, interpolated in \(J\);
ExponentialGapLaw: the modified exponential-gap law of Koszykowski et al. (1987), in which a Q-branch line’s width is the sum of the rates of rotational energy transfer out of its level.

Widths are in Hz, densities in amagat (AMAGAT), temperatures in K. The laws are per line; the overlap of lines at high density (line mixing, the collapse of a Q branch) is not modelled.

Module Attributes

`AMAGAT`	One amagat, m^-3: the ideal-gas number density at 1 atm and 0 degrees C.
`WAVENUMBER_TO_HZ`	Factor from a wavenumber in cm^-1 to a frequency in Hz.

Classes

`Broadening`(coefficient, reference_temperature)	A broadening coefficient and its temperature law.
`ExponentialGapLaw`(alpha, beta, delta, a, ...)	Q-branch widths from the modified exponential-gap law (Koszykowski et al. 1987).
`LineTable`(j, coefficient, ...)	Broadening coefficients per rotational level, from one tabulation.
`LineTableLaw`(tables[, substitutes, dicke, ...])	Per-line widths from per-level tables, one per perturber.
`LinewidthLaw`([constant, dicke, ...])	One width law for every line of a band.

optmat.gas.linewidths.AMAGAT: float = 2.686780111798444e+25#: One amagat, m^-3: the ideal-gas number density at 1 atm and 0 degrees C.

class optmat.gas.linewidths.Broadening(coefficient, reference_temperature, exponent=0.0, linear=0.0, quadratic=0.0)[source]#

Bases: object

A broadening coefficient and its temperature law.

gamma(T) = [coefficient + linear (T - T0) + quadratic (T - T0)^2] (T0 / T)^exponent in Hz/amagat (FWHM).

Attributes:

coefficientfloat: gamma(T0), Hz/amagat.
reference_temperaturefloat: T0, K.
exponentfloat: The power-law exponent N.
linear, quadraticfloat: Polynomial terms, Hz/amagat/K and Hz/amagat/K^2 (Bischel & Dyer’s law).

at(temperature)[source]#

Return the coefficient at a temperature.

Parameters:

temperaturefloat: Temperature, K, > 0.

Returns:

float: gamma(T), Hz/amagat.

coefficient: float#

exponent: float = 0.0#

linear: float = 0.0#

quadratic: float = 0.0#

reference_temperature: float#

class optmat.gas.linewidths.ExponentialGapLaw(alpha, beta, delta, a, exponent, reference_temperature, b, d=0.0, j_max=60, foreign_as_self=(), sources=())[source]#

Bases: object

Q-branch widths from the modified exponential-gap law (Koszykowski et al. 1987).

The rate of transfer up from level i to j (E_j > E_i, the same nuclear spin symmetry) is gamma_ji = alpha p [(1 + a E_i / (k T delta)) / (1 + a E_i / (k T))]^2 exp(-beta (E_j - E_i) / (k T)), the rate down follows by detailed balance, gamma_ij = (2 J_i + 1) / (2 J_j + 1) gamma_ji exp((E_j - E_i) / (k T)), and the FWHM of line J is 2 sum_{i != J} gamma_iJ (their eqs. 4.1-4.4). alpha scales with temperature as (T0 / T)^n; p is the pressure in atm.

Attributes:

alphafloat: alpha at T0, cm^-1/atm.
beta, delta, afloat: The law’s constants.
exponentfloat: n.
reference_temperaturefloat: T0, K.
b, dfloat: The rotational constants of the levels, cm^-1.
j_maxint: The highest level summed over.
foreign_as_selftuple of str: Collision partners whose broadening is taken as the self-broadening (a declared approximation, where no foreign coefficients are curated).
sourcestuple of str: The formula record and the constants’ datasets.

a: float#

alpha: float#

b: float#

beta: float#

d: float = 0.0#

delta: float#

exponent: float#

foreign_as_self: tuple[str, ...] = ()#

fwhm(j, temperature, density, fractions=None)[source]#

Return the linewidths at a state.

Parameters:

jnumpy.ndarray: The lines’ rotational levels.
temperaturefloat: Temperature, K, > 0.
densityfloat: Total number density, m^-3, > 0 (converted to a pressure with the ideal gas law).
fractionsmapping of str to float, optional: The pure gas, or partners listed in foreign_as_self: the law is for self-broadening.

Returns:

numpy.ndarray: FWHM of each line, Hz.

Raises:

DataError: If a foreign perturber is asked for.

j_max: int = 60#

reference_temperature: float#

sources: tuple[str, ...] = ()#

class optmat.gas.linewidths.LineTable(j, coefficient, reference_temperature, exponent)[source]#

Bases: object

Broadening coefficients per rotational level, from one tabulation.

Between tabulated levels the coefficient and exponent are interpolated linearly in J; beyond the ends they are held at the end values (the linewidths of the highest levels change slowly, and their populations are small).

Attributes:

jtuple of int: The tabulated levels.
coefficienttuple of float: gamma(J, T0), Hz/amagat.
reference_temperaturefloat: T0, K.
exponenttuple of float: The power-law exponent of each level ((T0 / T)^N).

at(j, temperature)[source]#

Return the coefficients of levels j at a temperature.

Parameters:

jnumpy.ndarray: Rotational levels.
temperaturefloat: Temperature, K, > 0.

Returns:

numpy.ndarray: gamma(J, T), Hz/amagat.

coefficient: tuple[float, ...]#

exponent: tuple[float, ...]#

j: tuple[int, ...]#

reference_temperature: float#

class optmat.gas.linewidths.LineTableLaw(tables, substitutes=<factory>, dicke=0.0, sources=())[source]#

Bases: object

Per-line widths from per-level tables, one per perturber.

Attributes:

tablesdict of str to LineTable: By perturber.
substitutesdict of str to str: Perturbers without a table, and the perturber whose table stands in.
dickefloat: A Dicke coefficient A for every line, Hz amagat (H2’s rotational lines).
sourcestuple of str: The curated datasets the tables come from.

dicke: float = 0.0#

fwhm(j, temperature, density, fractions=None)[source]#

Return the linewidths at a state.

Parameters:

jnumpy.ndarray: The lines’ lower rotational levels.
temperaturefloat: Temperature, K, > 0.
densityfloat: Total number density, m^-3, > 0.
fractionsmapping of str to float, optional: Mole fractions by perturber; the pure gas by default.

Returns:

numpy.ndarray: FWHM of each line, Hz.

sources: tuple[str, ...] = ()#

substitutes: Mapping[str, str]#

tables: Mapping[str, LineTable]#

class optmat.gas.linewidths.LinewidthLaw(constant=0.0, dicke=0.0, dicke_temperature=298.0, dicke_exponent=0.0, broadening=<factory>, substitutes=<factory>, sources=())[source]#

Bases: object

One width law for every line of a band.

Attributes:

constantfloat: Density-independent width, Hz.
dickefloat: Dicke coefficient A at dicke_temperature, Hz amagat.
dicke_temperaturefloat: The temperature of dicke, K.
dicke_exponentfloat: n_A in A(T) = A (T / T_A)^n_A.
broadeningdict of str to Broadening: By perturber (self, a gas id, or air).
substitutesdict of str to str: Perturbers without coefficients, and the perturber whose coefficient stands in for each (a declared approximation).
sourcestuple of str: The curated datasets and formulas the law comes from.

broadening: Mapping[str, Broadening]#

constant: float = 0.0#

dicke: float = 0.0#

dicke_exponent: float = 0.0#

dicke_temperature: float = 298.0#

fwhm(j, temperature, density, fractions=None)[source]#

Return the linewidths at a state.

Parameters:

jnumpy.ndarray: The lines’ rotational levels (the width is the same for all).
temperaturefloat: Temperature, K, > 0.
densityfloat: Total number density, m^-3, > 0.
fractionsmapping of str to float, optional: Mole fractions by perturber; the pure gas by default.

Returns:

numpy.ndarray: FWHM of each line, Hz.

sources: tuple[str, ...] = ()#

substitutes: Mapping[str, str]#

optmat.gas.linewidths.WAVENUMBER_TO_HZ: float = 29979245800.0#: Factor from a wavenumber in cm^-1 to a frequency in Hz.